"""
Symbolic Expressions
RELATIONAL EXPRESSIONS:
We create a relational expression::
sage: x = var('x')
sage: eqn = (x-1)^2 <= x^2 - 2*x + 3
sage: eqn.subs(x == 5)
16 <= 18
Notice that squaring the relation squares both sides.
::
sage: eqn^2
(x - 1)^4 <= (x^2 - 2*x + 3)^2
sage: eqn.expand()
x^2 - 2*x + 1 <= x^2 - 2*x + 3
The can transform a true relational into a false one::
sage: eqn = SR(-5) < SR(-3); eqn
-5 < -3
sage: bool(eqn)
True
sage: eqn^2
25 < 9
sage: bool(eqn^2)
False
We can do arithmetic with relationals::
sage: e = x+1 <= x-2
sage: e + 2
x + 3 <= x
sage: e - 1
x <= x - 3
sage: e*(-1)
-x - 1 <= -x + 2
sage: (-2)*e
-2*x - 2 <= -2*x + 4
sage: e*5
5*x + 5 <= 5*x - 10
sage: e/5
1/5*x + 1/5 <= 1/5*x - 2/5
sage: 5/e
5/(x + 1) <= 5/(x - 2)
sage: e/(-2)
-1/2*x - 1/2 <= -1/2*x + 1
sage: -2/e
-2/(x + 1) <= -2/(x - 2)
We can even add together two relations, so long as the operators are
the same::
sage: (x^3 + x <= x - 17) + (-x <= x - 10)
x^3 <= 2*x - 27
Here they aren't::
sage: (x^3 + x <= x - 17) + (-x >= x - 10)
Traceback (most recent call last):
...
TypeError: incompatible relations
ARBITRARY SAGE ELEMENTS:
You can work symbolically with any Sage data type. This can lead to
nonsense if the data type is strange, e.g., an element of a finite
field (at present).
We mix Singular variables with symbolic variables::
sage: R.<u,v> = QQ[]
sage: var('a,b,c')
(a, b, c)
sage: expand((u + v + a + b + c)^2)
a^2 + 2*a*b + b^2 + 2*a*c + 2*b*c + c^2 + 2*a*u + 2*b*u + 2*c*u + u^2 + 2*a*v + 2*b*v + 2*c*v + 2*u*v + v^2
TESTS:
Test Jacobian on Pynac expressions. #5546 ::
sage: var('x,y')
(x, y)
sage: f = x + y
sage: jacobian(f, [x,y])
[1 1]
Test if matrices work #5546 ::
sage: var('x,y,z')
(x, y, z)
sage: M = matrix(2,2,[x,y,z,x])
sage: v = vector([x,y])
sage: M * v
(x^2 + y^2, x*y + x*z)
sage: v*M
(x^2 + y*z, 2*x*y)
Test if comparison bugs from #6256 are fixed::
sage: t = exp(sqrt(x)); u = 1/t
sage: t*u
1
sage: t + u
e^(-sqrt(x)) + e^sqrt(x)
sage: t
e^sqrt(x)
Test if #9947 is fixed::
sage: real_part(1+2*(sqrt(2)+1)*(sqrt(2)-1))
3
sage: a=(sqrt(4*(sqrt(3) - 5)*(sqrt(3) + 5) + 48) + 4*sqrt(3))/ (sqrt(3) + 5)
sage: a.real_part()
4*sqrt(3)/(sqrt(3) + 5)
sage: a.imag_part()
sqrt(abs(4*(sqrt(3) + 5)*(sqrt(3) - 5) + 48))/(sqrt(3) + 5)
"""
include "sage/ext/interrupt.pxi"
include "sage/ext/stdsage.pxi"
include "sage/ext/cdefs.pxi"
include "sage/ext/python.pxi"
import operator
import ring
import sage.rings.integer
import sage.rings.rational
from sage.structure.element cimport ModuleElement, RingElement, Element
from sage.symbolic.getitem cimport OperandsWrapper
from sage.symbolic.function import get_sfunction_from_serial, SymbolicFunction
from sage.rings.rational import Rational
from sage.misc.derivative import multi_derivative
from sage.rings.infinity import AnInfinity, infinity, minus_infinity, unsigned_infinity
from sage.misc.decorators import rename_keyword
from sage.misc.superseded import deprecated_function_alias
from sage.structure.dynamic_class import dynamic_class
LOG_TEN_TWO_PLUS_EPSILON = 3.321928094887363
cpdef bint is_Expression(x):
"""
Returns True if *x* is a symbolic Expression.
EXAMPLES::
sage: from sage.symbolic.expression import is_Expression
sage: is_Expression(x)
True
sage: is_Expression(2)
False
sage: is_Expression(SR(2))
True
"""
return isinstance(x, Expression)
cpdef bint is_SymbolicEquation(x):
"""
Returns True if *x* is a symbolic equation.
EXAMPLES:
The following two examples are symbolic equations::
sage: from sage.symbolic.expression import is_SymbolicEquation
sage: is_SymbolicEquation(sin(x) == x)
True
sage: is_SymbolicEquation(sin(x) < x)
True
sage: is_SymbolicEquation(x)
False
This is not, since ``2==3`` evaluates to the boolean
``False``::
sage: is_SymbolicEquation(2 == 3)
False
However here since both 2 and 3 are coerced to be symbolic, we
obtain a symbolic equation::
sage: is_SymbolicEquation(SR(2) == SR(3))
True
"""
return isinstance(x, Expression) and is_a_relational((<Expression>x)._gobj)
cdef class Expression(CommutativeRingElement):
cpdef object pyobject(self):
"""
Get the underlying Python object.
OUTPUT:
The Python object corresponding to this expression, assuming
this expression is a single numerical value or an infinity
representable in Python. Otherwise, a ``TypeError`` is raised.
EXAMPLES::
sage: var('x')
x
sage: b = -17/3
sage: a = SR(b)
sage: a.pyobject()
-17/3
sage: a.pyobject() is b
True
TESTS::
sage: SR(oo).pyobject()
+Infinity
sage: SR(-oo).pyobject()
-Infinity
sage: SR(unsigned_infinity).pyobject()
Infinity
sage: SR(I*oo).pyobject()
Traceback (most recent call last):
...
TypeError: Python infinity cannot have complex phase.
"""
cdef GConstant* c
if is_a_constant(self._gobj):
from sage.symbolic.constants import constants_name_table
return constants_name_table[GEx_to_str(&self._gobj)]
if is_a_infinity(self._gobj):
if (ex_to_infinity(self._gobj).is_unsigned_infinity()): return unsigned_infinity
if (ex_to_infinity(self._gobj).is_plus_infinity()): return infinity
if (ex_to_infinity(self._gobj).is_minus_infinity()): return minus_infinity
raise TypeError('Python infinity cannot have complex phase.')
if not is_a_numeric(self._gobj):
raise TypeError, "self must be a numeric expression"
return py_object_from_numeric(self._gobj)
def __init__(self, SR, x=0):
"""
Nearly all expressions are created by calling new_Expression_from_*,
but we need to make sure this at least doesn't leave self._gobj
uninitialized and segfault.
TESTS::
sage: sage.symbolic.expression.Expression(SR)
0
sage: sage.symbolic.expression.Expression(SR, 5)
5
We test subclassing ``Expression``::
sage: from sage.symbolic.expression import Expression
sage: class exp_sub(Expression): pass
sage: f = function('f')
sage: t = f(x)
sage: u = exp_sub(SR, t)
sage: u.operator()
f
"""
self._parent = SR
cdef Expression exp = self.coerce_in(x)
GEx_construct_ex(&self._gobj, exp._gobj)
def __dealloc__(self):
"""
Delete memory occupied by this expression.
"""
GEx_destruct(&self._gobj)
def __getstate__(self):
"""
Returns a tuple describing the state of this expression for pickling.
This should return all information that will be required to unpickle
the object. The functionality for unpickling is implemented in
__setstate__().
In order to pickle Expression objects, we return a tuple containing
* 0 - as pickle version number
in case we decide to change the pickle format in the feature
* names of symbols of this expression
* a string representation of self stored in a Pynac archive.
TESTS::
sage: var('x,y,z')
(x, y, z)
sage: t = 2*x*y^z+3
sage: s = dumps(t)
sage: t.__getstate__()
(0,
['x', 'y', 'z'],
...)
"""
cdef GArchive ar
ar.archive_ex(self._gobj, "sage_ex")
ar_str = GArchive_to_str(&ar)
return (0, map(repr, self.variables()), ar_str)
def _dbgprint(self):
r"""
Print pynac debug output to ``stderr``.
EXAMPLES::
sage: (1+x)._dbgprint()
x + 1
"""
self._gobj.dbgprint()
def _dbgprinttree(self):
r"""
Print pynac expression tree debug output to ``stderr``.
EXAMPLES:
The expression tree is composed of Ginac primitives
and functions, organized by the tree, with various
other memory and hash information which will vary::
sage: (1+x+exp(x+1))._dbgprinttree() # not tested
add @0x65e5960, hash=0x4727e01a, flags=0x3, nops=3
x (symbol) @0x6209150, serial=6, hash=0x2057b15e, flags=0xf, domain=0
1 (numeric) @0x3474cf0, hash=0x0, flags=0x7
-----
function exp @0x24835d0, hash=0x765c2165, flags=0xb, nops=1
add @0x65df570, hash=0x420740d2, flags=0xb, nops=2
x (symbol) @0x6209150, serial=6, hash=0x2057b15e, flags=0xf, domain=0
1 (numeric) @0x3474cf0, hash=0x0, flags=0x7
-----
overall_coeff
1 (numeric) @0x65e4df0, hash=0x7fd3, flags=0x7
=====
=====
1 (numeric) @0x3474cf0, hash=0x0, flags=0x7
-----
overall_coeff
1 (numeric) @0x663cc40, hash=0x7fd3, flags=0x7
=====
TESTS:
This test is just to make sure the function is working::
sage: (1+x+exp(x+1))._dbgprinttree()
add @...
x (symbol) ...
1 (numeric) ...
...
overall_coeff
1 (numeric) ...
=====
"""
self._gobj.dbgprinttree();
def __setstate__(self, state):
"""
Initializes the state of the object from data saved in a pickle.
During unpickling __init__ methods of classes are not called, the saved
data is passed to the class via this function instead.
TESTS::
sage: var('x,y,z')
(x, y, z)
sage: t = 2*x*y^z+3
sage: u = loads(dumps(t)) # indirect doctest
sage: u
2*x*y^z + 3
sage: bool(t == u)
True
sage: u.subs(x=z)
2*y^z*z + 3
sage: loads(dumps(x.parent()(2)))
2
"""
if state[0] != 0 or len(state) != 3:
raise ValueError, "unknown state information"
self._set_parent(ring.SR)
cdef GExList sym_lst
for name in state[1]:
sym_lst.append_sym(\
ex_to_symbol((<Expression>ring.SR.symbol(name))._gobj))
cdef GArchive ar
GArchive_from_str(&ar, state[2], len(state[2]))
GEx_construct_ex(&self._gobj, ar.unarchive_ex(sym_lst, <unsigned>0))
def __copy__(self):
"""
TESTS::
sage: copy(x)
x
"""
return new_Expression_from_GEx(self._parent, self._gobj)
def _repr_(self):
"""
Return string representation of this symbolic expression.
EXAMPLES::
sage: var("x y")
(x, y)
sage: repr(x+y)
'x + y'
TESTS::
# printing of modular number equal to -1 as coefficient
sage: k.<a> = GF(9); k(2)*x
2*x
sage: (x+1)*Mod(6,7)
6*x + 6
#printing rational functions
sage: x/y
x/y
sage: x/2/y
1/2*x/y
sage: .5*x/y
0.500000000000000*x/y
sage: x^(-1)
1/x
sage: x^(-5)
x^(-5)
sage: x^(-y)
x^(-y)
sage: 2*x^(-1)
2/x
sage: i*x
I*x
sage: -x.parent(i)
-I
sage: y + 3*(x^(-1))
y + 3/x
Printing the exp function::
sage: x.parent(1).exp()
e
sage: x.exp()
e^x
Powers::
sage: _ = var('A,B,n'); (A*B)^n
(A*B)^n
sage: (A/B)^n
(A/B)^n
sage: n*x^(n-1)
n*x^(n - 1)
sage: (A*B)^(n+1)
(A*B)^(n + 1)
sage: (A/B)^(n-1)
(A/B)^(n - 1)
sage: n*x^(n+1)
n*x^(n + 1)
sage: n*x^(n-1)
n*x^(n - 1)
sage: n*(A/B)^(n+1)
n*(A/B)^(n + 1)
sage: (n+A/B)^(n+1)
(n + A/B)^(n + 1)
Powers where the base or exponent is a Python object::
sage: (2/3)^x
(2/3)^x
sage: x^CDF(1,2)
x^(1.0 + 2.0*I)
sage: (2/3)^(2/3)
(2/3)^(2/3)
sage: (-x)^(1/4)
(-x)^(1/4)
sage: k.<a> = GF(9)
sage: SR(a+1)^x
(a + 1)^x
Check if #7876 is fixed::
sage: (1/2-1/2*I )*sqrt(2)
-(1/2*I - 1/2)*sqrt(2)
sage: latex((1/2-1/2*I )*sqrt(2))
-\left(\frac{1}{2} i - \frac{1}{2}\right) \, \sqrt{2}
Check if :trac:`9632` is fixed::
sage: zeta(x) + cos(x)
cos(x) + zeta(x)
sage: psi(1,1/3)*log(3)
log(3)*psi(1, 1/3)
"""
return self._parent._repr_element_(self)
def _ascii_art_(self):
"""
TESTS::
sage: i = var('i')
sage: ascii_art(sum(i^2/pi*x^i, i, 0, oo))
2
x + x
-------------------------------
3 2
- pi*x + 3*pi*x - 3*pi*x + pi
sage: ascii_art(integral(exp(x + x^2)/(x+1), x))
/
|
| 2
| x + x
| e
| ------- dx
| x + 1
|
/
"""
from sympy import pretty, sympify
from sage.misc.ascii_art import AsciiArt
try:
s = pretty(sympify(self), use_unicode=False)
except Exception:
s = self
return AsciiArt(str(s).splitlines())
def _interface_(self, I):
"""
EXAMPLES::
sage: f = sin(e + 2)
sage: f._interface_(sage.calculus.calculus.maxima)
sin(%e+2)
"""
if is_a_constant(self._gobj):
return self.pyobject()._interface_(I)
return super(Expression, self)._interface_(I)
def _maxima_(self, session=None):
"""
EXAMPLES::
sage: f = sin(e + 2)
sage: f._maxima_()
sin(%e+2)
sage: _.parent() is sage.calculus.calculus.maxima
True
"""
if session is None:
from sage.calculus.calculus import maxima
return super(Expression, self)._interface_(maxima)
else:
return super(Expression, self)._interface_(session)
def _interface_init_(self, I):
"""
EXAMPLES::
sage: a = (pi + 2).sin()
sage: a._maxima_init_()
'sin((%pi)+(2))'
sage: a = (pi + 2).sin()
sage: a._maple_init_()
'sin((pi)+(2))'
sage: a = (pi + 2).sin()
sage: a._mathematica_init_()
'Sin[(Pi)+(2)]'
sage: f = pi + I*e
sage: f._pari_init_()
'(Pi)+((exp(1))*(I))'
TESTS:
Check if complex numbers are converted to Maxima correctly
:trac:`7557`::
sage: SR(1.5*I)._maxima_init_()
'1.5000000000000000*%i'
sage: SR(CC.0)._maxima_init_()
'1.0000000000000000*%i'
sage: SR(CDF.0)._maxima_init_()
'1.0000000000000000*%i'
"""
from sage.symbolic.expression_conversions import InterfaceInit
return InterfaceInit(I)(self)
def _gap_init_(self):
"""
Conversion of symbolic object to GAP always results in a GAP
string.
EXAMPLES::
sage: gap(e + pi^2 + x^3)
x^3 + pi^2 + e
"""
return '"%s"'%repr(self)
def _singular_init_(self):
"""
Conversion of a symbolic object to Singular always results in a
Singular string.
EXAMPLES::
sage: singular(e + pi^2 + x^3)
x^3 + pi^2 + e
"""
return '"%s"'%repr(self)
def _magma_init_(self, magma):
"""
Return string representation in Magma of this symbolic expression.
Since Magma has no notation of symbolic calculus, this simply
returns something that evaluates in Magma to a a Magma string.
EXAMPLES::
sage: x = var('x')
sage: f = sin(cos(x^2) + log(x))
sage: f._magma_init_(magma)
'"sin(cos(x^2) + log(x))"'
sage: magma(f) # optional - magma
sin(log(x) + cos(x^2))
sage: magma(f).Type() # optional - magma
MonStgElt
"""
return '"%s"'%repr(self)
def _latex_(self):
r"""
Return string representation of this symbolic expression.
TESTS::
sage: var('x,y,z')
(x, y, z)
sage: latex(y + 3*(x^(-1)))
y + \frac{3}{x}
sage: latex(x^(y+z^(1/y)))
x^{y + z^{\left(\frac{1}{y}\right)}}
sage: latex(1/sqrt(x+y))
\frac{1}{\sqrt{x + y}}
sage: latex(sin(x*(z+y)^x))
\sin\left(x {\left(y + z\right)}^{x}\right)
sage: latex(3/2*(x+y)/z/y)
\frac{3 \, {\left(x + y\right)}}{2 \, y z}
sage: latex((2^(x^y)))
2^{\left(x^{y}\right)}
sage: latex(abs(x))
{\left| x \right|}
sage: latex((x*y).conjugate())
\overline{x} \overline{y}
sage: latex(x*(1/(x^2)+sqrt(x^7)))
x {\left(\sqrt{x^{7}} + \frac{1}{x^{2}}\right)}
Check spacing of coefficients of mul expressions (#3202)::
sage: latex(2*3^x)
2 \, 3^{x}
Powers::
sage: _ = var('A,B,n')
sage: latex((n+A/B)^(n+1))
{\left(n + \frac{A}{B}\right)}^{n + 1}
sage: latex((A*B)^n)
\left(A B\right)^{n}
sage: latex((A*B)^(n-1))
\left(A B\right)^{n - 1}
Powers where the base or exponent is a Python object::
sage: latex((2/3)^x)
\left(\frac{2}{3}\right)^{x}
sage: latex(x^CDF(1,2))
x^{1.0 + 2.0i}
sage: latex((2/3)^(2/3))
\left(\frac{2}{3}\right)^{\frac{2}{3}}
sage: latex((-x)^(1/4))
\left(-x\right)^{\frac{1}{4}}
sage: k.<a> = GF(9)
sage: latex(SR(a+1)^x)
\left(a + 1\right)^{x}
More powers, #7406::
sage: latex((x^pi)^e)
{\left(x^{\pi}\right)}^{e}
sage: latex((x^(pi+1))^e)
{\left(x^{\pi + 1}\right)}^{e}
sage: a,b,c = var('a b c')
sage: latex(a^(b^c))
a^{\left(b^{c}\right)}
sage: latex((a^b)^c)
{\left(a^{b}\right)}^{c}
Separate coefficients to numerator and denominator, #7363::
sage: latex(2/(x+1))
\frac{2}{x + 1}
sage: latex(1/2/(x+1))
\frac{1}{2 \, {\left(x + 1\right)}}
Check if rational function coefficients without a ``numerator()`` method
are printed correctly. #8491::
sage: latex(6.5/x)
\frac{6.50000000000000}{x}
sage: latex(Mod(2,7)/x)
\frac{2}{x}
Check if we avoid extra parenthesis in rational functions (#8688)::
sage: latex((x+2)/(x^3+1))
\frac{x + 2}{x^{3} + 1}
sage: latex((x+2)*(x+1)/(x^3+1))
\frac{{\left(x + 2\right)} {\left(x + 1\right)}}{x^{3} + 1}
sage: latex((x+2)/(x^3+1)/(x+1))
\frac{x + 2}{{\left(x^{3} + 1\right)} {\left(x + 1\right)}}
Check that the sign is correct (#9086)::
sage: latex(-1/x)
-\frac{1}{x}
sage: latex(1/-x)
-\frac{1}{x}
More tests for the sign (#9314)::
sage: latex(-2/x)
-\frac{2}{x}
sage: latex(-x/y)
-\frac{x}{y}
sage: latex(-x*z/y)
-\frac{x z}{y}
sage: latex(-x/z/y)
-\frac{x}{y z}
Check if #9394 is fixed::
sage: var('n')
n
sage: latex( e^(2*I*pi*n*x - 2*I*pi*n) )
e^{\left(2 i \, \pi n x - 2 i \, \pi n\right)}
sage: latex( e^(2*I*pi*n*x - (2*I+1)*pi*n) )
e^{\left(2 i \, \pi n x - \left(2 i + 1\right) \, \pi n\right)}
sage: x+(1-2*I)*y
x - (2*I - 1)*y
sage: latex(x+(1-2*I)*y)
x - \left(2 i - 1\right) \, y
Check if complex coefficients with denominators are displayed
correctly #10769::
sage: var('a x')
(a, x)
sage: latex(1/2*I/x)
\frac{i}{2 \, x}
sage: ratio = i/2* x^2/a
sage: latex(ratio)
\frac{i \, x^{2}}{2 \, a}
Parenthesis in powers, #13262::
sage: latex(1+x^(2/3)+x^(-2/3))
x^{\frac{2}{3}} + \frac{1}{x^{\frac{2}{3}}} + 1
"""
return self._parent._latex_element_(self)
def _mathml_(self):
"""
Returns a MathML representation of this object.
EXAMPLES::
sage: mathml(pi)
<mi>π</mi>
sage: mathml(pi+2)
MATHML version of the string pi + 2
"""
from sage.misc.all import mathml
try:
obj = self.pyobject()
except TypeError:
return mathml(repr(self))
return mathml(obj)
def _integer_(self, ZZ=None):
"""
EXAMPLES::
sage: f = x^3 + 17*x -3
sage: ZZ(f.coeff(x^3))
1
sage: ZZ(f.coeff(x))
17
sage: ZZ(f.coeff(x,0))
-3
sage: type(ZZ(f.coeff(x,0)))
<type 'sage.rings.integer.Integer'>
Coercion is done if necessary::
sage: f = x^3 + 17/1*x
sage: ZZ(f.coeff(x))
17
sage: type(ZZ(f.coeff(x)))
<type 'sage.rings.integer.Integer'>
If the symbolic expression is just a wrapper around an integer,
that very same integer is returned::
sage: n = 17; SR(n)._integer_() is n
True
"""
try:
n = self.pyobject()
except TypeError:
raise TypeError, "unable to convert x (=%s) to an integer"%(self)
if isinstance(n, sage.rings.integer.Integer):
return n
return sage.rings.integer.Integer(n)
def __int__(self):
"""
EXAMPLES::
sage: int(log(8)/log(2))
3
sage: int(-log(8)/log(2))
-3
sage: int(sin(2)*100)
90
sage: int(-sin(2)*100)
-90
sage: int(SR(3^64)) == 3^64
True
sage: int(SR(10^100)) == 10^100
True
sage: int(SR(10^100-10^-100)) == 10^100 - 1
True
sage: int(sqrt(-3))
Traceback (most recent call last):
...
ValueError: cannot convert sqrt(-3) to int
"""
from sage.functions.all import floor, ceil
try:
rif_self = sage.rings.all.RIF(self)
except TypeError:
raise ValueError, "cannot convert %s to int"%(self)
if rif_self > 0 or (rif_self.contains_zero() and self > 0):
result = floor(self)
else:
result = ceil(self)
if not isinstance(result, sage.rings.integer.Integer):
raise ValueError, "cannot convert %s to int"%(self)
else:
return int(result)
def __long__(self):
"""
EXAMPLES::
sage: long(sin(2)*100)
90L
"""
return long(int(self))
def _rational_(self):
"""
EXAMPLES::
sage: f = x^3 + 17/1*x - 3/8
sage: QQ(f.coeff(x^2))
0
sage: QQ(f.coeff(x^3))
1
sage: a = QQ(f.coeff(x)); a
17
sage: type(a)
<type 'sage.rings.rational.Rational'>
sage: QQ(f.coeff(x,0))
-3/8
If the symbolic expression is just a wrapper around a rational,
that very same rational is returned::
sage: n = 17/1; SR(n)._rational_() is n
True
"""
try:
n = self.pyobject()
except TypeError:
raise TypeError, "unable to convert %s to a rational"%self
if isinstance(n, sage.rings.rational.Rational):
return n
return sage.rings.rational.Rational(n)
cpdef _eval_self(self, R):
"""
Evaluate this expression numerically.
This function is used to convert symbolic expressions to ``RR``,
``CC``, ``float``, ``complex``, ``CIF`` and ``RIF``.
EXAMPLES::
sage: var('x,y,z')
(x, y, z)
sage: sin(x).subs(x=5)._eval_self(RR)
-0.958924274663138
sage: gamma(x).subs(x=I)._eval_self(CC)
-0.154949828301811 - 0.498015668118356*I
sage: x._eval_self(CC)
Traceback (most recent call last):
...
TypeError: Cannot evaluate symbolic expression to a numeric value.
Check if we can compute a real evaluation even if the expression
contains complex coefficients::
sage: RR((I - sqrt(2))*(I+sqrt(2)))
-3.00000000000000
sage: cos(I)._eval_self(RR)
1.54308063481524
sage: float(cos(I))
1.5430806348152437
"""
cdef GEx res
try:
res = self._gobj.evalf(0, R)
except TypeError as err:
if R is float:
R_complex = complex
else:
try:
R_complex = R.complex_field()
except (TypeError, AttributeError):
raise err
res = self._gobj.evalf(0, R_complex)
if is_a_numeric(res):
return R(py_object_from_numeric(res))
else:
raise TypeError, "Cannot evaluate symbolic expression to a numeric value."
cpdef _convert(self, R):
"""
Convert all the numeric coefficients and constants in this expression
to the given ring `R`. This results in an expression which contains
only variables, and functions whose arguments contain a variable.
EXAMPLES::
sage: f = sqrt(2) * cos(3); f
sqrt(2)*cos(3)
sage: f._convert(RDF)
-1.40006081534
sage: f._convert(float)
-1.40006081533995
There is nothing to convert for variables::
sage: x._convert(CC)
x
Note that the output is not meant to be in the in the given ring `R`.
Since the results of some functions will still be floating point
approximations::
sage: t = log(10); t
log(10)
sage: t._convert(QQ)
2.30258509299405
::
sage: (0.25 / (log(5.74 /x^0.9, 10))^2 / 4)._convert(QQ)
0.331368631904900/log(287/50/x^0.900000000000000)^2
sage: (0.25 / (log(5.74 /x^0.9, 10))^2 / 4)._convert(CC)
0.331368631904900/log(5.74000000000000/x^0.900000000000000)^2
When converting to an exact domain, powers remain unevaluated::
sage: f = sqrt(2) * cos(3); f
sqrt(2)*cos(3)
sage: f._convert(int)
-0.989992496600445*sqrt(2)
"""
cdef GEx res = self._gobj.evalf(0, R)
return new_Expression_from_GEx(self._parent, res)
def _mpfr_(self, R):
"""
Return a numerical approximation of this symbolic expression in the RealField R.
The precision of the approximation is determined by the precision of
the input R.
EXAMPLES::
0.090909090909090909090909090909090909090909090909090909090909
sage: a = sin(3); a
sin(3)
sage: RealField(200)(a)
0.14112000805986722210074480280811027984693326425226558415188
sage: a._mpfr_(RealField(100))
0.14112000805986722210074480281
"""
return self._eval_self(R)
def _real_mpfi_(self, R):
"""
Returns this expression as a real interval.
EXAMPLES::
sage: RIF(sqrt(2))
1.414213562373095?
"""
try:
return self._eval_self(R)
except TypeError:
raise TypeError, "unable to simplify to a real interval approximation"
def _complex_mpfi_(self, R):
"""
Returns this expression as a complex interval.
EXAMPLES::
sage: CIF(pi)
3.141592653589794?
"""
try:
return self._eval_self(R)
except TypeError:
raise TypeError, "unable to simplify to a complex interval approximation"
def _real_double_(self, R):
"""
EXAMPLES::
sage: RDF(sin(3))
0.14112000806
"""
return self._eval_self(R)
def _complex_mpfr_field_(self, R):
"""
Return a numerical approximation to this expression in the given
ComplexField R.
The precision of the approximation is determined by the precision of
the input R.
EXAMPLES::
sage: ComplexField(200)(SR(1/11))
0.090909090909090909090909090909090909090909090909090909090909
sage: zeta(x).subs(x=I)._complex_mpfr_field_(ComplexField(70))
0.0033002236853241028742 - 0.41815544914132167669*I
sage: gamma(x).subs(x=I)._complex_mpfr_field_(ComplexField(60))
-0.1549498283018106... - 0.49801566811835604*I
sage: log(x).subs(x=I)._complex_mpfr_field_(ComplexField(50))
1.5707963267949*I
sage: CC(sqrt(2))
1.41421356237309
sage: a = sqrt(-2); a
sqrt(-2)
sage: CC(a).imag()
1.41421356237309
sage: ComplexField(200)(a).imag()
1.4142135623730950488016887242096980785696718753769480731767
sage: ComplexField(100)((-1)^(1/10))
0.95105651629515357211643933338 + 0.30901699437494742410229341718*I
sage: CC(x*sin(0))
0.000000000000000
"""
return self._eval_self(R)
def _complex_double_(self, R):
"""
Return a numerical approximation to this expression in the given
Complex Double Field R.
EXAMPLES::
sage: CDF(SR(1/11))
0.0909090909091
sage: zeta(x).subs(x=I)._complex_double_(CDF)
0.00330022368532 - 0.418155449141*I
sage: gamma(x).subs(x=I)._complex_double_(CDF)
-0.154949828302 - 0.498015668118*I
sage: log(x).subs(x=I)._complex_double_(CDF)
1.57079632679*I
sage: CDF((-1)^(1/3))
0.5 + 0.866025403784*I
"""
return self._eval_self(R)
def __float__(self):
"""
Return float conversion of self, assuming self is constant.
Otherwise, raise a TypeError.
OUTPUT:
A ``float``. Double precision evaluation of self.
EXAMPLES::
sage: float(SR(12))
12.0
sage: float(SR(2/3))
0.6666666666666666
sage: float(sqrt(SR(2)))
1.4142135623730951
sage: float(x^2 + 1)
Traceback (most recent call last):
...
TypeError: unable to simplify to float approximation
sage: float(SR(RIF(2)))
Traceback (most recent call last):
...
TypeError: unable to simplify to float approximation
"""
try:
return float(self._eval_self(float))
except TypeError:
raise TypeError, "unable to simplify to float approximation"
def __complex__(self):
"""
EXAMPLES::
sage: complex(I)
1j
sage: complex(erf(3*I))
1629.9946226015657j
"""
try:
return self._eval_self(complex)
except TypeError:
raise TypeError, "unable to simplify to complex approximation"
def _sympy_(self):
"""
Returns a Sympy version of this object.
EXAMPLES::
sage: pi._sympy_()
pi
sage: type(_)
<class 'sympy.core.numbers.Pi'>
"""
from sage.symbolic.expression_conversions import sympy
return sympy(self)
def _algebraic_(self, field):
"""
Convert a symbolic expression to an algebraic number.
EXAMPLES::
sage: QQbar(sqrt(2) + sqrt(8))
4.242640687119285?
sage: AA(sqrt(2) ^ 4) == 4
True
sage: AA(-golden_ratio)
-1.618033988749895?
sage: QQbar((2*I)^(1/2))
1 + 1*I
sage: QQbar(e^(pi*I/3))
0.500000000000000? + 0.866025403784439?*I
sage: QQbar(sqrt(2))
1.414213562373095?
sage: AA(abs(1+I))
1.414213562373095?
sage: golden_ratio._algebraic_(QQbar)
1.618033988749895?
sage: QQbar(golden_ratio)
1.618033988749895?
sage: AA(x*sin(0))
0
sage: QQbar(x*sin(0))
0
"""
from sage.symbolic.expression_conversions import algebraic
return algebraic(self, field)
def __hash__(self):
"""
Return hash of this expression.
EXAMPLES::
The hash of an object in Python or its coerced version into
the symbolic ring is the same::
sage: hash(SR(3/1))
3
sage: hash(SR(19/23))
4
sage: hash(19/23)
4
The hash for symbolic expressions are unfortunately random. Here we
only test that the hash() function returns without error, and that
the return type is correct::
sage: x, y = var("x y")
sage: t = hash(x); type(t)
<type 'int'>
sage: t = hash(x^y); type(t)
<type 'int'>
sage: type(hash(x+y))
<type 'int'>
sage: d = {x+y: 5}
sage: d
{x + y: 5}
In this example hashing is important otherwise the answer is
wrong::
sage: uniq([x-x, -x+x])
[0]
Test if exceptions during hashing are handled properly::
sage: t = SR(matrix(2,2,range(4)))
sage: hash(t)
Traceback (most recent call last):
...
TypeError: mutable matrices are unhashable
TESTS:
Test if hashes for fderivatives with different parameters collide.
#6243::
sage: f = function('f'); t = f(x,y)
sage: u = t.derivative(x); v = t.derivative(y)
sage: hash(u) == hash(v)
False
sage: d = {u: 3, v: 5}; sorted(d.values())
[3, 5]
More checks for fderivative hashes #6851 ::
sage: hash(f(x).derivative(x)) == hash(f(x).derivative(x,2))
False
sage: d = dict( (f(x).derivative(x, i), i) for i in range(1,6) )
sage: len(d.keys())
5
We create a function with 10 arguments and test if there are hash
collisions between any of its derivatives of order at most 7. #7508::
sage: num_vars = 10; max_order=7
sage: X = var(' '.join(['x'+str(i) for i in range(num_vars)]))
sage: f = function('f',*X)
sage: hashes=set()
sage: for length in range(1,max_order+1): # long time (4s on sage.math, 2012)
... for s in UnorderedTuples(X, length):
... deriv = f.diff(*s)
... h = hash(deriv)
... if h in hashes:
... print "deriv: %s, hash:%s"%(deriv,h)
... else:
... hashes.add(n)
"""
return self._gobj.gethash()
def __richcmp__(left, right, int op):
"""
Create a formal symbolic inequality or equality.
EXAMPLES::
sage: var('x, y')
(x, y)
sage: x + 2/3 < y^2
x + 2/3 < y^2
sage: x^3 -y <= y + x
x^3 - y <= x + y
sage: x^3 -y == y + x
x^3 - y == x + y
sage: x^3 - y^10 >= y + x^10
-y^10 + x^3 >= x^10 + y
sage: x^2 > x
x^2 > x
Testing trac #11309 which changes the behavior of comparison of
comparisons::
sage: (-x + y < 0) in [x - y < 0]
False
sage: (x - 1 < 0) in [x - 2 < 0]
False
sage: Set([-x + y < 0, x - y < 0])
{-x + y < 0, x - y < 0}
sage: (x < y) == (x > y)
False
sage: (x < 0) < (x < 1)
False
sage: (x < y) != (y > x)
False
sage: (x >= y) == (y <= x)
True
sage: (x > y) == (y <= x)
False
sage: (x < x) == (x < x)
True
sage: (y > y) != (y > y)
False
sage: (x < y) != x
True
sage: (x == y) == (y == x)
True
sage: (x != y) != (y != x)
False
sage: (x == y) != (x != y)
True
sage: (x == y) == (y != x)
False
sage: x == (x == x)
False
"""
return (<Element>left)._richcmp(right, op)
cdef _richcmp_c_impl(left, Element right, int op):
cdef Expression l, r
l = left
r = right
if is_a_relational(l._gobj):
if (op != Py_EQ and op != Py_NE):
return False
if is_a_relational(r._gobj):
if l.operator() == r.operator():
e2 = (
( l._gobj.lhs().is_equal(r._gobj.lhs()) and
l._gobj.rhs().is_equal(r._gobj.rhs()) ) or
( ( l.operator() == operator.eq or
l.operator() == operator.ne ) and
l._gobj.lhs().is_equal(r._gobj.rhs()) and
l._gobj.rhs().is_equal(r._gobj.lhs()) ))
else:
e2 = (
( ( l.operator() == operator.lt and
r.operator() == operator.gt ) or
( l.operator() == operator.gt and
r.operator() == operator.lt ) or
( l.operator() == operator.le and
r.operator() == operator.ge ) or
( l.operator() == operator.ge and
r.operator() == operator.le ) ) and
l._gobj.lhs().is_equal(r._gobj.rhs()) and
l._gobj.rhs().is_equal(r._gobj.lhs()) )
else:
e2 = False
if op == Py_EQ:
return e2
else:
return not e2
elif is_a_relational(r._gobj):
return op == Py_NE
cdef GEx e
if op == Py_LT:
e = g_lt(l._gobj, r._gobj)
elif op == Py_EQ:
e = g_eq(l._gobj, r._gobj)
elif op == Py_GT:
e = g_gt(l._gobj, r._gobj)
elif op == Py_LE:
e = g_le(l._gobj, r._gobj)
elif op == Py_NE:
e = g_ne(l._gobj, r._gobj)
elif op == Py_GE:
e = g_ge(l._gobj, r._gobj)
else:
raise TypeError
return new_Expression_from_GEx(l._parent, e)
def assume(self):
r"""
Assume that this equation holds. This is relevant for symbolic
integration, among other things.
EXAMPLES: We call the assume method to assume that `x>2`::
sage: (x > 2).assume()
Bool returns True below if the inequality is *definitely* known to
be True.
::
sage: bool(x > 0)
True
sage: bool(x < 0)
False
This may or may not be True, so bool returns False::
sage: bool(x > 3)
False
If you make inconsistent or meaningless assumptions,
Sage will let you know::
sage: forget()
sage: assume(x<0)
sage: assume(x>0)
Traceback (most recent call last):
...
ValueError: Assumption is inconsistent
sage: assumptions()
[x < 0]
sage: forget()
TESTS::
sage: v,c = var('v,c')
sage: assume(c != 0)
sage: integral((1+v^2/c^2)^3/(1-v^2/c^2)^(3/2),v)
83/8*v/sqrt(-v^2/c^2 + 1) - 17/8*v^3/(c^2*sqrt(-v^2/c^2 + 1)) - 1/4*v^5/(c^4*sqrt(-v^2/c^2 + 1)) - 75/8*arcsin(v/(c^2*sqrt(c^(-2))))/sqrt(c^(-2))
sage: forget()
"""
from sage.symbolic.assumptions import _assumptions
from sage.calculus.calculus import maxima
if not self.is_relational():
raise TypeError, "self (=%s) must be a relational expression"%self
if not self in _assumptions:
m = self._maxima_init_assume_()
s = maxima.assume(m)
if str(s._sage_()[0]) in ['meaningless','inconsistent','redundant']:
raise ValueError, "Assumption is %s"%str(s._sage_()[0])
_assumptions.append(self)
def forget(self):
"""
Forget the given constraint.
EXAMPLES::
sage: var('x,y')
(x, y)
sage: forget()
sage: assume(x>0, y < 2)
sage: assumptions()
[x > 0, y < 2]
sage: forget(y < 2)
sage: assumptions()
[x > 0]
TESTS:
Check if #7507 is fixed::
sage: forget()
sage: n = var('n')
sage: foo=sin((-1)*n*pi)
sage: foo.simplify()
-sin(pi*n)
sage: assume(n, 'odd')
sage: assumptions()
[n is odd]
sage: foo.simplify()
0
sage: forget(n, 'odd')
sage: assumptions()
[]
sage: foo.simplify()
-sin(pi*n)
"""
from sage.symbolic.assumptions import _assumptions
from sage.calculus.calculus import maxima
if not self.is_relational():
raise TypeError, "self (=%s) must be a relational expression"%self
m = self._maxima_init_assume_()
maxima.forget(m)
try:
_assumptions.remove(self)
except ValueError:
pass
def _maxima_init_assume_(self):
"""
Return string that when evaluated in Maxima defines the assumption
determined by this expression.
EXAMPLES::
sage: f = x+2 > sqrt(3)
sage: f._maxima_init_assume_()
'((x)+(2))>((3/1)^(1/2))'
"""
from sage.calculus.calculus import maxima
l = self.lhs()._assume_str()
r = self.rhs()._assume_str()
op = self.operator()
if op is operator.eq:
m = 'equal(%s, %s)'%(l, r)
elif op is operator.ne:
m = 'notequal(%s, %s)'%(l, r)
else:
m = '(%s)%s(%s)' % (l, maxima._relation_symbols()[op], r)
return m
def _assume_str(self):
"""
TESTS::
sage: x = var('x')
sage: x._assume_str()
'x'
sage: y = function('y', x)
sage: y._assume_str()
'y'
sage: abs(x)._assume_str()
'abs(x)'
"""
if is_a_function(self._gobj) and self.nops() == 1 and \
is_a_symbol(self._gobj.op(0)):
op = self.operator()
if isinstance(op, SymbolicFunction):
return self.operator().name()
return self._maxima_init_()
_is_real = deprecated_function_alias(10859, is_real)
_is_positive = deprecated_function_alias(10859, is_positive)
_is_negative = deprecated_function_alias(10859, is_negative)
_is_integer = deprecated_function_alias(10859, is_integer)
_is_symbol = deprecated_function_alias(10859, is_symbol)
_is_constant = deprecated_function_alias(10859, is_constant)
_is_numeric = deprecated_function_alias(10859, is_numeric)
def is_real(self):
"""
Returns True if this expression is known to be a real number.
EXAMPLES::
sage: t0 = SR.symbol("t0", domain='real')
sage: t0.is_real()
True
sage: t0.is_positive()
False
sage: t1 = SR.symbol("t1", domain='positive')
sage: (t0+t1).is_real()
True
sage: (t0+x).is_real()
False
sage: (t0*t1).is_real()
True
sage: (t0*x).is_real()
False
The following is real, but we can't deduce that.::
sage: (x*x.conjugate()).is_real()
False
"""
return self._gobj.info(info_real)
def is_positive(self):
"""
Returns True if this expression is known to be positive.
EXAMPLES::
sage: t0 = SR.symbol("t0", domain='positive')
sage: t0.is_positive()
True
sage: t0.is_negative()
False
sage: t0.is_real()
True
sage: t1 = SR.symbol("t1", domain='positive')
sage: (t0*t1).is_positive()
True
sage: (t0 + t1).is_positive()
True
sage: (t0*x).is_positive()
False
"""
return self._gobj.info(info_positive)
def is_negative(self):
"""
Return True if this expression is known to be negative.
EXAMPLES::
sage: SR(-5).is_negative()
True
Check if we can correctly deduce negativity of mul objects::
sage: t0 = SR.symbol("t0", domain='positive')
sage: t0.is_negative()
False
sage: (-t0).is_negative()
True
sage: (-pi).is_negative()
True
"""
return self._gobj.info(info_negative)
def is_integer(self):
"""
Return True if this expression is known to be an integer.
EXAMPLES::
sage: SR(5).is_integer()
True
"""
return self._gobj.info(info_integer)
def is_symbol(self):
"""
Return True if this symbolic expression consists of only a symbol, i.e.,
a symbolic variable.
EXAMPLES::
sage: x.is_symbol()
True
sage: var('y')
y
sage: y.is_symbol()
True
sage: (x*y).is_symbol()
False
sage: pi.is_symbol()
False
::
sage: ((x*y)/y).is_symbol()
True
sage: (x^y).is_symbol()
False
"""
return is_a_symbol(self._gobj)
def is_constant(self):
"""
Return True if this symbolic expression is a constant.
This function is intended to provide an interface to query the internal
representation of the expression. In this sense, the word ``constant``
doesn't reflect the mathematical properties of the expression.
Expressions which have no variables may return ``False``.
EXAMPLES::
sage: pi.is_constant()
True
sage: x.is_constant()
False
sage: SR(1).is_constant()
False
Note that the complex I is not a constant::
sage: I.is_constant()
False
sage: I.is_numeric()
True
"""
return is_a_constant(self._gobj)
def is_numeric(self):
"""
Return True if this expression only consists of a numeric object.
EXAMPLES::
sage: SR(1).is_numeric()
True
sage: x.is_numeric()
False
sage: pi.is_numeric()
False
"""
return is_a_numeric(self._gobj)
def is_series(self):
"""
Return True if ``self`` is a series.
Series are special kinds of symbolic expressions that are
constructed via the :meth:`series` method. They usually have
an ``Order()`` term unless the series representation is exact,
see :meth:`is_terminating_series`.
OUTPUT:
Boolean. Whether ``self`` is a series symbolic
expression. Usually, this means that it was constructed by the
:meth:`series` method.
Returns ``False`` if only a subexpression of the symbolic
expression is a series.
EXAMPLES::
sage: SR(5).is_series()
False
sage: var('x')
x
sage: x.is_series()
False
sage: exp(x).is_series()
False
sage: exp(x).series(x,10).is_series()
True
Laurent series are series, too::
sage: laurent_series = (cos(x)/x).series(x, 5)
sage: laurent_series
1*x^(-1) + (-1/2)*x + 1/24*x^3 + Order(x^5)
sage: laurent_series.is_series()
True
Something only containing a series as a subexpression is not a
series::
sage: sum_expr = 1 + exp(x).series(x,5); sum_expr
(1 + 1*x + 1/2*x^2 + 1/6*x^3 + 1/24*x^4 + Order(x^5)) + 1
sage: sum_expr.is_series()
False
"""
return is_a_series(self._gobj)
def is_terminating_series(self):
"""
Return True if ``self`` is a series without order term.
A series is terminating if it can be represented exactly,
without requiring an order term. See also :meth:`is_series`
for general series.
OUTPUT:
Boolean. Whether ``self`` was constructed by :meth:`series`
and has no order term.
EXAMPLES::
sage: (x^5+x^2+1).series(x,10)
1 + 1*x^2 + 1*x^5
sage: (x^5+x^2+1).series(x,10).is_series()
True
sage: (x^5+x^2+1).series(x,10).is_terminating_series()
True
sage: SR(5).is_terminating_series()
False
sage: var('x')
x
sage: x.is_terminating_series()
False
sage: exp(x).series(x,10).is_terminating_series()
False
"""
return g_is_a_terminating_series(self._gobj)
cpdef bint is_polynomial(self, var):
"""
Return True if self is a polynomial in the given variable.
EXAMPLES::
sage: var('x,y,z')
(x, y, z)
sage: t = x^2 + y; t
x^2 + y
sage: t.is_polynomial(x)
True
sage: t.is_polynomial(y)
True
sage: t.is_polynomial(z)
True
sage: t = sin(x) + y; t
y + sin(x)
sage: t.is_polynomial(x)
False
sage: t.is_polynomial(y)
True
sage: t.is_polynomial(sin(x))
True
TESTS:
Check if we can handle derivatives. #6523::
sage: f(x) = function('f',x)
sage: f(x).diff(x).is_zero()
False
Check if #11352 is fixed::
sage: el = -1/2*(2*x^2 - sqrt(2*x - 1)*sqrt(2*x + 1) - 1)
sage: el.is_polynomial(x)
False
"""
cdef Expression symbol0 = self.coerce_in(var)
return self._gobj.is_polynomial(symbol0._gobj)
cpdef bint is_relational(self):
"""
Return True if self is a relational expression.
EXAMPLES::
sage: x = var('x')
sage: eqn = (x-1)^2 == x^2 - 2*x + 3
sage: eqn.is_relational()
True
sage: sin(x).is_relational()
False
"""
return is_a_relational(self._gobj)
cpdef bint is_infinity(self):
"""
Return True if self is an infinite expression.
EXAMPLES::
sage: SR(oo).is_infinity()
True
sage: x.is_infinity()
False
"""
return is_a_infinity(self._gobj)
def left_hand_side(self):
"""
If self is a relational expression, return the left hand side
of the relation. Otherwise, raise a ValueError.
EXAMPLES::
sage: x = var('x')
sage: eqn = (x-1)^2 == x^2 - 2*x + 3
sage: eqn.left_hand_side()
(x - 1)^2
sage: eqn.lhs()
(x - 1)^2
sage: eqn.left()
(x - 1)^2
"""
if not self.is_relational():
raise ValueError, "self must be a relational expression"
return new_Expression_from_GEx(self._parent, self._gobj.lhs())
lhs = left = left_hand_side
def right_hand_side(self):
"""
If self is a relational expression, return the right hand side
of the relation. Otherwise, raise a ValueError.
EXAMPLES::
sage: x = var('x')
sage: eqn = (x-1)^2 <= x^2 - 2*x + 3
sage: eqn.right_hand_side()
x^2 - 2*x + 3
sage: eqn.rhs()
x^2 - 2*x + 3
sage: eqn.right()
x^2 - 2*x + 3
"""
if not self.is_relational():
raise ValueError, "self must be a relation"
return new_Expression_from_GEx(self._parent, self._gobj.rhs())
rhs = right = right_hand_side
def is_trivial_zero(self):
"""
Check if this expression is trivially equal to zero without any
simplification.
This method is intended to be used in library code where trying to
obtain a mathematically correct result by applying potentially
expensive rewrite rules is not desirable.
EXAMPLES::
sage: SR(0).is_trivial_zero()
True
sage: SR(0.0).is_trivial_zero()
True
sage: SR(float(0.0)).is_trivial_zero()
True
sage: (SR(1)/2^1000).is_trivial_zero()
False
sage: SR(1./2^10000).is_trivial_zero()
False
The :meth:`~sage.structure.element.Element.is_zero` method
is more capable::
sage: t = pi + (pi - 1)*pi - pi^2
sage: t.is_trivial_zero()
False
sage: t.is_zero()
True
sage: u = sin(x)^2 + cos(x)^2 - 1
sage: u.is_trivial_zero()
False
sage: u.is_zero()
True
"""
return self._gobj.is_zero()
def __nonzero__(self):
"""
Return True unless this symbolic expression can be shown by Sage
to be zero. Note that deciding if an expression is zero is
undecidable in general.
EXAMPLES::
sage: x = var('x')
sage: forget()
sage: SR(0).__nonzero__()
False
sage: SR(1).__nonzero__()
True
sage: bool(abs(x))
True
sage: bool(x/x - 1)
False
This is called by :meth:`is_zero`::
sage: k = var('k')
sage: pol = 1/(k-1) - 1/k - 1/k/(k-1)
sage: pol.is_zero()
True
sage: f = sin(x)^2 + cos(x)^2 - 1
sage: f.is_zero()
True
TESTS:
First, a bunch of tests of nonzero (which is called by bool)
for symbolic relations::
sage: x = var('x')
sage: bool((x-1)^2 == x^2 - 2*x + 1)
True
sage: bool(((x-1)^2 == x^2 - 2*x + 1).expand())
True
sage: bool(((x-1)^2 == x^2 - 2*x + 3).expand())
False
sage: bool(2 + x < 3 + x)
True
sage: bool(2 + x < 1 + x)
False
sage: bool(2 + x > 1 + x)
True
sage: bool(1 + x > 1 + x)
False
sage: bool(1 + x >= 1 + x)
True
sage: bool(1 + x < 1 + x)
False
sage: bool(1 + x <= 1 + x)
True
sage: bool(1 + x^2 != 1 + x*x)
False
sage: bool(1 + x^2 != 2 + x*x)
True
sage: bool(SR(oo) == SR(oo))
True
sage: bool(-SR(oo) == SR(oo))
False
sage: bool(-SR(oo) != SR(oo))
True
Next, tests to ensure assumptions are correctly used::
sage: x, y, z = var('x, y, z')
sage: assume(x>=y,y>=z,z>=x)
sage: bool(x==z)
True
sage: bool(z<x)
False
sage: bool(z>y)
False
sage: bool(y==z)
True
sage: bool(y<=z)
True
sage: forget()
sage: assume(x>=1,x<=1)
sage: bool(x==1)
True
sage: bool(x != 1)
False
sage: bool(x>1)
False
sage: forget()
sage: assume(x>0)
sage: bool(x==0)
False
sage: bool(x != 0)
True
sage: bool(x == 1)
False
The following must be true, even though we don't
know for sure that x isn't 1, as symbolic comparisons
elsewhere rely on x!=y unless we are sure it is not
true; there is no equivalent of Maxima's ``unknown``.
Since it is False that x==1, it is True that x != 1.
::
sage: bool(x != 1)
True
sage: forget()
sage: assume(x>y)
sage: bool(x==y)
False
sage: bool(x != y)
True
sage: bool(x != y) # The same comment as above applies here as well
True
sage: forget()
Comparisons of infinities::
sage: assert( (1+I)*oo == (2+2*I)*oo )
sage: assert( SR(unsigned_infinity) == SR(unsigned_infinity) )
sage: assert( SR(I*oo) == I*oo )
sage: assert( SR(-oo) <= SR(oo) )
sage: assert( SR(oo) >= SR(-oo) )
sage: assert( SR(oo) != SR(-oo) )
sage: assert( sqrt(2)*oo != I*oo )
"""
if self.is_relational():
if is_a_constant(self._gobj.lhs()) and is_a_constant(self._gobj.rhs()):
return self.operator()(self.lhs().pyobject(), self.rhs().pyobject())
pynac_result = relational_to_bool(self._gobj)
if is_a_infinity(self._gobj.lhs()) or is_a_infinity(self._gobj.rhs()):
return pynac_result
if pynac_result:
if self.operator() == operator.ne:
m = self._maxima_()
s = m.parent()._eval_line('is (notequal(%s,%s))'%(repr(m.lhs()),repr(m.rhs())))
if s == 'false':
return False
else:
return True
else:
return True
need_assumptions = False
from sage.symbolic.assumptions import assumptions
assumption_list = assumptions()
if assumption_list:
vars = self.variables()
if vars:
assumption_var_list = []
for eqn in assumption_list:
try:
assumption_var_list.append(eqn.variables())
except AttributeError:
assumption_var_list.append((eqn._var,))
assumption_vars = set(sum(assumption_var_list, ()))
if set(vars).intersection(assumption_vars):
need_assumptions = True
if not need_assumptions:
res = self.test_relation()
if res is True:
return True
elif res is False:
return False
from sage.symbolic.relation import test_relation_maxima
return test_relation_maxima(self)
self_is_zero = self._gobj.is_zero()
if self_is_zero:
return False
else:
return not bool(self == self._parent.zero_element())
def test_relation(self, int ntests=20, domain=None, proof=True):
"""
Test this relation at several random values, attempting to find
a contradiction. If this relation has no variables, it will also
test this relation after casting into the domain.
Because the interval fields never return false positives, we can be
assured that if True or False is returned (and proof is False) then
the answer is correct.
INPUT:
- ``ntests`` -- (default ``20``) the number of iterations to run
- ``domain`` -- (optional) the domain from which to draw the random
values defaults to ``CIF`` for equality testing and ``RIF`` for
order testing
- ``proof`` -- (default ``True``) if ``False`` and the domain is an
interval field, regard overlapping (potentially equal) intervals as
equal, and return ``True`` if all tests succeeded.
OUTPUT:
Boolean or ``NotImplemented``, meaning
- ``True`` -- this relation holds in the domain and has no variables.
- ``False`` -- a contradiction was found.
- ``NotImplemented`` -- no contradiction found.
EXAMPLES::
sage: (3 < pi).test_relation()
True
sage: (0 >= pi).test_relation()
False
sage: (exp(pi) - pi).n()
19.9990999791895
sage: (exp(pi) - pi == 20).test_relation()
False
sage: (sin(x)^2 + cos(x)^2 == 1).test_relation()
NotImplemented
sage: (sin(x)^2 + cos(x)^2 == 1).test_relation(proof=False)
True
sage: (x == 1).test_relation()
False
sage: var('x,y')
(x, y)
sage: (x < y).test_relation()
False
TESTS::
sage: all_relations = [op for name, op in sorted(operator.__dict__.items()) if len(name) == 2]
sage: all_relations
[<built-in function eq>, <built-in function ge>, <built-in function gt>, <built-in function le>, <built-in function lt>, <built-in function ne>]
sage: [op(3, pi).test_relation() for op in all_relations]
[False, False, False, True, True, True]
sage: [op(pi, pi).test_relation() for op in all_relations]
[True, True, False, True, False, False]
sage: s = 'some_very_long_variable_name_which_will_definitely_collide_if_we_use_a_reasonable_length_bound_for_a_hash_that_respects_lexicographic_order'
sage: t1, t2 = var(','.join([s+'1',s+'2']))
sage: (t1 == t2).test_relation()
False
sage: (cot(pi + x) == 0).test_relation()
NotImplemented
"""
cdef int k, eq_count = 0
cdef bint is_interval
if not self.is_relational():
raise ValueError, "self must be a relation"
cdef operators op = relational_operator(self._gobj)
from sage.rings.real_mpfi import is_RealIntervalField
from sage.rings.complex_interval_field import is_ComplexIntervalField
from sage.rings.all import RIF, CIF
if domain is None:
is_interval = True
if op == equal or op == not_equal:
domain = CIF
else:
domain = RIF
else:
is_interval = is_RealIntervalField(domain) or is_ComplexIntervalField(domain)
zero = domain(0)
diff = self.lhs() - self.rhs()
vars = diff.variables()
if op == equal:
falsify = operator.ne
elif op == not_equal:
falsify = operator.eq
elif op == less:
falsify = operator.ge
elif op == less_or_equal:
falsify = operator.gt
elif op == greater:
falsify = operator.le
elif op == greater_or_equal:
falsify = operator.lt
cdef bint equality_ok = op in [equal, less_or_equal, greater_or_equal]
cdef int errors = 0
val = None
if len(vars) == 0:
try:
val = domain(diff)
except (TypeError, ValueError, ArithmeticError), ex:
pass
else:
if self.operator()(val, zero):
return True
elif falsify(val, zero):
return False
if is_interval and not proof:
if val.contains_zero():
return equality_ok
else:
return not equality_ok
else:
for k in range(ntests):
try:
if is_interval:
if val and k > 4 and val.contains_zero() and domain.prec() < 1000:
domain = domain.to_prec(int(domain.prec() * 1.5))
var_dict = dict([(v, domain.random_element() * domain.random_element(-2,6).exp()) for v in vars])
else:
var_dict = dict([(v, domain.random_element()) for v in vars])
val = domain(diff.subs(var_dict))
if falsify(val, zero):
return False
if is_interval:
eq_count += <bint>val.contains_zero()
except (TypeError, ValueError, ArithmeticError, AttributeError), ex:
errors += 1
if k == errors > 3 and is_ComplexIntervalField(domain):
domain = RIF.to_prec(domain.prec())
eq_count += equality_ok
if not proof:
if not equality_ok:
return eq_count == 0
elif op == equal and is_interval:
return eq_count == ntests
else:
return True
return NotImplemented
def negation(self):
"""
Returns the negated version of self, that is the relation that is
False iff self is True.
EXAMPLES::
sage: (x < 5).negation()
x >= 5
sage: (x == sin(3)).negation()
x != sin(3)
sage: (2*x >= sqrt(2)).negation()
2*x < sqrt(2)
"""
if not self.is_relational():
raise ValueError, "self must be a relation"
cdef operators op = relational_operator(self._gobj)
if op == equal:
falsify = operator.ne
elif op == not_equal:
falsify = operator.eq
elif op == less:
falsify = operator.ge
elif op == less_or_equal:
falsify = operator.gt
elif op == greater:
falsify = operator.le
elif op == greater_or_equal:
falsify = operator.lt
return falsify(self.lhs(), self.rhs())
def contradicts(self, soln):
"""
Returns ``True`` if this relation is violated by the given variable assignment(s).
EXAMPLES::
sage: (x<3).contradicts(x==0)
False
sage: (x<3).contradicts(x==3)
True
sage: (x<=3).contradicts(x==3)
False
sage: y = var('y')
sage: (x<y).contradicts(x==30)
False
sage: (x<y).contradicts({x: 30, y: 20})
True
"""
return bool(self.negation().subs(soln))
def is_unit(self):
"""
Return True if this expression is a unit of the symbolic ring.
EXAMPLES::
sage: SR(1).is_unit()
True
sage: SR(-1).is_unit()
True
sage: SR(0).is_unit()
False
"""
if not not self:
return True
if self == 0:
return False
raise NotImplementedError
cdef Expression coerce_in(self, z):
"""
Quickly coerce z to be an Expression.
"""
cdef Expression w
try:
w = z
return w
except TypeError:
return self._parent._coerce_(z)
cpdef ModuleElement _add_(left, ModuleElement right):
"""
Add left and right.
EXAMPLES::
sage: var("x y")
(x, y)
sage: x + y + y + x
2*x + 2*y
# adding relational expressions
sage: ( (x+y) > x ) + ( x > y )
2*x + y > x + y
sage: ( (x+y) > x ) + x
2*x + y > 2*x
TESTS::
sage: x + ( (x+y) > x )
2*x + y > 2*x
sage: ( x > y) + (y < x)
Traceback (most recent call last):
...
TypeError: incompatible relations
sage: (x < 1) + (y <= 2)
x + y < 3
sage: x + oo
+Infinity
sage: x - oo
-Infinity
sage: x + unsigned_infinity
Infinity
sage: x - unsigned_infinity
Infinity
sage: nsr = x.parent()
sage: nsr(oo) + nsr(oo)
+Infinity
sage: nsr(-oo) + nsr(-oo)
-Infinity
sage: nsr(oo) - nsr(oo)
Traceback (most recent call last):
...
RuntimeError: indeterminate expression: infinity - infinity encountered.
sage: nsr(-oo) - nsr(-oo)
Traceback (most recent call last):
...
RuntimeError: indeterminate expression: infinity - infinity encountered.
sage: nsr(unsigned_infinity) + nsr(oo)
Traceback (most recent call last):
...
RuntimeError: indeterminate expression: unsigned_infinity +- infinity encountered.
sage: nsr(unsigned_infinity) - nsr(oo)
Traceback (most recent call last):
...
RuntimeError: indeterminate expression: unsigned_infinity +- infinity encountered.
sage: nsr(oo) + nsr(unsigned_infinity)
Traceback (most recent call last):
...
RuntimeError: indeterminate expression: unsigned_infinity +- infinity encountered.
sage: nsr(oo) - nsr(unsigned_infinity)
Traceback (most recent call last):
...
RuntimeError: indeterminate expression: unsigned_infinity +- infinity encountered.
sage: nsr(unsigned_infinity) + nsr(unsigned_infinity)
Infinity
"""
cdef GEx x
cdef Expression _right = <Expression>right
cdef operators op
if is_a_relational(left._gobj):
if is_a_relational(_right._gobj):
op = compatible_relation(relational_operator(left._gobj),
relational_operator(_right._gobj))
x = relational(gadd(left._gobj.lhs(), _right._gobj.lhs()),
gadd(left._gobj.rhs(), _right._gobj.rhs()),
op)
else:
x = relational(gadd(left._gobj.lhs(), _right._gobj),
gadd(left._gobj.rhs(), _right._gobj),
relational_operator(left._gobj))
elif is_a_relational(_right._gobj):
x = relational(gadd(left._gobj, _right._gobj.lhs()),
gadd(left._gobj, _right._gobj.rhs()),
relational_operator(_right._gobj))
else:
x = gadd(left._gobj, _right._gobj)
return new_Expression_from_GEx(left._parent, x)
cpdef ModuleElement _sub_(left, ModuleElement right):
"""
EXAMPLES::
sage: var("x y")
(x, y)
sage: x - y
x - y
# subtracting relational expressions
sage: ( (x+y) > x ) - ( x > y )
y > x - y
sage: ( (x+y) > x ) - x
y > 0
TESTS::
sage: x - ( (x+y) > x )
-y > 0
sage: ( x > y) - (y < x)
Traceback (most recent call last):
...
TypeError: incompatible relations
sage: x - oo
-Infinity
sage: oo - x
+Infinity
"""
cdef GEx x
cdef Expression _right = <Expression>right
if is_a_relational(left._gobj):
if is_a_relational(_right._gobj):
op = compatible_relation(relational_operator(left._gobj),
relational_operator(_right._gobj))
x = relational(gsub(left._gobj.lhs(), _right._gobj.lhs()),
gsub(left._gobj.rhs(), _right._gobj.rhs()),
op)
else:
x = relational(gsub(left._gobj.lhs(), _right._gobj),
gsub(left._gobj.rhs(), _right._gobj),
relational_operator(left._gobj))
elif is_a_relational(_right._gobj):
x = relational(gsub(left._gobj, _right._gobj.lhs()),
gsub(left._gobj, _right._gobj.rhs()),
relational_operator(_right._gobj))
else:
x = gsub(left._gobj, _right._gobj)
return new_Expression_from_GEx(left._parent, x)
cpdef RingElement _mul_(left, RingElement right):
"""
Multiply left and right.
EXAMPLES::
sage: var("x y")
(x, y)
sage: x*y*y
x*y^2
# multiplying relational expressions
sage: ( (x+y) > x ) * ( x > y )
(x + y)*x > x*y
sage: ( (x+y) > x ) * x
(x + y)*x > x^2
sage: ( (x+y) > x ) * -1
-x - y > -x
TESTS::
sage: x * ( (x+y) > x )
(x + y)*x > x^2
sage: ( x > y) * (y < x)
Traceback (most recent call last):
...
TypeError: incompatible relations
sage: a = 1000 + 300*x + x^3 + 30*x^2
sage: a*Mod(1,7)
x^3 + 2*x^2 + 6*x + 6
sage: var('z')
z
sage: 3*(x+y)/z
3*(x + y)/z
sage: (-x+z)*(3*x-3*z)
-3*(x - z)^2
# check if comparison of constant terms in Pynac add objects work
sage: (y-1)*(y-2)
(y - 1)*(y - 2)
Check if Pynac can compute inverses of Python longs (:trac:`13107`)::
sage: SR(4L)*SR(2L)^(-1)
2.0
Check for simplifications when multiplying instances of exp::
sage: exp(x)*exp(y)
e^(x + y)
sage: exp(x)^2*exp(y)
e^(2*x + y)
sage: x^y*exp(x+y)*exp(-y)
x^y*e^x
sage: x^y*exp(x+y)*(x+y)*(2*x+2*y)*exp(-y)
2*(x + y)^2*x^y*e^x
sage: x^y*exp(x+y)*(x+y)*(2*x+2*y)*exp(-y)*exp(z)^2
2*(x + y)^2*x^y*e^(x + 2*z)
sage: 1/exp(x)
e^(-x)
sage: exp(x)/exp(y)
e^(x - y)
sage: A = exp(I*pi/5)
sage: t = A*A*A*A; t
e^(4/5*I*pi)
sage: t*A
-1
sage: b = -x*A; c = b*b; c
x^2*e^(2/5*I*pi)
sage: u = -t*A; u
1
Products of non integer powers of exp are not simplified::
sage: exp(x)^I*exp(z)^(2.5)
(e^x)^I*(e^z)^2.50000000000000
::
sage: x*oo
Traceback (most recent call last):
...
RuntimeError: indeterminate expression: infinity * f(x) encountered.
sage: x*unsigned_infinity
Traceback (most recent call last):
...
ValueError: oo times number < oo not defined
sage: SR(oo)*SR(oo)
+Infinity
sage: SR(-oo)*SR(oo)
-Infinity
sage: SR(oo)*SR(-oo)
-Infinity
sage: SR(unsigned_infinity)*SR(oo)
Infinity
Check if we are returning informative error messages in case of
nonsensical arithmetic :trac:`13739`::
sage: t = GF(5)(3)
sage: u = GF(7)(4)
sage: var('y')
y
sage: e = t*x + u*y
sage: t*e
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for '*': 'Finite Field
of size 7' and 'Finite Field of size 5'
The same issue (with a different test case) was reported in
:trac:`10960`::
sage: K.<b> = FiniteField(9)
sage: i*b
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for '*': 'Number Field
in I with defining polynomial x^2 + 1' and 'Finite Field in b of
size 3^2'
"""
cdef GEx x
cdef Expression _right = <Expression>right
cdef operators o
if is_a_relational(left._gobj):
if is_a_relational(_right._gobj):
op = compatible_relation(relational_operator(left._gobj),
relational_operator(_right._gobj))
x = relational(gmul(left._gobj.lhs(), _right._gobj.lhs()),
gmul(left._gobj.rhs(), _right._gobj.rhs()),
op)
else:
o = relational_operator(left._gobj)
x = relational(gmul(left._gobj.lhs(), _right._gobj),
gmul(left._gobj.rhs(), _right._gobj),
o)
elif is_a_relational(_right._gobj):
o = relational_operator(_right._gobj)
x = relational(gmul(left._gobj, _right._gobj.lhs()),
gmul(left._gobj, _right._gobj.rhs()),
o)
else:
x = gmul(left._gobj, _right._gobj)
return new_Expression_from_GEx(left._parent, x)
cpdef RingElement _div_(left, RingElement right):
"""
Divide left and right.
EXAMPLES::
sage: var("x y")
(x, y)
sage: x/y/y
x/y^2
# dividing relational expressions
sage: ( (x+y) > x ) / ( x > y )
(x + y)/x > x/y
sage: ( (x+y) > x ) / x
(x + y)/x > 1
sage: ( (x+y) > x ) / -1
-x - y > -x
TESTS::
sage: x / ( (x+y) > x )
x/(x + y) > 1
sage: ( x > y) / (y < x)
Traceback (most recent call last):
...
TypeError: incompatible relations
sage: x/oo
0
sage: oo/x
Traceback (most recent call last):
...
RuntimeError: indeterminate expression: infinity * f(x) encountered.
sage: SR(oo)/SR(oo)
Traceback (most recent call last):
...
RuntimeError: indeterminate expression: 0 * infinity encountered.
sage: SR(-oo)/SR(oo)
Traceback (most recent call last):
...
RuntimeError: indeterminate expression: 0 * infinity encountered.
sage: SR(oo)/SR(-oo)
Traceback (most recent call last):
...
RuntimeError: indeterminate expression: 0 * infinity encountered.
sage: SR(oo)/SR(unsigned_infinity)
Traceback (most recent call last):
...
RuntimeError: indeterminate expression: 0 * infinity encountered.
sage: SR(unsigned_infinity)/SR(oo)
Traceback (most recent call last):
...
RuntimeError: indeterminate expression: 0 * infinity encountered.
sage: SR(0)/SR(oo)
0
sage: SR(0)/SR(unsigned_infinity)
0
sage: x/0
Traceback (most recent call last):
...
ZeroDivisionError: Symbolic division by zero
Check if Pynac can compute divisions of Python longs (:trac:`13107`)::
sage: SR(1L)/SR(2L)
0.5
"""
cdef GEx x
cdef Expression _right = <Expression>right
cdef operators o
try:
if is_a_relational(left._gobj):
if is_a_relational(_right._gobj):
op = compatible_relation(relational_operator(left._gobj),
relational_operator(_right._gobj))
x = relational(gdiv(left._gobj.lhs(), _right._gobj.lhs()),
gdiv(left._gobj.rhs(), _right._gobj.rhs()),
op)
else:
o = relational_operator(left._gobj)
x = relational(gdiv(left._gobj.lhs(), _right._gobj),
gdiv(left._gobj.rhs(), _right._gobj),
o)
elif is_a_relational(_right._gobj):
o = relational_operator(_right._gobj)
x = relational(gdiv(left._gobj, _right._gobj.lhs()),
gdiv(left._gobj, _right._gobj.rhs()),
o)
else:
x = gdiv(left._gobj, _right._gobj)
return new_Expression_from_GEx(left._parent, x)
except Exception, msg:
if 'division by zero' in str(msg):
raise ZeroDivisionError, "Symbolic division by zero"
else:
raise
def __invert__(self):
"""
Return the inverse of this symbolic expression.
EXAMPLES::
sage: ~x
1/x
sage: ~SR(3)
1/3
sage: v1=var('v1'); a = (2*erf(2*v1*arcsech(0))/v1); ~a
1/2*v1/erf(2*v1*arcsech(0))
"""
return 1/self
def __cmp__(left, right):
"""
Compare self and right, returning -1, 0, or 1, depending on if
self < right, self == right, or self > right, respectively.
Use this instead of the operators <=, <, etc. to compare symbolic
expressions when you do not want to get a formal inequality back.
IMPORTANT: Both self and right *must* have the same type, or
this function won't be called.
EXAMPLES::
sage: x,y = var('x,y')
sage: x.__cmp__(y)
1
sage: x < y
x < y
sage: cmp(x,y)
1
sage: cmp(SR(0.5), SR(0.7))
-1
sage: SR(0.5) < SR(0.7)
0.500000000000000 < 0.700000000000000
sage: cmp(SR(0.5), 0.7)
-1
sage: cmp(sin(SR(2)), sin(SR(1)))
1
sage: float(sin(SR(2)))
0.9092974268256817
sage: float(sin(SR(1)))
0.8414709848078965
TESTS:
Check that :trac:`9880` is fixed::
sage: b = [var('b_%s'%i) for i in range(4)]
sage: precomp = (2^b_2 + 2)*(2^b_1 + 2^(-b_1) + 2^b_1*2^b_0 - \
2^b_1*2^(-b_0) - 2^(-b_1)*2^b_0 - 2^(-b_1)*2^(-b_0) + \
2^b_0 + 2^(-b_0) - 9) + (2^b_1 + 2^(-b_1) + \
2^b_1*2^b_0 - 2^b_1*2^(-b_0) - 2^(-b_1)*2^b_0 - \
2^(-b_1)*2^(-b_0) + 2^b_0 + 2^(-b_0) - 9)/2^b_2
sage: repl_dict = {b_0: b_0, b_3: b_1, b_2: b_3, b_1: b_2}
sage: P = precomp.substitute(repl_dict)
sage: P.expand()
-2^(-b_0)*2^(-b_2)*2^b_3 - 2^b_0*2^(-b_2)*2^b_3 -
2^(-b_0)*2^b_2*2^b_3 + 2^b_0*2^b_2*2^b_3 - 2*2^(-b_0)*2^(-b_2)
- 2*2^b_0*2^(-b_2) - 2*2^(-b_0)*2^b_2 + 2*2^b_0*2^b_2 +
2^(-b_0)*2^b_3 + 2^b_0*2^b_3 + 2^(-b_2)*2^b_3 + 2^b_2*2^b_3 +
2*2^(-b_0) + 2*2^b_0 + 2*2^(-b_2) + 2*2^b_2 - 9*2^b_3 -
2^(-b_0)*2^(-b_2)/2^b_3 - 2^b_0*2^(-b_2)/2^b_3 -
2^(-b_0)*2^b_2/2^b_3 + 2^b_0*2^b_2/2^b_3 + 2^(-b_0)/2^b_3 +
2^b_0/2^b_3 + 2^(-b_2)/2^b_3 + 2^b_2/2^b_3 - 9/2^b_3 - 18
sage: _0,b_1,b_2=var('b_0,b_1,b_2')
sage: f = 1/27*b_2^2/(2^b_2)^2 + 1/27*b_1^2/(2^b_1)^2 + \
1/27*b_0^2/(2^b_0)^2 + 1/27*b_2/(2^b_2)^2 - 2/81/(2^b_2)^2 + \
1/27*b_1/(2^b_1)^2 + 8/243/(2^b_2)^2 - 1/81*b_0/(2^b_0)^2 - \
1/27*b_1^2/((2^b_2)^2*(2^b_1)^2) - \
1/27*b_0^2/((2^b_2)^2*(2^b_0)^2) - 20/243/(2^b_1)^2 + 1/9/2^b_0 \
+ 4/81*b_0/(2^b_0)^2 - 8/243/(2^b_2)^2 - 2/9/(2^b_2*2^b_1) - \
2/9/(2^b_2*2^b_0) + 8/243/(2^b_1)^2 - 1/9/2^b_0 + \
2/9/(2^b_2*2^b_1) + 2/9/(2^b_2*2^b_0) - \
2/27*b_1*b_2/((2^b_2)^2*(2^b_1)^2) - \
1/27*b_2^2/((2^b_2)^2*(2^b_1)^2) - \
2/27*b_0*b_2/((2^b_2)^2*(2^b_0)^2) - \
1/27*b_2^2/((2^b_2)^2*(2^b_0)^2) + 2/81/(2^b_1)^2 - \
1/27*b_0^2/((2^b_1)^2*(2^b_0)^2) - \
2/27*b_0*b_1/((2^b_1)^2*(2^b_0)^2) - \
1/27*b_1^2/((2^b_1)^2*(2^b_0)^2) - 2/81/(2^b_0)^2 + \
5/27*b_1/((2^b_2)^2*(2^b_1)^2) + 5/27*b_2/((2^b_2)^2*(2^b_1)^2) \
+ 5/27*b_0/((2^b_2)^2*(2^b_0)^2) + \
5/27*b_2/((2^b_2)^2*(2^b_0)^2) + 5/27*b_0/((2^b_1)^2*(2^b_0)^2) \
+ 5/27*b_1/((2^b_1)^2*(2^b_0)^2) - 4/81/((2^b_2)^2*(2^b_1)^2) + \
1/27*b_0^2/((2^b_2)^2*(2^b_1)^2*(2^b_0)^2) + \
2/27*b_0*b_1/((2^b_2)^2*(2^b_1)^2*(2^b_0)^2) + \
2/27*b_0*b_2/((2^b_2)^2*(2^b_1)^2*(2^b_0)^2) + \
1/27*b_1^2/((2^b_2)^2*(2^b_1)^2*(2^b_0)^2) + \
2/27*b_1*b_2/((2^b_2)^2*(2^b_1)^2*(2^b_0)^2) + \
1/27*b_2^2/((2^b_2)^2*(2^b_1)^2*(2^b_0)^2) - \
4/81/((2^b_2)^2*(2^b_0)^2) - 4/81/((2^b_1)^2*(2^b_0)^2) - \
11/27*b_0/((2^b_2)^2*(2^b_1)^2*(2^b_0)^2) - \
11/27*b_1/((2^b_2)^2*(2^b_1)^2*(2^b_0)^2) - \
11/27*b_2/((2^b_2)^2*(2^b_1)^2*(2^b_0)^2) + \
64/81/((2^b_2)^2*(2^b_1)^2*(2^b_0)^2) + 35/81 \
sage: f.nops()
38
sage: x,y,z = var('x y z');
sage: print (-x+z)*(3*x-3*z)
-3*(x - z)^2
sage: t = var('t')
sage: (x-t)^3
-(t - x)^3
sage: (-t+x)^3
-(t - x)^3
sage: (-x+t)^3
(t - x)^3
This example is from :trac:`10833`::
sage: R.<x,c> = PolynomialRing(QQ,2)
sage: phi(x) = x^2 + c
sage: def iterkate(n):
....: pol = x
....: for i in range(1,n):
....: pol = phi(pol)
....: return pol
....:
sage: g = expand(iterkate(7))
sage: g.nops()
480
"""
return (<Element>left)._cmp(right)
cdef int _cmp_c_impl(left, Element right) except -2:
"""
Compare ``left`` and ``right``.
INPUT:
- ``right`` -- A :class:`Expression` instance.
OUTPUT:
Boolean.
EXAMPLES::
sage: a = sqrt(3)
sage: b = x^2+1
sage: a.__cmp__(b) # indirect doctest
-1
"""
return print_order_compare(left._gobj, (<Expression>right)._gobj)
cpdef int _cmp_add(Expression left, Expression right) except -2:
"""
Compare ``left`` and ``right`` in the print order.
INPUT:
- ``right`` -- A :class:`Expression` instance.
OUTPUT:
Boolean.
EXAMPLES::
sage: a = sqrt(3)
sage: b = x^2+1
sage: a._cmp_add(b)
-1
sage: b._cmp_add(a)
1
sage: b._cmp_add(1)
Traceback (most recent call last):
...
TypeError: Argument 'right' has incorrect type (expected
sage.symbolic.expression.Expression, got sage.rings.integer.Integer)
"""
return print_order_compare(left._gobj, right._gobj)
cpdef int _cmp_mul(Expression left, Expression right) except -2:
"""
Compare ``left`` and ``right`` in the print order for products.
INPUT:
- ``right`` -- A :class:`Expression` instance.
OUTPUT:
Boolean.
EXAMPLES::
sage: a = sqrt(3)
sage: b = x^2+1
sage: a._cmp_mul(b)
-1
sage: b._cmp_mul(a)
1
sage: b._cmp_mul(1)
Traceback (most recent call last):
...
TypeError: Argument 'right' has incorrect type (expected
sage.symbolic.expression.Expression, got sage.rings.integer.Integer)
"""
return print_order_compare_mul(left._gobj, right._gobj)
def __pow__(self, exp, ignored):
"""
Return self raised to the power of exp.
INPUT:
- ``exp`` -- something that coerces to a symbolic expressions.
- ``ignored`` -- the second argument that should accept a modulus
is actually ignored.
OUTPUT:
A symbolic expression.
EXAMPLES::
sage: var('x,y')
(x, y)
sage: x.__pow__(y)
x^y
sage: x^(3/5)
x^(3/5)
sage: x^sin(x)^cos(y)
x^(sin(x)^cos(y))
TESTS::
sage: (Mod(2,7)*x^2 + Mod(2,7))^7
(2*x^2 + 2)^7
The leading coefficient in the result above is 1 since::
sage: t = Mod(2,7); gcd(t, t)^7
1
sage: gcd(t,t).parent()
Ring of integers modulo 7
::
sage: k = GF(7)
sage: f = expand((k(1)*x^5 + k(1)*x^2 + k(2))^7); f
x^35 + x^14 + 2
sage: x^oo
Traceback (most recent call last):
...
ValueError: power::eval(): pow(f(x), infinity) is not defined.
sage: SR(oo)^2
+Infinity
sage: SR(-oo)^2
+Infinity
sage: SR(-oo)^3
-Infinity
sage: SR(unsigned_infinity)^2
Infinity
Test powers of exp::
sage: exp(2)^5
e^10
sage: exp(x)^5
e^(5*x)
Test base a Python numeric type::
sage: int(2)^x
2^x
sage: float(2.3)^(x^3 - x^2 + 1/3)
2.3^(x^3 - x^2 + 1/3)
sage: complex(1,3)^(sqrt(2))
(1+3j)^sqrt(2)
Test complex numeric powers::
sage: I^0.5
0.707106781186548 + 0.707106781186547*I
sage: (I + 1) ^ (0.5 + I)
0.400667052375828 + 0.365310866736929*I
sage: I^I
I^I
sage: I^x
I^x
sage: I^(1/2)
sqrt(I)
sage: I^(2/3)
I^(2/3)
sage: 2^(1/2)
sqrt(2)
sage: (2*I)^(1/2)
sqrt(2*I)
Test if we can take powers of elements of Q(i) #8659::
sage: t = I.pyobject().parent()(8)
sage: t^(1/2)
2*sqrt(2)
sage: (t^2)^(1/4)
2*4^(1/4)
Test if we can compute inverses of Python longs (:trac:`13107`)::
sage: SR(2L)^(-1)
0.5
"""
cdef Expression base, nexp
try:
base = self
nexp = base.coerce_in(exp)
except TypeError:
nexp = exp
base = nexp.coerce_in(self)
cdef GEx x
if is_a_relational(base._gobj):
x = relational(g_pow(base._gobj.lhs(), nexp._gobj),
g_pow(base._gobj.rhs(), nexp._gobj),
relational_operator(base._gobj))
else:
x = g_pow(base._gobj, nexp._gobj)
return new_Expression_from_GEx(base._parent, x)
def derivative(self, *args):
"""
Returns the derivative of this expressions with respect to the
variables supplied in args.
Multiple variables and iteration counts may be supplied; see
documentation for the global
:meth:`~sage.calculus.functional.derivative` function for more
details.
.. seealso::
This is implemented in the `_derivative` method (see the
source code).
EXAMPLES::
sage: var("x y")
(x, y)
sage: t = (x^2+y)^2
sage: t.derivative(x)
4*(x^2 + y)*x
sage: t.derivative(x, 2)
12*x^2 + 4*y
sage: t.derivative(x, 2, y)
4
sage: t.derivative(y)
2*x^2 + 2*y
If the function depends on only one variable, you may omit the
variable. Giving just a number (for the order of the derivative)
also works::
sage: f(x) = x^3 + sin(x)
sage: f.derivative()
x |--> 3*x^2 + cos(x)
sage: f.derivative(2)
x |--> 6*x - sin(x)
::
sage: t = sin(x+y^2)*tan(x*y)
sage: t.derivative(x)
(tan(x*y)^2 + 1)*y*sin(y^2 + x) + cos(y^2 + x)*tan(x*y)
sage: t.derivative(y)
(tan(x*y)^2 + 1)*x*sin(y^2 + x) + 2*y*cos(y^2 + x)*tan(x*y)
::
sage: h = sin(x)/cos(x)
sage: derivative(h,x,x,x)
8*sin(x)^2/cos(x)^2 + 6*sin(x)^4/cos(x)^4 + 2
sage: derivative(h,x,3)
8*sin(x)^2/cos(x)^2 + 6*sin(x)^4/cos(x)^4 + 2
::
sage: var('x, y')
(x, y)
sage: u = (sin(x) + cos(y))*(cos(x) - sin(y))
sage: derivative(u,x,y)
-cos(x)*cos(y) + sin(x)*sin(y)
sage: f = ((x^2+1)/(x^2-1))^(1/4)
sage: g = derivative(f, x); g # this is a complex expression
-1/2*((x^2 + 1)*x/(x^2 - 1)^2 - x/(x^2 - 1))/((x^2 + 1)/(x^2 - 1))^(3/4)
sage: g.factor()
-x/((x + 1)^2*(x - 1)^2*((x^2 + 1)/(x^2 - 1))^(3/4))
::
sage: y = var('y')
sage: f = y^(sin(x))
sage: derivative(f, x)
y^sin(x)*cos(x)*log(y)
::
sage: g(x) = sqrt(5-2*x)
sage: g_3 = derivative(g, x, 3); g_3(2)
-3
::
sage: f = x*e^(-x)
sage: derivative(f, 100)
x*e^(-x) - 100*e^(-x)
::
sage: g = 1/(sqrt((x^2-1)*(x+5)^6))
sage: derivative(g, x)
-((x + 5)^6*x + 3*(x^2 - 1)*(x + 5)^5)/((x^2 - 1)*(x + 5)^6)^(3/2)
TESTS::
sage: t.derivative()
Traceback (most recent call last):
...
ValueError: No differentiation variable specified.
"""
return multi_derivative(self, args)
diff = differentiate = derivative
def _derivative(self, symb=None, deg=1):
"""
Return the deg-th (partial) derivative of self with respect to symb.
EXAMPLES::
sage: var("x y")
(x, y)
sage: b = (x+y)^5
sage: b._derivative(x, 2)
20*(x + y)^3
sage: foo = function('foo',nargs=2)
sage: foo(x^2,x^2)._derivative(x)
2*x*D[0](foo)(x^2, x^2) + 2*x*D[1](foo)(x^2, x^2)
sage: SR(1)._derivative()
0
If the expression is a callable symbolic expression, and no
variables are specified, then calculate the gradient::
sage: f(x,y)=x^2+y
sage: f.diff() # gradient
(x, y) |--> (2*x, 1)
TESTS:
Raise error if no variable is specified and there are multiple
variables::
sage: b._derivative()
Traceback (most recent call last):
...
ValueError: No differentiation variable specified.
Check if #6524 is fixed::
sage: f = function('f')
sage: f(x)*f(x).derivative(x)*f(x).derivative(x,2)
f(x)*D[0](f)(x)*D[0, 0](f)(x)
sage: g = f(x).diff(x)
sage: h = f(x).diff(x)*sin(x)
sage: h/g
sin(x)
"""
if symb is None:
vars = self.variables()
if len(vars) == 1:
symb = vars[0]
elif len(vars) == 0:
return self._parent(0)
elif sage.symbolic.callable.is_CallableSymbolicExpression(self):
return self.gradient()
else:
raise ValueError, "No differentiation variable specified."
if not isinstance(deg, (int, long, sage.rings.integer.Integer)) \
or deg < 1:
raise TypeError, "argument deg should be an integer >= 1."
cdef Expression symbol = self.coerce_in(symb)
if not is_a_symbol(symbol._gobj):
raise TypeError, "argument symb must be a symbol"
cdef GEx x
sig_on()
try:
x = self._gobj.diff(ex_to_symbol(symbol._gobj), deg)
finally:
sig_off()
return new_Expression_from_GEx(self._parent, x)
def gradient(self, variables=None):
r"""
Compute the gradient of a symbolic function.
This function returns a vector whose components are the derivatives
of the original function with respect to the arguments of the
original function. Alternatively, you can specify the variables as
a list.
EXAMPLES::
sage: x,y = var('x y')
sage: f = x^2+y^2
sage: f.gradient()
(2*x, 2*y)
sage: g(x,y) = x^2+y^2
sage: g.gradient()
(x, y) |--> (2*x, 2*y)
sage: n = var('n')
sage: f(x,y) = x^n+y^n
sage: f.gradient()
(x, y) |--> (n*x^(n - 1), n*y^(n - 1))
sage: f.gradient([y,x])
(x, y) |--> (n*y^(n - 1), n*x^(n - 1))
"""
from sage.modules.free_module_element import vector
if variables is None:
variables = self.arguments()
return vector([self.derivative(x) for x in variables])
def hessian(self):
r"""
Compute the hessian of a function. This returns a matrix components
are the 2nd partial derivatives of the original function.
EXAMPLES::
sage: x,y = var('x y')
sage: f = x^2+y^2
sage: f.hessian()
[2 0]
[0 2]
sage: g(x,y) = x^2+y^2
sage: g.hessian()
[(x, y) |--> 2 (x, y) |--> 0]
[(x, y) |--> 0 (x, y) |--> 2]
"""
from sage.matrix.constructor import matrix
return matrix([[g.derivative(x) for x in self.arguments()]
for g in self.gradient()])
def series(self, symbol, int order):
r"""
Return the power series expansion of self in terms of the
given variable to the given order.
INPUT:
- ``symbol`` - a symbolic variable or symbolic equality
such as ``x == 5``; if an equality is given, the
expansion is around the value on the right hand side
of the equality
- ``order`` - an integer
OUTPUT:
A power series.
To truncate the power series and obtain a normal expression, use the
:meth:`truncate` command.
EXAMPLES:
We expand a polynomial in `x` about 0, about `1`, and also truncate
it back to a polynomial::
sage: var('x,y')
(x, y)
sage: f = (x^3 - sin(y)*x^2 - 5*x + 3); f
x^3 - x^2*sin(y) - 5*x + 3
sage: g = f.series(x, 4); g
3 + (-5)*x + (-sin(y))*x^2 + 1*x^3
sage: g.truncate()
x^3 - x^2*sin(y) - 5*x + 3
sage: g = f.series(x==1, 4); g
(-sin(y) - 1) + (-2*sin(y) - 2)*(x - 1) + (-sin(y) + 3)*(x - 1)^2 + 1*(x - 1)^3
sage: h = g.truncate(); h
(x - 1)^3 - (x - 1)^2*(sin(y) - 3) - 2*(x - 1)*(sin(y) + 1) - sin(y) - 1
sage: h.expand()
x^3 - x^2*sin(y) - 5*x + 3
We computer another series expansion of an analytic function::
sage: f = sin(x)/x^2
sage: f.series(x,7)
1*x^(-1) + (-1/6)*x + 1/120*x^3 + (-1/5040)*x^5 + Order(x^7)
sage: f.series(x==1,3)
(sin(1)) + (cos(1) - 2*sin(1))*(x - 1) + (-2*cos(1) + 5/2*sin(1))*(x - 1)^2 + Order((x - 1)^3)
sage: f.series(x==1,3).truncate().expand()
-2*x^2*cos(1) + 5/2*x^2*sin(1) + 5*x*cos(1) - 7*x*sin(1) - 3*cos(1) + 11/2*sin(1)
Following the GiNaC tutorial, we use John Machin's amazing
formula `\pi = 16 \tan^{-1}(1/5) - 4 \tan^{-1}(1/239)` to compute
digits of `\pi`. We expand the arc tangent around 0 and insert
the fractions 1/5 and 1/239.
::
sage: x = var('x')
sage: f = atan(x).series(x, 10); f
1*x + (-1/3)*x^3 + 1/5*x^5 + (-1/7)*x^7 + 1/9*x^9 + Order(x^10)
sage: float(16*f.subs(x==1/5) - 4*f.subs(x==1/239))
3.1415926824043994
TESTS:
Check if #8943 is fixed::
sage: ((1+arctan(x))**(1/x)).series(x==0, 3)
(e) + (-1/2*e)*x + (1/8*e)*x^2 + Order(x^3)
"""
cdef Expression symbol0 = self.coerce_in(symbol)
cdef GEx x
sig_on()
try:
x = self._gobj.series(symbol0._gobj, order, 0)
finally:
sig_off()
return new_Expression_from_GEx(self._parent, x)
def taylor(self, *args):
r"""
Expands this symbolic expression in a truncated Taylor or
Laurent series in the variable `v` around the point `a`,
containing terms through `(x - a)^n`. Functions in more
variables is also supported.
INPUT:
- ``*args`` - the following notation is supported
- ``x, a, n`` - variable, point, degree
- ``(x, a), (y, b), n`` - variables with points, degree of polynomial
EXAMPLES::
sage: var('a, x, z')
(a, x, z)
sage: taylor(a*log(z), z, 2, 3)
1/24*a*(z - 2)^3 - 1/8*a*(z - 2)^2 + 1/2*a*(z - 2) + a*log(2)
::
sage: taylor(sqrt (sin(x) + a*x + 1), x, 0, 3)
1/48*(3*a^3 + 9*a^2 + 9*a - 1)*x^3 - 1/8*(a^2 + 2*a + 1)*x^2 + 1/2*(a + 1)*x + 1
::
sage: taylor (sqrt (x + 1), x, 0, 5)
7/256*x^5 - 5/128*x^4 + 1/16*x^3 - 1/8*x^2 + 1/2*x + 1
::
sage: taylor (1/log (x + 1), x, 0, 3)
-19/720*x^3 + 1/24*x^2 - 1/12*x + 1/x + 1/2
::
sage: taylor (cos(x) - sec(x), x, 0, 5)
-1/6*x^4 - x^2
::
sage: taylor ((cos(x) - sec(x))^3, x, 0, 9)
-1/2*x^8 - x^6
::
sage: taylor (1/(cos(x) - sec(x))^3, x, 0, 5)
-15377/7983360*x^4 - 6767/604800*x^2 + 11/120/x^2 + 1/2/x^4 - 1/x^6 - 347/15120
TESTS:
Check that ticket #7472 is fixed (Taylor polynomial in more variables)::
sage: x,y=var('x y'); taylor(x*y^3,(x,1),(y,1),4)
(x - 1)*(y - 1)^3 + 3*(x - 1)*(y - 1)^2 + (y - 1)^3 + 3*(x - 1)*(y - 1) + 3*(y - 1)^2 + x + 3*y - 3
sage: expand(_)
x*y^3
"""
from sage.all import SR, Integer
A=args
try:
if isinstance(A[0],tuple):
B=[]
B.append([SR(A[i][0]) for i in range(len(A)-1)])
B.append([A[i][1] for i in range(len(A)-1)])
else:
B=[A[0],SR(A[1])]
B.append(Integer(A[len(A)-1]))
except Exception:
raise NotImplementedError, "Wrong arguments passed to taylor. See taylor? for more details."
l = self._maxima_().taylor(B)
return self.parent()(l)
def truncate(self):
"""
Given a power series or expression, return the corresponding
expression without the big oh.
INPUT:
- ``self`` -- a series as output by the :meth:`series` command.
OUTPUT:
A symbolic expression.
EXAMPLES::
sage: f = sin(x)/x^2
sage: f.truncate()
sin(x)/x^2
sage: f.series(x,7)
1*x^(-1) + (-1/6)*x + 1/120*x^3 + (-1/5040)*x^5 + Order(x^7)
sage: f.series(x,7).truncate()
-1/5040*x^5 + 1/120*x^3 - 1/6*x + 1/x
sage: f.series(x==1,3).truncate().expand()
-2*x^2*cos(1) + 5/2*x^2*sin(1) + 5*x*cos(1) - 7*x*sin(1) - 3*cos(1) + 11/2*sin(1)
"""
if not is_a_series(self._gobj):
return self
return new_Expression_from_GEx(self._parent, series_to_poly(self._gobj))
def expand(Expression self, side=None):
"""
Expand this symbolic expression. Products of sums and exponentiated
sums are multiplied out, numerators of rational expressions which
are sums are split into their respective terms, and multiplications
are distributed over addition at all levels.
EXAMPLES:
We expand the expression `(x-y)^5` using both
method and functional notation.
::
sage: x,y = var('x,y')
sage: a = (x-y)^5
sage: a.expand()
x^5 - 5*x^4*y + 10*x^3*y^2 - 10*x^2*y^3 + 5*x*y^4 - y^5
sage: expand(a)
x^5 - 5*x^4*y + 10*x^3*y^2 - 10*x^2*y^3 + 5*x*y^4 - y^5
We expand some other expressions::
sage: expand((x-1)^3/(y-1))
x^3/(y - 1) - 3*x^2/(y - 1) + 3*x/(y - 1) - 1/(y - 1)
sage: expand((x+sin((x+y)^2))^2)
x^2 + 2*x*sin((x + y)^2) + sin((x + y)^2)^2
We can expand individual sides of a relation::
sage: a = (16*x-13)^2 == (3*x+5)^2/2
sage: a.expand()
256*x^2 - 416*x + 169 == 9/2*x^2 + 15*x + 25/2
sage: a.expand('left')
256*x^2 - 416*x + 169 == 1/2*(3*x + 5)^2
sage: a.expand('right')
(16*x - 13)^2 == 9/2*x^2 + 15*x + 25/2
TESTS::
sage: var('x,y')
(x, y)
sage: ((x + (2/3)*y)^3).expand()
x^3 + 2*x^2*y + 4/3*x*y^2 + 8/27*y^3
sage: expand( (x*sin(x) - cos(y)/x)^2 )
x^2*sin(x)^2 - 2*cos(y)*sin(x) + cos(y)^2/x^2
sage: f = (x-y)*(x+y); f
(x + y)*(x - y)
sage: f.expand()
x^2 - y^2
"""
if side is not None:
if not is_a_relational(self._gobj):
raise ValueError, "expansion on sides only makes sense for relations"
if side == 'left':
return self.operator()(self.lhs().expand(), self.rhs())
elif side == 'right':
return self.operator()(self.lhs(), self.rhs().expand())
else:
raise ValueError, "side must be 'left', 'right', or None"
cdef GEx x
sig_on()
try:
x = self._gobj.expand(0)
finally:
sig_off()
return new_Expression_from_GEx(self._parent, x)
expand_rational = rational_expand = expand
def expand_trig(self, full=False, half_angles=False, plus=True, times=True):
"""
Expands trigonometric and hyperbolic functions of sums of angles
and of multiple angles occurring in self. For best results, self
should already be expanded.
INPUT:
- ``full`` - (default: False) To enhance user control
of simplification, this function expands only one level at a time
by default, expanding sums of angles or multiple angles. To obtain
full expansion into sines and cosines immediately, set the optional
parameter full to True.
- ``half_angles`` - (default: False) If True, causes
half-angles to be simplified away.
- ``plus`` - (default: True) Controls the sum rule;
expansion of sums (e.g. 'sin(x + y)') will take place only if plus
is True.
- ``times`` - (default: True) Controls the product
rule, expansion of products (e.g. sin(2\*x)) will take place only
if times is True.
OUTPUT:
A symbolic expression.
EXAMPLES::
sage: sin(5*x).expand_trig()
5*cos(x)^4*sin(x) - 10*cos(x)^2*sin(x)^3 + sin(x)^5
sage: cos(2*x + var('y')).expand_trig()
cos(2*x)*cos(y) - sin(2*x)*sin(y)
We illustrate various options to this function::
sage: f = sin(sin(3*cos(2*x))*x)
sage: f.expand_trig()
sin((3*cos(cos(2*x))^2*sin(cos(2*x)) - sin(cos(2*x))^3)*x)
sage: f.expand_trig(full=True)
sin((3*(cos(cos(x)^2)*cos(sin(x)^2) + sin(cos(x)^2)*sin(sin(x)^2))^2*(cos(sin(x)^2)*sin(cos(x)^2) - cos(cos(x)^2)*sin(sin(x)^2)) - (cos(sin(x)^2)*sin(cos(x)^2) - cos(cos(x)^2)*sin(sin(x)^2))^3)*x)
sage: sin(2*x).expand_trig(times=False)
sin(2*x)
sage: sin(2*x).expand_trig(times=True)
2*cos(x)*sin(x)
sage: sin(2 + x).expand_trig(plus=False)
sin(x + 2)
sage: sin(2 + x).expand_trig(plus=True)
cos(x)*sin(2) + cos(2)*sin(x)
sage: sin(x/2).expand_trig(half_angles=False)
sin(1/2*x)
sage: sin(x/2).expand_trig(half_angles=True)
(-1)^floor(1/2*x/pi)*sqrt(-1/2*cos(x) + 1/2)
ALIASES:
:meth:`trig_expand` and :meth:`expand_trig` are the same
"""
from sage.calculus.calculus import maxima_options
M = self._maxima_()
P = M.parent()
opt = maxima_options(trigexpand=full, halfangles=half_angles,
trigexpandplus=plus, trigexpandtimes=times)
cmd = 'trigexpand(%s), %s'%(M.name(), opt)
ans = P(cmd)
return self.parent()(ans)
trig_expand = expand_trig
def reduce_trig(self, var=None):
r"""
Combines products and powers of trigonometric and hyperbolic
sin's and cos's of x into those of multiples of x. It also
tries to eliminate these functions when they occur in
denominators.
INPUT:
- ``self`` - a symbolic expression
- ``var`` - (default: None) the variable which is used for
these transformations. If not specified, all variables are
used.
OUTPUT:
A symbolic expression.
EXAMPLES::
sage: y=var('y')
sage: f=sin(x)*cos(x)^3+sin(y)^2
sage: f.reduce_trig()
-1/2*cos(2*y) + 1/8*sin(4*x) + 1/4*sin(2*x) + 1/2
To reduce only the expressions involving x we use optional parameter::
sage: f.reduce_trig(x)
sin(y)^2 + 1/8*sin(4*x) + 1/4*sin(2*x)
ALIASES: :meth:`trig_reduce` and :meth:`reduce_trig` are the same
"""
M = self._maxima_()
P = M.parent()
if var is None:
cmd = 'trigreduce(%s)'%(M.name())
else:
cmd = 'trigreduce(%s,%s)'%(M.name(),str(var))
ans = P(cmd)
return self.parent()(ans)
trig_reduce = reduce_trig
def match(self, pattern):
"""
Check if self matches the given pattern.
INPUT:
- ``pattern`` -- a symbolic expression, possibly containing wildcards
to match for
OUTPUT:
One of
``None`` if there is no match, or a dictionary mapping the
wildcards to the matching values if a match was found. Note
that the dictionary is empty if there were no wildcards in the
given pattern.
See also http://www.ginac.de/tutorial/Pattern-matching-and-advanced-substitutions.html
EXAMPLES::
sage: var('x,y,z,a,b,c,d,f,g')
(x, y, z, a, b, c, d, f, g)
sage: w0 = SR.wild(0); w1 = SR.wild(1); w2 = SR.wild(2)
sage: ((x+y)^a).match((x+y)^a) # no wildcards, so empty dict
{}
sage: print ((x+y)^a).match((x+y)^b)
None
sage: t = ((x+y)^a).match(w0^w1)
sage: t[w0], t[w1]
(x + y, a)
sage: print ((x+y)^a).match(w0^w0)
None
sage: ((x+y)^(x+y)).match(w0^w0)
{$0: x + y}
sage: t = ((a+b)*(a+c)).match((a+w0)*(a+w1))
sage: t[w0], t[w1]
(c, b)
sage: ((a+b)*(a+c)).match((w0+b)*(w0+c))
{$0: a}
sage: t = ((a+b)*(a+c)).match((w0+w1)*(w0+w2))
sage: t[w0], t[w1], t[w2]
(a, c, b)
sage: print ((a+b)*(a+c)).match((w0+w1)*(w1+w2))
None
sage: t = (a*(x+y)+a*z+b).match(a*w0+w1)
sage: t[w0], t[w1]
(x + y, a*z + b)
sage: print (a+b+c+d+f+g).match(c)
None
sage: (a+b+c+d+f+g).has(c)
True
sage: (a+b+c+d+f+g).match(c+w0)
{$0: a + b + d + f + g}
sage: (a+b+c+d+f+g).match(c+g+w0)
{$0: a + b + d + f}
sage: (a+b).match(a+b+w0)
{$0: 0}
sage: print (a*b^2).match(a^w0*b^w1)
None
sage: (a*b^2).match(a*b^w1)
{$1: 2}
sage: (x*x.arctan2(x^2)).match(w0*w0.arctan2(w0^2))
{$0: x}
Beware that behind-the-scenes simplification can lead to
surprising results in matching::
sage: print (x+x).match(w0+w1)
None
sage: t = x+x; t
2*x
sage: t.operator()
<built-in function mul>
Since asking to match w0+w1 looks for an addition operator,
there is no match.
"""
cdef Expression p = self.coerce_in(pattern)
cdef GExList mlst
cdef bint res = self._gobj.match(p._gobj, mlst)
if not res:
return None
cdef dict rdict = {}
cdef GExListIter itr = mlst.begin()
cdef GExListIter lstend = mlst.end()
while itr.is_not_equal(lstend):
key = new_Expression_from_GEx(self._parent, itr.obj().lhs())
val = new_Expression_from_GEx(self._parent, itr.obj().rhs())
rdict[key] = val
itr.inc()
return rdict
def find(self, pattern):
"""
Find all occurrences of the given pattern in this expression.
Note that once a subexpression matches the pattern, the search doesn't
extend to subexpressions of it.
EXAMPLES::
sage: var('x,y,z,a,b')
(x, y, z, a, b)
sage: w0 = SR.wild(0); w1 = SR.wild(1)
sage: (sin(x)*sin(y)).find(sin(w0))
[sin(y), sin(x)]
sage: ((sin(x)+sin(y))*(a+b)).expand().find(sin(w0))
[sin(y), sin(x)]
sage: (1+x+x^2+x^3).find(x)
[x]
sage: (1+x+x^2+x^3).find(x^w0)
[x^2, x^3]
sage: (1+x+x^2+x^3).find(y)
[]
# subexpressions of a match are not listed
sage: ((x^y)^z).find(w0^w1)
[(x^y)^z]
"""
cdef Expression p = self.coerce_in(pattern)
cdef GExList found
self._gobj.find(p._gobj, found)
res = []
cdef GExListIter itr = found.begin()
while itr.is_not_equal(found.end()):
res.append(new_Expression_from_GEx(self._parent, itr.obj()))
itr.inc()
res.sort(cmp)
return res
def has(self, pattern):
"""
EXAMPLES::
sage: var('x,y,a'); w0 = SR.wild(); w1 = SR.wild()
(x, y, a)
sage: (x*sin(x + y + 2*a)).has(y)
True
Here "x+y" is not a subexpression of "x+y+2*a" (which has the
subexpressions "x", "y" and "2*a")::
sage: (x*sin(x + y + 2*a)).has(x+y)
False
sage: (x*sin(x + y + 2*a)).has(x + y + w0)
True
The following fails because "2*(x+y)" automatically gets converted to
"2*x+2*y" of which "x+y" is not a subexpression::
sage: (x*sin(2*(x+y) + 2*a)).has(x+y)
False
Although x^1==x and x^0==1, neither "x" nor "1" are actually of the
form "x^something"::
sage: (x+1).has(x^w0)
False
Here is another possible pitfall, where the first expression
matches because the term "-x" has the form "(-1)*x" in GiNaC. To check
whether a polynomial contains a linear term you should use the
coeff() function instead.
::
sage: (4*x^2 - x + 3).has(w0*x)
True
sage: (4*x^2 + x + 3).has(w0*x)
False
sage: (4*x^2 + x + 3).has(x)
True
sage: (4*x^2 - x + 3).coeff(x,1)
-1
sage: (4*x^2 + x + 3).coeff(x,1)
1
"""
cdef Expression p = self.coerce_in(pattern)
return self._gobj.has(p._gobj)
def substitute(self, in_dict=None, **kwds):
"""
EXAMPLES::
sage: var('x,y,z,a,b,c,d,f,g')
(x, y, z, a, b, c, d, f, g)
sage: w0 = SR.wild(0); w1 = SR.wild(1)
sage: t = a^2 + b^2 + (x+y)^3
# substitute with keyword arguments (works only with symbols)
sage: t.subs(a=c)
(x + y)^3 + b^2 + c^2
# substitute with a dictionary argument
sage: t.subs({a^2: c})
(x + y)^3 + b^2 + c
sage: t.subs({w0^2: w0^3})
a^3 + b^3 + (x + y)^3
# substitute with a relational expression
sage: t.subs(w0^2 == w0^3)
a^3 + b^3 + (x + y)^3
sage: t.subs(w0==w0^2)
(x^2 + y^2)^18 + a^16 + b^16
# more than one keyword argument is accepted
sage: t.subs(a=b, b=c)
(x + y)^3 + b^2 + c^2
# using keyword arguments with a dictionary is allowed
sage: t.subs({a:b}, b=c)
(x + y)^3 + b^2 + c^2
# in this case keyword arguments override the dictionary
sage: t.subs({a:b}, a=c)
(x + y)^3 + b^2 + c^2
sage: t.subs({a:b, b:c})
(x + y)^3 + b^2 + c^2
TESTS::
sage: # no arguments return the same expression
sage: t.subs()
(x + y)^3 + a^2 + b^2
# similarly for an empty dictionary argument
sage: t.subs({})
(x + y)^3 + a^2 + b^2
# non keyword or dictionary argument returns error
sage: t.subs(5)
Traceback (most recent call last):
...
TypeError: subs takes either a set of keyword arguments, a dictionary, or a symbolic relational expression
# substitutions with infinity
sage: (x/y).subs(y=oo)
0
sage: (x/y).subs(x=oo)
Traceback (most recent call last):
...
RuntimeError: indeterminate expression: infinity * f(x) encountered.
sage: (x*y).subs(x=oo)
Traceback (most recent call last):
...
RuntimeError: indeterminate expression: infinity * f(x) encountered.
sage: (x^y).subs(x=oo)
Traceback (most recent call last):
...
ValueError: power::eval(): pow(Infinity, f(x)) is not defined.
sage: (x^y).subs(y=oo)
Traceback (most recent call last):
...
ValueError: power::eval(): pow(f(x), infinity) is not defined.
sage: (x+y).subs(x=oo)
+Infinity
sage: (x-y).subs(y=oo)
-Infinity
sage: gamma(x).subs(x=-1)
Infinity
sage: 1/gamma(x).subs(x=-1)
0
# verify that this operation does not modify the passed dictionary (#6622)
sage: var('v t')
(v, t)
sage: f = v*t
sage: D = {v: 2}
sage: f(D, t=3)
6
sage: D
{v: 2}
Check if #9891 is fixed::
sage: exp(x).subs(x=log(x))
x
Check if :trac:`13587` is fixed::
sage: t = tan(x)^2 - tan(x)
sage: t.subs(x=pi/2)
Infinity
sage: u = gamma(x) - gamma(x-1)
sage: u.subs(x=-1)
Infinity
"""
cdef dict sdict = {}
if in_dict is not None:
if isinstance(in_dict, Expression):
return self._subs_expr(in_dict)
if not isinstance(in_dict, dict):
raise TypeError, "subs takes either a set of keyword arguments, a dictionary, or a symbolic relational expression"
sdict.update(in_dict)
if kwds:
for k, v in kwds.iteritems():
k = self._parent.var(k)
sdict[k] = v
cdef GExMap smap
for k, v in sdict.iteritems():
smap.insert(make_pair((<Expression>self.coerce_in(k))._gobj,
(<Expression>self.coerce_in(v))._gobj))
return new_Expression_from_GEx(self._parent, self._gobj.subs_map(smap))
subs = substitute
cpdef Expression _subs_expr(self, expr):
"""
EXAMPLES::
sage: var('x,y,z,a,b,c,d,f')
(x, y, z, a, b, c, d, f)
sage: w0 = SR.wild(0); w1 = SR.wild(1)
sage: (a^2 + b^2 + (x+y)^2)._subs_expr(w0^2 == w0^3)
a^3 + b^3 + (x + y)^3
sage: (a^4 + b^4 + (x+y)^4)._subs_expr(w0^2 == w0^3)
a^4 + b^4 + (x + y)^4
sage: (a^2 + b^4 + (x+y)^4)._subs_expr(w0^2 == w0^3)
b^4 + (x + y)^4 + a^3
sage: ((a+b+c)^2)._subs_expr(a+b == x)
(a + b + c)^2
sage: ((a+b+c)^2)._subs_expr(a+b+w0 == x+w0)
(c + x)^2
sage: (a+2*b)._subs_expr(a+b == x)
a + 2*b
sage: (a+2*b)._subs_expr(a+b+w0 == x+w0)
a + 2*b
sage: (a+2*b)._subs_expr(a+w0*b == x)
x
sage: (a+2*b)._subs_expr(a+b+w0*b == x+w0*b)
a + 2*b
sage: (4*x^3-2*x^2+5*x-1)._subs_expr(x==a)
4*a^3 - 2*a^2 + 5*a - 1
sage: (4*x^3-2*x^2+5*x-1)._subs_expr(x^w0==a^w0)
4*a^3 - 2*a^2 + 5*x - 1
sage: (4*x^3-2*x^2+5*x-1)._subs_expr(x^w0==a^(2*w0))._subs_expr(x==a)
4*a^6 - 2*a^4 + 5*a - 1
sage: sin(1+sin(x))._subs_expr(sin(w0)==cos(w0))
cos(cos(x) + 1)
sage: (sin(x)^2 + cos(x)^2)._subs_expr(sin(w0)^2+cos(w0)^2==1)
1
sage: (1 + sin(x)^2 + cos(x)^2)._subs_expr(sin(w0)^2+cos(w0)^2==1)
cos(x)^2 + sin(x)^2 + 1
sage: (17*x + sin(x)^2 + cos(x)^2)._subs_expr(w1 + sin(w0)^2+cos(w0)^2 == w1 + 1)
17*x + 1
sage: ((x-1)*(sin(x)^2 + cos(x)^2)^2)._subs_expr(sin(w0)^2+cos(w0)^2 == 1)
x - 1
"""
cdef Expression p = self.coerce_in(expr)
return new_Expression_from_GEx(self._parent, self._gobj.subs(p._gobj))
def substitute_expression(self, *equations):
"""
Given a dictionary of key:value pairs, substitute all occurrences
of key for value in self. The substitutions can also be given
as a number of symbolic equalities key == value; see the
examples.
.. warning::
This is a formal pattern substitution, which may or may not
have any mathematical meaning. The exact rules used at
present in Sage are determined by Maxima's subst
command. Sometimes patterns are not replaced even though
one would think they should be - see examples below.
EXAMPLES::
sage: f = x^2 + 1
sage: f.subs_expr(x^2 == x)
x + 1
::
sage: var('x,y,z'); f = x^3 + y^2 + z
(x, y, z)
sage: f.subs_expr(x^3 == y^2, z == 1)
2*y^2 + 1
Or the same thing giving the substitutions as a dictionary::
sage: f.subs_expr({x^3:y^2, z:1})
2*y^2 + 1
sage: f = x^2 + x^4
sage: f.subs_expr(x^2 == x)
x^4 + x
sage: f = cos(x^2) + sin(x^2)
sage: f.subs_expr(x^2 == x)
cos(x) + sin(x)
::
sage: f(x,y,t) = cos(x) + sin(y) + x^2 + y^2 + t
sage: f.subs_expr(y^2 == t)
(x, y, t) |--> x^2 + 2*t + cos(x) + sin(y)
The following seems really weird, but it *is* what Maple does::
sage: f.subs_expr(x^2 + y^2 == t)
(x, y, t) |--> x^2 + y^2 + t + cos(x) + sin(y)
sage: maple.eval('subs(x^2 + y^2 = t, cos(x) + sin(y) + x^2 + y^2 + t)') # optional - maple
'cos(x)+sin(y)+x^2+y^2+t'
sage: maxima.quit()
sage: maxima.eval('cos(x) + sin(y) + x^2 + y^2 + t, x^2 + y^2 = t')
'sin(y)+y^2+cos(x)+x^2+t'
Actually Mathematica does something that makes more sense::
sage: mathematica.eval('Cos[x] + Sin[y] + x^2 + y^2 + t /. x^2 + y^2 -> t') # optional - mathematica
2 t + Cos[x] + Sin[y]
"""
if isinstance(equations[0], dict):
eq_dict = equations[0]
equations = [ x == eq_dict[x] for x in eq_dict.keys() ]
if not all([is_SymbolicEquation(eq) for eq in equations]):
raise TypeError, "each expression must be an equation"
d = dict([(eq.lhs(), eq.rhs()) for eq in equations])
return self.subs(d)
subs_expr = substitute_expression
def substitute_function(self, original, new):
"""
Returns this symbolic expressions all occurrences of the
function *original* replaced with the function *new*.
EXAMPLES::
sage: x,y = var('x,y')
sage: foo = function('foo'); bar = function('bar')
sage: f = foo(x) + 1/foo(pi*y)
sage: f.substitute_function(foo, bar)
1/bar(pi*y) + bar(x)
"""
from sage.symbolic.expression_conversions import SubstituteFunction
return SubstituteFunction(self, original, new)()
def __call__(self, *args, **kwds):
"""
Calls the :meth:`subs` on this expression.
EXAMPLES::
sage: var('x,y,z')
(x, y, z)
sage: (x+y)(x=z^2, y=x^y)
z^2 + x^y
"""
return self._parent._call_element_(self, *args, **kwds)
def variables(self):
"""
Return sorted tuple of variables that occur in this expression.
EXAMPLES::
sage: (x,y,z) = var('x,y,z')
sage: (x+y).variables()
(x, y)
sage: (2*x).variables()
(x,)
sage: (x^y).variables()
(x, y)
sage: sin(x+y^z).variables()
(x, y, z)
"""
from sage.symbolic.ring import SR
cdef GExSet sym_set
g_list_symbols(self._gobj, sym_set)
res = []
cdef GExSetIter itr = sym_set.begin()
while itr.is_not_equal(sym_set.end()):
res.append(new_Expression_from_GEx(SR, itr.obj()))
itr.inc()
res.sort(cmp=lambda x,y: -cmp(x,y))
return tuple(res)
def arguments(self):
"""
EXAMPLES::
sage: x,y = var('x,y')
sage: f = x + y
sage: f.arguments()
(x, y)
sage: g = f.function(x)
sage: g.arguments()
(x,)
"""
try:
return self._parent.arguments()
except AttributeError:
return self.variables()
args = arguments
def number_of_arguments(self):
"""
EXAMPLES::
sage: x,y = var('x,y')
sage: f = x + y
sage: f.number_of_arguments()
2
sage: g = f.function(x)
sage: g.number_of_arguments()
1
::
sage: x,y,z = var('x,y,z')
sage: (x+y).number_of_arguments()
2
sage: (x+1).number_of_arguments()
1
sage: (sin(x)+1).number_of_arguments()
1
sage: (sin(z)+x+y).number_of_arguments()
3
sage: (sin(x+y)).number_of_arguments()
2
::
sage: ( 2^(8/9) - 2^(1/9) )(x-1)
Traceback (most recent call last):
...
ValueError: the number of arguments must be less than or equal to 0
"""
return len(self.arguments())
def number_of_operands(self):
"""
Returns the number of arguments of this expression.
EXAMPLES::
sage: var('a,b,c,x,y')
(a, b, c, x, y)
sage: a.number_of_operands()
0
sage: (a^2 + b^2 + (x+y)^2).number_of_operands()
3
sage: (a^2).number_of_operands()
2
sage: (a*b^2*c).number_of_operands()
3
"""
return self._gobj.nops()
nops = number_of_operands
def __len__(self):
"""
Returns the number of arguments of this expression.
EXAMPLES::
sage: var('a,b,c,x,y')
(a, b, c, x, y)
sage: len(a)
0
sage: len((a^2 + b^2 + (x+y)^2))
3
sage: len((a^2))
2
sage: len(a*b^2*c)
3
"""
return self.number_of_operands()
def operands(self):
"""
Returns a list containing the operands of this expression.
EXAMPLES::
sage: var('a,b,c,x,y')
(a, b, c, x, y)
sage: (a^2 + b^2 + (x+y)^2).operands()
[a^2, b^2, (x + y)^2]
sage: (a^2).operands()
[a, 2]
sage: (a*b^2*c).operands()
[a, b^2, c]
"""
from sage.symbolic.ring import SR
return [new_Expression_from_GEx(SR, self._gobj.op(i)) \
for i from 0 <= i < self._gobj.nops()]
def operator(self):
"""
Returns the topmost operator in this expression.
EXAMPLES::
sage: x,y,z = var('x,y,z')
sage: (x+y).operator()
<built-in function add>
sage: (x^y).operator()
<built-in function pow>
sage: (x^y * z).operator()
<built-in function mul>
sage: (x < y).operator()
<built-in function lt>
sage: abs(x).operator()
abs
sage: r = gamma(x).operator(); type(r)
<class 'sage.functions.other.Function_gamma'>
sage: psi = function('psi', nargs=1)
sage: psi(x).operator()
psi
sage: r = psi(x).operator()
sage: r == psi
True
sage: f = function('f', nargs=1, conjugate_func=lambda self, x: 2*x)
sage: nf = f(x).operator()
sage: nf(x).conjugate()
2*x
sage: f = function('f')
sage: a = f(x).diff(x); a
D[0](f)(x)
sage: a.operator()
D[0](f)
TESTS::
sage: (x <= y).operator()
<built-in function le>
sage: (x == y).operator()
<built-in function eq>
sage: (x != y).operator()
<built-in function ne>
sage: (x > y).operator()
<built-in function gt>
sage: (x >= y).operator()
<built-in function ge>
"""
cdef operators o
cdef unsigned serial
import operator
if is_a_add(self._gobj):
return operator.add
elif is_a_mul(self._gobj) or is_a_ncmul(self._gobj):
return operator.mul
elif is_a_power(self._gobj):
return operator.pow
elif is_a_relational(self._gobj):
o = relational_operator(self._gobj)
if o == equal:
return operator.eq
elif o == not_equal:
return operator.ne
elif o == less:
return operator.lt
elif o == less_or_equal:
return operator.le
elif o == greater:
return operator.gt
elif o == greater_or_equal:
return operator.ge
else:
raise RuntimeError, "operator type not known, please report this as a bug"
elif is_a_function(self._gobj):
serial = ex_to_function(self._gobj).get_serial()
res = get_sfunction_from_serial(serial)
if res is None:
raise RuntimeError, "cannot find SFunction in table"
if is_a_fderivative(self._gobj):
from sage.symbolic.pynac import paramset_from_Expression
from sage.symbolic.operators import FDerivativeOperator
parameter_set = paramset_from_Expression(self)
res = FDerivativeOperator(res, parameter_set)
return res
return None
def __index__(self):
"""
EXAMPLES::
sage: a = range(10)
sage: a[:SR(5)]
[0, 1, 2, 3, 4]
"""
return int(self._integer_())
def iterator(self):
"""
Return an iterator over the operands of this expression.
EXAMPLES::
sage: x,y,z = var('x,y,z')
sage: list((x+y+z).iterator())
[x, y, z]
sage: list((x*y*z).iterator())
[x, y, z]
sage: list((x^y*z*(x+y)).iterator())
[x + y, x^y, z]
Note that symbols, constants and numeric objects don't have operands,
so the iterator function raises an error in these cases::
sage: x.iterator()
Traceback (most recent call last):
...
ValueError: expressions containing only a numeric coefficient, constant or symbol have no operands
sage: pi.iterator()
Traceback (most recent call last):
...
ValueError: expressions containing only a numeric coefficient, constant or symbol have no operands
sage: SR(5).iterator()
Traceback (most recent call last):
...
ValueError: expressions containing only a numeric coefficient, constant or symbol have no operands
"""
if is_a_symbol(self._gobj) or is_a_constant(self._gobj) or \
is_a_numeric(self._gobj):
raise ValueError, "expressions containing only a numeric coefficient, constant or symbol have no operands"
return new_ExpIter_from_Expression(self)
property op:
def __get__(self):
"""
Provide access to the operands of an expression through a property.
EXAMPLES::
sage: t = 1+x+x^2
sage: t.op
Operands of x^2 + x + 1
sage: x.op
Traceback (most recent call last):
...
TypeError: expressions containing only a numeric coefficient, constant or symbol have no operands
sage: t.op[0]
x^2
Indexing directly with ``t[1]`` causes problems with numpy types.
sage: t[1]
Traceback (most recent call last):
...
TypeError: 'sage.symbolic.expression.Expression' object does not support indexing
"""
if is_a_symbol(self._gobj) or is_a_constant(self._gobj) or \
is_a_numeric(self._gobj):
raise TypeError, "expressions containing only a numeric coefficient, constant or symbol have no operands"
cdef OperandsWrapper res = OperandsWrapper.__new__(OperandsWrapper)
res._expr = self
return res
def _numerical_approx(self, prec=None, digits=None):
"""
Return a numerical approximation this symbolic expression as
either a real or complex number with at least the requested
number of bits or digits of precision.
EXAMPLES::
sage: sin(x).subs(x=5).n()
-0.958924274663138
sage: sin(x).subs(x=5).n(100)
-0.95892427466313846889315440616
sage: sin(x).subs(x=5).n(digits=50)
-0.95892427466313846889315440615599397335246154396460
sage: zeta(x).subs(x=2).numerical_approx(digits=50)
1.6449340668482264364724151666460251892189499012068
sage: cos(3).numerical_approx(200)
-0.98999249660044545727157279473126130239367909661558832881409
sage: numerical_approx(cos(3),200)
-0.98999249660044545727157279473126130239367909661558832881409
sage: numerical_approx(cos(3), digits=10)
-0.9899924966
sage: (i + 1).numerical_approx(32)
1.00000000 + 1.00000000*I
sage: (pi + e + sqrt(2)).numerical_approx(100)
7.2740880444219335226246195788
TESTS:
We test the evaluation of different infinities available in Pynac::
sage: t = x - oo; t
-Infinity
sage: t.n()
-infinity
sage: t = x + oo; t
+Infinity
sage: t.n()
+infinity
sage: t = x - unsigned_infinity; t
Infinity
sage: t.n()
+infinity
Some expressions can't be evaluated numerically::
sage: n(sin(x))
Traceback (most recent call last):
...
TypeError: cannot evaluate symbolic expression numerically
sage: a = var('a')
sage: (x^2 + 2*x + 2).subs(x=a).n()
Traceback (most recent call last):
...
TypeError: cannot evaluate symbolic expression numerically
Make sure we've rounded up log(10,2) enough to guarantee
sufficient precision (trac #10164)::
sage: ks = 4*10**5, 10**6
sage: all(len(str(e.n(digits=k)))-1 >= k for k in ks)
True
"""
if prec is None:
if digits is None:
prec = 53
else:
prec = int((digits+1) * LOG_TEN_TWO_PLUS_EPSILON) + 1
from sage.rings.real_mpfr import RealField
R = RealField(prec)
cdef Expression x
try:
x = self._convert(R)
except TypeError:
pass
R = R.complex_field()
x = self._convert(R)
if is_a_numeric(x._gobj):
res = py_object_from_numeric(x._gobj)
elif is_a_constant(x._gobj):
res = x.pyobject()
else:
raise TypeError("cannot evaluate symbolic expression numerically")
if isinstance(res, AnInfinity):
return res.n(prec=prec,digits=digits)
return res
numerical_approx =_numerical_approx
n=_numerical_approx
N=_numerical_approx
def round(self):
"""
Round this expression to the nearest integer.
EXAMPLES::
sage: u = sqrt(43203735824841025516773866131535024)
sage: u.round()
207855083711803945
sage: t = sqrt(Integer('1'*1000)).round(); print str(t)[-10:]
3333333333
sage: (-sqrt(110)).round()
-10
sage: (-sqrt(115)).round()
-11
sage: (sqrt(-3)).round()
Traceback (most recent call last):
...
ValueError: could not convert sqrt(-3) to a real number
"""
try:
return self.pyobject().round()
except (TypeError, AttributeError):
pass
from sage.functions.all import floor, ceil
try:
rif_self = sage.rings.all.RIF(self)
except TypeError:
raise ValueError, "could not convert %s to a real number"%(self)
half = 1 / sage.rings.integer.Integer(2)
if rif_self < 0 or (rif_self.contains_zero() and self < 0):
result = ceil(self - half)
else:
result = floor(self + half)
if not isinstance(result, sage.rings.integer.Integer):
raise ValueError, "could not convert %s to a real number"%(self)
else:
return result
def function(self, *args):
"""
Return a callable symbolic expression with the given variables.
EXAMPLES:
We will use several symbolic variables in the examples below::
sage: var('x, y, z, t, a, w, n')
(x, y, z, t, a, w, n)
::
sage: u = sin(x) + x*cos(y)
sage: g = u.function(x,y)
sage: g(x,y)
x*cos(y) + sin(x)
sage: g(t,z)
t*cos(z) + sin(t)
sage: g(x^2, x^y)
x^2*cos(x^y) + sin(x^2)
::
sage: f = (x^2 + sin(a*w)).function(a,x,w); f
(a, x, w) |--> x^2 + sin(a*w)
sage: f(1,2,3)
sin(3) + 4
Using the :meth:`function` method we can obtain the above function
`f`, but viewed as a function of different variables::
sage: h = f.function(w,a); h
(w, a) |--> x^2 + sin(a*w)
This notation also works::
sage: h(w,a) = f
sage: h
(w, a) |--> x^2 + sin(a*w)
You can even make a symbolic expression `f` into a function
by writing ``f(x,y) = f``::
sage: f = x^n + y^n; f
x^n + y^n
sage: f(x,y) = f
sage: f
(x, y) |--> x^n + y^n
sage: f(2,3)
3^n + 2^n
"""
from sage.symbolic.callable import CallableSymbolicExpressionRing
from sage.symbolic.ring import is_SymbolicVariable
for i in args:
if not is_SymbolicVariable(i):
break
else:
R = CallableSymbolicExpressionRing(args, check=False)
return R(self)
raise TypeError, "Must construct a function with a tuple (or list) of symbolic variables."
def power(self, exp, hold=False):
"""
Returns the current expression to the power ``exp``.
To prevent automatic evaluation use the ``hold`` argument.
EXAMPLES::
sage: (x^2).power(2)
x^4
sage: (x^2).power(2, hold=True)
(x^2)^2
To then evaluate again, we currently must use Maxima via
:meth:`simplify`::
sage: a = (x^2).power(2, hold=True); a.simplify()
x^4
"""
cdef Expression nexp = self.coerce_in(exp)
return new_Expression_from_GEx(self._parent,
g_hold2_wrapper(g_power_construct, self._gobj, nexp._gobj,
hold))
def add(self, *args, hold=False):
"""
Return the sum of the current expression and the given arguments.
To prevent automatic evaluation use the ``hold`` argument.
EXAMPLES::
sage: x.add(x)
2*x
sage: x.add(x, hold=True)
x + x
sage: x.add(x, (2+x), hold=True)
(x + 2) + x + x
sage: x.add(x, (2+x), x, hold=True)
(x + 2) + x + x + x
sage: x.add(x, (2+x), x, 2*x, hold=True)
(x + 2) + 2*x + x + x + x
To then evaluate again, we currently must use Maxima via
:meth:`simplify`::
sage: a = x.add(x, hold=True); a.simplify()
2*x
"""
nargs = [self.coerce_in(x) for x in args]
cdef GExVector vec
cdef Py_ssize_t i
vec.push_back(self._gobj)
for i in range(len(args)):
vec.push_back((<Expression>nargs[i])._gobj)
return new_Expression_from_GEx(self._parent, g_add_construct(vec, hold))
def mul(self, *args, hold=False):
"""
Return the product of the current expression and the given arguments.
To prevent automatic evaluation use the ``hold`` argument.
EXAMPLES::
sage: x.mul(x)
x^2
sage: x.mul(x, hold=True)
x*x
sage: x.mul(x, (2+x), hold=True)
(x + 2)*x*x
sage: x.mul(x, (2+x), x, hold=True)
(x + 2)*x*x*x
sage: x.mul(x, (2+x), x, 2*x, hold=True)
(2*x)*(x + 2)*x*x*x
To then evaluate again, we currently must use Maxima via
:meth:`simplify`::
sage: a = x.mul(x, hold=True); a.simplify()
x^2
"""
nargs = [self.coerce_in(x) for x in args]
cdef GExVector vec
cdef Py_ssize_t i
vec.push_back(self._gobj)
for i in range(len(args)):
vec.push_back((<Expression>nargs[i])._gobj)
return new_Expression_from_GEx(self._parent, g_mul_construct(vec, hold))
def coefficient(self, s, int n=1):
"""
Returns the coefficient of `s^n` in this symbolic expression.
INPUT:
- ``s`` - expression
- ``n`` - integer, default 1
OUTPUT:
A symbolic expression. The coefficient of `s^n`.
Sometimes it may be necessary to expand or factor first, since this
is not done automatically.
EXAMPLES::
sage: var('x,y,a')
(x, y, a)
sage: f = 100 + a*x + x^3*sin(x*y) + x*y + x/y + 2*sin(x*y)/x; f
x^3*sin(x*y) + a*x + x*y + x/y + 2*sin(x*y)/x + 100
sage: f.collect(x)
x^3*sin(x*y) + (a + y + 1/y)*x + 2*sin(x*y)/x + 100
sage: f.coefficient(x,0)
100
sage: f.coefficient(x,-1)
2*sin(x*y)
sage: f.coefficient(x,1)
a + y + 1/y
sage: f.coefficient(x,2)
0
sage: f.coefficient(x,3)
sin(x*y)
sage: f.coefficient(x^3)
sin(x*y)
sage: f.coefficient(sin(x*y))
x^3 + 2/x
sage: f.collect(sin(x*y))
a*x + x*y + (x^3 + 2/x)*sin(x*y) + x/y + 100
sage: var('a, x, y, z')
(a, x, y, z)
sage: f = (a*sqrt(2))*x^2 + sin(y)*x^(1/2) + z^z
sage: f.coefficient(sin(y))
sqrt(x)
sage: f.coefficient(x^2)
sqrt(2)*a
sage: f.coefficient(x^(1/2))
sin(y)
sage: f.coefficient(1)
0
sage: f.coefficient(x, 0)
sqrt(x)*sin(y) + z^z
"""
cdef Expression ss = self.coerce_in(s)
return new_Expression_from_GEx(self._parent, self._gobj.coeff(ss._gobj, n))
coeff = coefficient
def coefficients(self, x=None):
r"""
Coefficients of this symbolic expression as a polynomial in x.
INPUT:
- ``x`` -- optional variable.
OUTPUT:
A list of pairs ``(expr, n)``, where ``expr`` is a symbolic
expression and ``n`` is a power.
EXAMPLES::
sage: var('x, y, a')
(x, y, a)
sage: p = x^3 - (x-3)*(x^2+x) + 1
sage: p.coefficients()
[[1, 0], [3, 1], [2, 2]]
sage: p = expand((x-a*sqrt(2))^2 + x + 1); p
-2*sqrt(2)*a*x + 2*a^2 + x^2 + x + 1
sage: p.coefficients(a)
[[x^2 + x + 1, 0], [-2*sqrt(2)*x, 1], [2, 2]]
sage: p.coefficients(x)
[[2*a^2 + 1, 0], [-2*sqrt(2)*a + 1, 1], [1, 2]]
A polynomial with wacky exponents::
sage: p = (17/3*a)*x^(3/2) + x*y + 1/x + x^x
sage: p.coefficients(x)
[[1, -1], [x^x, 0], [y, 1], [17/3*a, 3/2]]
"""
f = self._maxima_()
maxima = f.parent()
maxima._eval_line('load(coeflist)')
if x is None:
x = self.default_variable()
x = self.parent().var(repr(x))
G = f.coeffs(x)
from sage.calculus.calculus import symbolic_expression_from_maxima_string
S = symbolic_expression_from_maxima_string(repr(G))
return S[1:]
coeffs = coefficients
def leading_coefficient(self, s):
"""
Return the leading coefficient of s in self.
EXAMPLES::
sage: var('x,y,a')
(x, y, a)
sage: f = 100 + a*x + x^3*sin(x*y) + x*y + x/y + 2*sin(x*y)/x; f
x^3*sin(x*y) + a*x + x*y + x/y + 2*sin(x*y)/x + 100
sage: f.leading_coefficient(x)
sin(x*y)
sage: f.leading_coefficient(y)
x
sage: f.leading_coefficient(sin(x*y))
x^3 + 2/x
"""
cdef Expression ss = self.coerce_in(s)
return new_Expression_from_GEx(self._parent, self._gobj.lcoeff(ss._gobj))
leading_coeff = leading_coefficient
def trailing_coefficient(self, s):
"""
Return the trailing coefficient of s in self, i.e., the coefficient
of the smallest power of s in self.
EXAMPLES::
sage: var('x,y,a')
(x, y, a)
sage: f = 100 + a*x + x^3*sin(x*y) + x*y + x/y + 2*sin(x*y)/x; f
x^3*sin(x*y) + a*x + x*y + x/y + 2*sin(x*y)/x + 100
sage: f.trailing_coefficient(x)
2*sin(x*y)
sage: f.trailing_coefficient(y)
x
sage: f.trailing_coefficient(sin(x*y))
a*x + x*y + x/y + 100
"""
cdef Expression ss = self.coerce_in(s)
return new_Expression_from_GEx(self._parent, self._gobj.tcoeff(ss._gobj))
trailing_coeff = trailing_coefficient
def low_degree(self, s):
"""
Return the exponent of the lowest nonpositive power of s in self.
OUTPUT:
An integer ``<= 0``.
EXAMPLES::
sage: var('x,y,a')
(x, y, a)
sage: f = 100 + a*x + x^3*sin(x*y) + x*y + x/y^10 + 2*sin(x*y)/x; f
x^3*sin(x*y) + a*x + x*y + 2*sin(x*y)/x + x/y^10 + 100
sage: f.low_degree(x)
-1
sage: f.low_degree(y)
-10
sage: f.low_degree(sin(x*y))
0
sage: (x^3+y).low_degree(x)
0
"""
cdef Expression ss = self.coerce_in(s)
return self._gobj.ldegree(ss._gobj)
def degree(self, s):
"""
Return the exponent of the highest nonnegative power of s in self.
OUTPUT:
An integer ``>= 0``.
EXAMPLES::
sage: var('x,y,a')
(x, y, a)
sage: f = 100 + a*x + x^3*sin(x*y) + x*y + x/y^10 + 2*sin(x*y)/x; f
x^3*sin(x*y) + a*x + x*y + 2*sin(x*y)/x + x/y^10 + 100
sage: f.degree(x)
3
sage: f.degree(y)
1
sage: f.degree(sin(x*y))
1
sage: (x^-3+y).degree(x)
0
"""
cdef Expression ss = self.coerce_in(s)
return self._gobj.degree(ss._gobj)
def unit(self, s):
"""
Return the unit of this expression when considered as a
polynomial in ``s``.
See also :meth:`content`, :meth:`primitive_part`, and
:meth:`unit_content_primitive`.
INPUT:
- ``s`` -- a symbolic expression.
OUTPUT:
The unit part of a polynomial as a symbolic expression. It is
defined as the sign of the leading coefficient.
EXAMPLES::
sage: (2*x+4).unit(x)
1
sage: (-2*x+1).unit(x)
-1
sage: (2*x+1/2).unit(x)
1
sage: var('y')
y
sage: (2*x - 4*sin(y)).unit(sin(y))
-1
"""
cdef Expression ss = self.coerce_in(s)
return new_Expression_from_GEx(self._parent, self._gobj.unit(ss._gobj))
def content(self, s):
"""
Return the content of this expression when considered as a
polynomial in ``s``.
See also :meth:`unit`, :meth:`primitive_part`, and
:meth:`unit_content_primitive`.
INPUT:
- ``s`` -- a symbolic expression.
OUTPUT:
The content part of a polynomial as a symbolic expression. It
is defined as the gcd of the coefficients.
.. warning::
The expression is considered to be a univariate polynomial
in ``s``. The output is different from the ``content()``
method provided by multivariate polynomial rings in Sage.
EXAMPLES::
sage: (2*x+4).content(x)
2
sage: (2*x+1).content(x)
1
sage: (2*x+1/2).content(x)
1/2
sage: var('y')
y
sage: (2*x + 4*sin(y)).content(sin(y))
2
"""
cdef Expression ss = self.coerce_in(s)
return new_Expression_from_GEx(self._parent, self._gobj.content(ss._gobj))
def primitive_part(self, s):
"""
Return the primitive polynomial of this expression when
considered as a polynomial in ``s``.
See also :meth:`unit`, :meth:`content`, and
:meth:`unit_content_primitive`.
INPUT:
- ``s`` -- a symbolic expression.
OUTPUT:
The primitive polynomial as a symbolic expression. It is
defined as the quotient by the :meth:`unit` and
:meth:`content` parts (with respect to the variable ``s``).
EXAMPLES::
sage: (2*x+4).primitive_part(x)
x + 2
sage: (2*x+1).primitive_part(x)
2*x + 1
sage: (2*x+1/2).primitive_part(x)
4*x + 1
sage: var('y')
y
sage: (2*x + 4*sin(y)).primitive_part(sin(y))
x + 2*sin(y)
"""
cdef Expression ss = self.coerce_in(s)
return new_Expression_from_GEx(self._parent, self._gobj.primpart(ss._gobj))
def unit_content_primitive(self, s):
"""
Return the factorization into unit, content, and primitive part.
INPUT:
- ``s`` -- a symbolic expression, usually a symbolic
variable. The whole symbolic expression ``self`` will be
considered as a univariate polynomial in ``s``.
OUTPUT:
A triple (unit, content, primitive polynomial)` containing the
:meth:`unit <unit>`, :meth:`content <content>`, and
:meth:`primitive polynomial <primitive_part>`. Their product equals
``self``.
EXAMPLES::
sage: var('x,y')
(x, y)
sage: ex = 9*x^3*y+3*y
sage: ex.unit_content_primitive(x)
(1, 3*y, 3*x^3 + 1)
sage: ex.unit_content_primitive(y)
(1, 9*x^3 + 3, y)
"""
cdef Expression ss = self.coerce_in(s)
cdef GEx unit, cont, prim
self._gobj.unitcontprim(ss._gobj, unit, cont, prim)
return (new_Expression_from_GEx(self._parent, unit),
new_Expression_from_GEx(self._parent, cont),
new_Expression_from_GEx(self._parent, prim))
def poly(self, x=None):
r"""
Express this symbolic expression as a polynomial in *x*. If
this is not a polynomial in *x*, then some coefficients may be
functions of *x*.
.. warning::
This is different from :meth:`polynomial` which returns
a Sage polynomial over a given base ring.
EXAMPLES::
sage: var('a, x')
(a, x)
sage: p = expand((x-a*sqrt(2))^2 + x + 1); p
-2*sqrt(2)*a*x + 2*a^2 + x^2 + x + 1
sage: p.poly(a)
-2*sqrt(2)*a*x + 2*a^2 + x^2 + x + 1
sage: bool(p.poly(a) == (x-a*sqrt(2))^2 + x + 1)
True
sage: p.poly(x)
2*a^2 - (2*sqrt(2)*a - 1)*x + x^2 + 1
"""
from sage.symbolic.ring import SR
f = self._maxima_()
P = f.parent()
P._eval_line('load(coeflist)')
if x is None:
x = self.default_variable()
x = self._parent.var(repr(x))
G = f.coeffs(x)
ans = None
for i in range(1, len(G)):
Z = G[i]
coeff = SR(Z[0])
n = SR(Z[1])
if repr(coeff) != '0':
if repr(n) == '0':
xpow = SR(1)
elif repr(n) == '1':
xpow = x
else:
xpow = x**n
if ans is None:
ans = coeff*xpow
else:
ans += coeff*xpow
return ans
def polynomial(self, base_ring, ring=None):
r"""
Return this symbolic expression as an algebraic polynomial
over the given base ring, if possible.
The point of this function is that it converts purely symbolic
polynomials into optimised algebraic polynomials over a given
base ring.
.. warning::
This is different from :meth:`poly` which is used to rewrite
self as a polynomial in terms of one of the variables.
INPUT:
- ``base_ring`` - a ring
EXAMPLES::
sage: f = x^2 -2/3*x + 1
sage: f.polynomial(QQ)
x^2 - 2/3*x + 1
sage: f.polynomial(GF(19))
x^2 + 12*x + 1
Polynomials can be useful for getting the coefficients of an
expression::
sage: g = 6*x^2 - 5
sage: g.coefficients()
[[-5, 0], [6, 2]]
sage: g.polynomial(QQ).list()
[-5, 0, 6]
sage: g.polynomial(QQ).dict()
{0: -5, 2: 6}
::
sage: f = x^2*e + x + pi/e
sage: f.polynomial(RDF)
2.71828182846*x^2 + x + 1.15572734979
sage: g = f.polynomial(RR); g
2.71828182845905*x^2 + x + 1.15572734979092
sage: g.parent()
Univariate Polynomial Ring in x over Real Field with 53 bits of precision
sage: f.polynomial(RealField(100))
2.7182818284590452353602874714*x^2 + x + 1.1557273497909217179100931833
sage: f.polynomial(CDF)
2.71828182846*x^2 + x + 1.15572734979
sage: f.polynomial(CC)
2.71828182845905*x^2 + x + 1.15572734979092
We coerce a multivariate polynomial with complex symbolic
coefficients::
sage: x, y, n = var('x, y, n')
sage: f = pi^3*x - y^2*e - I; f
pi^3*x - y^2*e - I
sage: f.polynomial(CDF)
(-2.71828182846)*y^2 + 31.0062766803*x - 1.0*I
sage: f.polynomial(CC)
(-2.71828182845905)*y^2 + 31.0062766802998*x - 1.00000000000000*I
sage: f.polynomial(ComplexField(70))
(-2.7182818284590452354)*y^2 + 31.006276680299820175*x - 1.0000000000000000000*I
Another polynomial::
sage: f = sum((e*I)^n*x^n for n in range(5)); f
x^4*e^4 - I*x^3*e^3 - x^2*e^2 + I*x*e + 1
sage: f.polynomial(CDF)
54.5981500331*x^4 - 20.0855369232*I*x^3 - 7.38905609893*x^2 + 2.71828182846*I*x + 1.0
sage: f.polynomial(CC)
54.5981500331442*x^4 - 20.0855369231877*I*x^3 - 7.38905609893065*x^2 + 2.71828182845905*I*x + 1.00000000000000
A multivariate polynomial over a finite field::
sage: f = (3*x^5 - 5*y^5)^7; f
(3*x^5 - 5*y^5)^7
sage: g = f.polynomial(GF(7)); g
3*x^35 + 2*y^35
sage: parent(g)
Multivariate Polynomial Ring in x, y over Finite Field of size 7
"""
from sage.symbolic.expression_conversions import polynomial
return polynomial(self, base_ring=base_ring, ring=ring)
def _polynomial_(self, R):
"""
Coerce this symbolic expression to a polynomial in `R`.
EXAMPLES::
sage: var('x,y,z,w')
(x, y, z, w)
::
sage: R = QQ[x,y,z]
sage: R(x^2 + y)
x^2 + y
sage: R = QQ[w]
sage: R(w^3 + w + 1)
w^3 + w + 1
sage: R = GF(7)[z]
sage: R(z^3 + 10*z)
z^3 + 3*z
.. note::
If the base ring of the polynomial ring is the symbolic ring,
then a constant polynomial is always returned.
::
sage: R = SR[x]
sage: a = R(sqrt(2) + x^3 + y)
sage: a
x^3 + y + sqrt(2)
sage: type(a)
<class 'sage.rings.polynomial.polynomial_element_generic.Polynomial_generic_dense_field'>
sage: a.degree()
0
We coerce to a double precision complex polynomial ring::
sage: f = e*x^3 + pi*y^3 + sqrt(2) + I; f
pi*y^3 + x^3*e + sqrt(2) + I
sage: R = CDF[x,y]
sage: R(f)
2.71828182846*x^3 + 3.14159265359*y^3 + 1.41421356237 + 1.0*I
We coerce to a higher-precision polynomial ring
::
sage: R = ComplexField(100)[x,y]
sage: R(f)
2.7182818284590452353602874714*x^3 + 3.1415926535897932384626433833*y^3 + 1.4142135623730950488016887242 + 1.0000000000000000000000000000*I
TESTS:
This shows that the issue at trac #5755 is fixed (attempting to
coerce a symbolic expression to a non-symbolic polynomial ring
caused an error::
sage: xx = var('xx')
sage: RDF['xx'](1.0*xx)
xx
sage: RDF['xx'](2.0*xx)
2.0*xx
sage: RR['xx'](1.0*xx)
xx
sage: RR['xx'](2.0*xx)
2.00000000000000*xx
This shows that the issue at trac #4246 is fixed (attempting to
coerce an expression containing at least one variable that's not in
`R` raises an error)::
sage: x, y = var('x y')
sage: S = PolynomialRing(Integers(4), 1, 'x')
sage: S(x)
x
sage: S(y)
Traceback (most recent call last):
...
TypeError: y is not a variable of Multivariate Polynomial Ring in x over Ring of integers modulo 4
sage: S(x+y)
Traceback (most recent call last):
...
TypeError: y is not a variable of Multivariate Polynomial Ring in x over Ring of integers modulo 4
sage: (x+y)._polynomial_(S)
Traceback (most recent call last):
...
TypeError: y is not a variable of Multivariate Polynomial Ring in x over Ring of integers modulo 4
"""
from sage.symbolic.all import SR
from sage.rings.polynomial.multi_polynomial_ring import is_MPolynomialRing
base_ring = R.base_ring()
if base_ring == SR:
if is_MPolynomialRing(R):
return R({tuple([0]*R.ngens()):self})
else:
return R([self])
return self.polynomial(None, ring=R)
def power_series(self, base_ring):
"""
Return algebraic power series associated to this symbolic
expression, which must be a polynomial in one variable, with
coefficients coercible to the base ring.
The power series is truncated one more than the degree.
EXAMPLES::
sage: theta = var('theta')
sage: f = theta^3 + (1/3)*theta - 17/3
sage: g = f.power_series(QQ); g
-17/3 + 1/3*theta + theta^3 + O(theta^4)
sage: g^3
-4913/27 + 289/9*theta - 17/9*theta^2 + 2602/27*theta^3 + O(theta^4)
sage: g.parent()
Power Series Ring in theta over Rational Field
"""
v = self.variables()
if len(v) != 1:
raise ValueError, "self must be a polynomial in one variable but it is in the variables %s"%tuple([v])
f = self.polynomial(base_ring)
from sage.rings.all import PowerSeriesRing
R = PowerSeriesRing(base_ring, names=f.parent().variable_names())
return R(f, f.degree()+1)
def gcd(self, b):
"""
Return the gcd of self and b, which must be integers or polynomials over
the rational numbers.
TODO: I tried the massive gcd from
http://trac.sagemath.org/sage_trac/ticket/694 on Ginac dies
after about 10 seconds. Singular easily does that GCD now.
Since Ginac only handles poly gcd over QQ, we should change
ginac itself to use Singular.
EXAMPLES::
sage: var('x,y')
(x, y)
sage: SR(10).gcd(SR(15))
5
sage: (x^3 - 1).gcd(x-1)
x - 1
sage: (x^3 - 1).gcd(x^2+x+1)
x^2 + x + 1
sage: (x^3 - sage.symbolic.constants.pi).gcd(x-sage.symbolic.constants.pi)
Traceback (most recent call last):
...
ValueError: gcd: arguments must be polynomials over the rationals
sage: gcd(x^3 - y^3, x-y)
-x + y
sage: gcd(x^100-y^100, x^10-y^10)
-x^10 + y^10
sage: gcd(expand( (x^2+17*x+3/7*y)*(x^5 - 17*y + 2/3) ), expand((x^13+17*x+3/7*y)*(x^5 - 17*y + 2/3)) )
1/7*x^5 - 17/7*y + 2/21
"""
cdef Expression r = self.coerce_in(b)
cdef GEx x
sig_on()
try:
x = g_gcd(self._gobj, r._gobj)
finally:
sig_off()
return new_Expression_from_GEx(self._parent, x)
def lcm(self, b):
"""
Return the lcm of self and b, which must be integers or
polynomials over the rational numbers. This is computed from
the gcd of self and b implicitly from the relation
self * b = gcd(self, b) * lcm(self, b).
.. NOTE::
In agreement with the convention in use for integers, if
self * b == 0, then gcd(self, b) == max(self, b) and
lcm(self, b) == 0.
EXAMPLES::
sage: var('x,y')
(x, y)
sage: SR(10).lcm(SR(15))
30
sage: (x^3 - 1).lcm(x-1)
x^3 - 1
sage: (x^3 - 1).lcm(x^2+x+1)
x^3 - 1
sage: (x^3 - sage.symbolic.constants.pi).lcm(x-sage.symbolic.constants.pi)
Traceback (most recent call last):
...
ValueError: lcm: arguments must be polynomials over the rationals
sage: lcm(x^3 - y^3, x-y)
-x^3 + y^3
sage: lcm(x^100-y^100, x^10-y^10)
-x^100 + y^100
sage: lcm(expand( (x^2+17*x+3/7*y)*(x^5 - 17*y + 2/3) ), expand((x^13+17*x+3/7*y)*(x^5 - 17*y + 2/3)) )
1/21*(21*x^18 - 357*x^13*y + 14*x^13 + 357*x^6 + 9*x^5*y -
6069*x*y - 153*y^2 + 238*x + 6*y)*(21*x^7 + 357*x^6 +
9*x^5*y - 357*x^2*y + 14*x^2 - 6069*x*y -
153*y^2 + 238*x + 6*y)/(3*x^5 - 51*y + 2)
TESTS:
Verify that x * y = gcd(x,y) * lcm(x,y)::
sage: x, y = var('x,y')
sage: LRs = [(SR(10), SR(15)), (x^3-1, x-1), (x^3-y^3, x-y), (x^3-1, x^2+x+1), (SR(0), x-y)]
sage: all((L.gcd(R) * L.lcm(R)) == L*R for L, R in LRs)
True
Make sure that the convention for what to do with the 0 is being respected::
sage: gcd(x, SR(0)), lcm(x, SR(0))
(x, 0)
sage: gcd(SR(0), SR(0)), lcm(SR(0), SR(0))
(0, 0)
"""
sb = self * b
try:
return 0 if sb == 0 else sb / self.gcd(b)
except ValueError:
raise ValueError("lcm: arguments must be polynomials over the rationals")
def collect(Expression self, s):
"""
Collect the coefficients of ``s`` into a group.
INPUT:
- ``s`` -- the symbol whose coefficients will be collected.
OUTPUT:
A new expression, equivalent to the original one, with the
coefficients of ``s`` grouped.
.. note::
The expression is not expanded or factored before the
grouping takes place. For best results, call :meth:`expand`
on the expression before :meth:`collect`.
EXAMPLES:
In the first term of `f`, `x` has a coefficient of `4y`. In
the second term, `x` has a coefficient of `z`. Therefore, if
we collect those coefficients, `x` will have a coefficient of
`4y+z`::
sage: x,y,z = var('x,y,z')
sage: f = 4*x*y + x*z + 20*y^2 + 21*y*z + 4*z^2 + x^2*y^2*z^2
sage: f.collect(x)
x^2*y^2*z^2 + x*(4*y + z) + 20*y^2 + 21*y*z + 4*z^2
Here we do the same thing for `y` and `z`; however, note that
we don't factor the `y^{2}` and `z^{2}` terms before
collecting coefficients::
sage: f.collect(y)
(x^2*z^2 + 20)*y^2 + (4*x + 21*z)*y + x*z + 4*z^2
sage: f.collect(z)
(x^2*y^2 + 4)*z^2 + 4*x*y + 20*y^2 + (x + 21*y)*z
Sometimes, we do have to call :meth:`expand()` on the
expression first to achieve the desired result::
sage: f = (x + y)*(x - z)
sage: f.collect(x)
x^2 + x*y - x*z - y*z
sage: f.expand().collect(x)
x^2 + x*(y - z) - y*z
TESTS:
The output should be equivalent to the input::
sage: polynomials = QQ['x']
sage: f = SR(polynomials.random_element())
sage: g = f.collect(x)
sage: bool(f == g)
True
If ``s`` is not present in the given expression, the
expression should not be modified. The variable `z` will not
be present in `f` below since `f` is a random polynomial of
maximum degree 10 in `x` and `y`::
sage: z = var('z')
sage: polynomials = QQ['x,y']
sage: f = SR(polynomials.random_element(10))
sage: g = f.collect(z)
sage: bool(str(f) == str(g))
True
Check if :trac:`9046` is fixed::
sage: var('a b x y z')
(a, b, x, y, z)
sage: p = -a*x^3 - a*x*y^2 + 2*b*x^2*y + 2*y^3 + x^2*z + y^2*z + x^2 + y^2 + a*x
sage: p.collect(x)
-a*x^3 + (2*b*y + z + 1)*x^2 + 2*y^3 + y^2*z - (a*y^2 - a)*x + y^2
"""
cdef Expression s0 = self.coerce_in(s)
cdef GEx x
sig_on()
try:
x = self._gobj.collect(s0._gobj, False)
finally:
sig_off()
return new_Expression_from_GEx(self._parent, x)
def collect_common_factors(self):
"""
EXAMPLES::
sage: var('x')
x
sage: (x/(x^2 + x)).collect_common_factors()
1/(x + 1)
"""
cdef GEx x
sig_on()
try:
x = g_collect_common_factors(self._gobj)
finally:
sig_off()
return new_Expression_from_GEx(self._parent, x)
def __abs__(self):
"""
Return the absolute value of this expression.
EXAMPLES::
sage: var('x, y')
(x, y)
The absolute value of a symbolic expression::
sage: abs(x^2+y^2)
abs(x^2 + y^2)
The absolute value of a number in the symbolic ring::
sage: abs(SR(-5))
5
sage: type(abs(SR(-5)))
<type 'sage.symbolic.expression.Expression'>
Because this overrides a Python builtin function, we do not
currently support a ``hold`` parameter to prevent automatic
evaluation::
sage: abs(SR(-5),hold=True)
Traceback (most recent call last):
...
TypeError: abs() takes no keyword arguments
But this is possible using the method :meth:`abs`::
sage: SR(-5).abs(hold=True)
abs(-5)
TESTS:
Check if :trac:`11155` is fixed::
sage: abs(pi+i)
abs(pi + I)
"""
return new_Expression_from_GEx(self._parent, g_abs(self._gobj))
def abs(self, hold=False):
"""
Return the absolute value of this expression.
EXAMPLES::
sage: var('x, y')
(x, y)
sage: (x+y).abs()
abs(x + y)
Using the ``hold`` parameter it is possible to prevent automatic
evaluation::
sage: SR(-5).abs(hold=True)
abs(-5)
To then evaluate again, we currently must use Maxima via
:meth:`simplify`::
sage: a = SR(-5).abs(hold=True); a.simplify()
5
TESTS:
From :trac:`7557`::
sage: var('y', domain='real')
y
sage: abs(exp(1.1*y*I)).simplify()
1
sage: var('y', domain='complex') # reset the domain for other tests
y
"""
return new_Expression_from_GEx(self._parent,
g_hold_wrapper(g_abs, self._gobj, hold))
def step(self, hold=False):
"""
Return the value of the Heaviside step function, which is 0 for
negative x, 1/2 for 0, and 1 for positive x.
EXAMPLES::
sage: x = var('x')
sage: SR(1.5).step()
1
sage: SR(0).step()
1/2
sage: SR(-1/2).step()
0
sage: SR(float(-1)).step()
0
Using the ``hold`` parameter it is possible to prevent automatic
evaluation::
sage: SR(2).step()
1
sage: SR(2).step(hold=True)
step(2)
"""
return new_Expression_from_GEx(self._parent,
g_hold_wrapper(g_step, self._gobj, hold))
def csgn(self, hold=False):
"""
Return the sign of self, which is -1 if self < 0, 0 if self ==
0, and 1 if self > 0, or unevaluated when self is a nonconstant
symbolic expression.
If self is not real, return the complex half-plane (left or right)
in which the number lies. If self is pure imaginary, return the sign
of the imaginary part of self.
EXAMPLES::
sage: x = var('x')
sage: SR(-2).csgn()
-1
sage: SR(0.0).csgn()
0
sage: SR(10).csgn()
1
sage: x.csgn()
csgn(x)
sage: SR(CDF.0).csgn()
1
sage: SR(I).csgn()
1
sage: SR(-I).csgn()
-1
sage: SR(1+I).csgn()
1
sage: SR(1-I).csgn()
1
sage: SR(-1+I).csgn()
-1
sage: SR(-1-I).csgn()
-1
Using the ``hold`` parameter it is possible to prevent automatic
evaluation::
sage: SR(I).csgn(hold=True)
csgn(I)
"""
return new_Expression_from_GEx(self._parent,
g_hold_wrapper(g_csgn, self._gobj, hold))
def conjugate(self, hold=False):
"""
Return the complex conjugate of this symbolic expression.
EXAMPLES::
sage: a = 1 + 2*I
sage: a.conjugate()
-2*I + 1
sage: a = sqrt(2) + 3^(1/3)*I; a
sqrt(2) + I*3^(1/3)
sage: a.conjugate()
sqrt(2) - I*3^(1/3)
sage: SR(CDF.0).conjugate()
-1.0*I
sage: x.conjugate()
conjugate(x)
sage: SR(RDF(1.5)).conjugate()
1.5
sage: SR(float(1.5)).conjugate()
1.5
sage: SR(I).conjugate()
-I
sage: ( 1+I + (2-3*I)*x).conjugate()
(3*I + 2)*conjugate(x) - I + 1
Using the ``hold`` parameter it is possible to prevent automatic
evaluation::
sage: SR(I).conjugate(hold=True)
conjugate(I)
This also works in functional notation::
sage: conjugate(I)
-I
sage: conjugate(I,hold=True)
conjugate(I)
To then evaluate again, we currently must use Maxima via
:meth:`simplify`::
sage: a = SR(I).conjugate(hold=True); a.simplify()
-I
"""
return new_Expression_from_GEx(self._parent,
g_hold_wrapper(g_conjugate, self._gobj, hold))
def norm(self):
r"""
The complex norm of this symbolic expression, i.e.,
the expression times its complex conjugate. If `c = a + bi` is a
complex number, then the norm of `c` is defined as the product of
`c` and its complex conjugate
.. MATH::
\text{norm}(c)
=
\text{norm}(a + bi)
=
c \cdot \overline{c}
=
a^2 + b^2.
The norm of a complex number is different from its absolute value.
The absolute value of a complex number is defined to be the square
root of its norm. A typical use of the complex norm is in the
integral domain `\ZZ[i]` of Gaussian integers, where the norm of
each Gaussian integer `c = a + bi` is defined as its complex norm.
.. SEEALSO::
- :func:`sage.misc.functional.norm`
EXAMPLES::
sage: a = 1 + 2*I
sage: a.norm()
5
sage: a = sqrt(2) + 3^(1/3)*I; a
sqrt(2) + I*3^(1/3)
sage: a.norm()
3^(2/3) + 2
sage: CDF(a).norm()
4.08008382305
sage: CDF(a.norm())
4.08008382305
"""
return (self*self.conjugate()).expand()
def real_part(self, hold=False):
"""
Return the real part of this symbolic expression.
EXAMPLES::
sage: x = var('x')
sage: x.real_part()
real_part(x)
sage: SR(2+3*I).real_part()
2
sage: SR(CDF(2,3)).real_part()
2.0
sage: SR(CC(2,3)).real_part()
2.00000000000000
sage: f = log(x)
sage: f.real_part()
log(abs(x))
Using the ``hold`` parameter it is possible to prevent automatic
evaluation::
sage: SR(2).real_part()
2
sage: SR(2).real_part(hold=True)
real_part(2)
This also works using functional notation::
sage: real_part(I,hold=True)
real_part(I)
sage: real_part(I)
0
To then evaluate again, we currently must use Maxima via
:meth:`simplify`::
sage: a = SR(2).real_part(hold=True); a.simplify()
2
"""
return new_Expression_from_GEx(self._parent,
g_hold_wrapper(g_real_part, self._gobj, hold))
real = real_part
def imag_part(self, hold=False):
r"""
Return the imaginary part of this symbolic expression.
EXAMPLES::
sage: sqrt(-2).imag_part()
sqrt(2)
We simplify `\ln(\exp(z))` to `z`. This should only
be for `-\pi<{\rm Im}(z)<=\pi`, but Maxima does not
have a symbolic imaginary part function, so we cannot
use ``assume`` to assume that first::
sage: z = var('z')
sage: f = log(exp(z))
sage: f
log(e^z)
sage: f.simplify()
z
sage: forget()
A more symbolic example::
sage: var('a, b')
(a, b)
sage: f = log(a + b*I)
sage: f.imag_part()
arctan2(imag_part(a) + real_part(b), -imag_part(b) + real_part(a))
Using the ``hold`` parameter it is possible to prevent automatic
evaluation::
sage: I.imag_part()
1
sage: I.imag_part(hold=True)
imag_part(I)
This also works using functional notation::
sage: imag_part(I,hold=True)
imag_part(I)
sage: imag_part(I)
1
To then evaluate again, we currently must use Maxima via
:meth:`simplify`::
sage: a = I.imag_part(hold=True); a.simplify()
1
TESTS::
sage: x = var('x')
sage: x.imag_part()
imag_part(x)
sage: SR(2+3*I).imag_part()
3
sage: SR(CC(2,3)).imag_part()
3.00000000000000
sage: SR(CDF(2,3)).imag_part()
3.0
"""
return new_Expression_from_GEx(self._parent,
g_hold_wrapper(g_imag_part, self._gobj, hold))
imag = imag_part
def sqrt(self, hold=False):
"""
Return the square root of this expression
EXAMPLES::
sage: var('x, y')
(x, y)
sage: SR(2).sqrt()
sqrt(2)
sage: (x^2+y^2).sqrt()
sqrt(x^2 + y^2)
sage: (x^2).sqrt()
sqrt(x^2)
Using the ``hold`` parameter it is possible to prevent automatic
evaluation::
sage: SR(4).sqrt()
2
sage: SR(4).sqrt(hold=True)
sqrt(4)
To then evaluate again, we currently must use Maxima via
:meth:`simplify`::
sage: a = SR(4).sqrt(hold=True); a.simplify()
2
To use this parameter in functional notation, you must coerce to
the symbolic ring::
sage: sqrt(SR(4),hold=True)
sqrt(4)
sage: sqrt(4,hold=True)
Traceback (most recent call last):
...
TypeError: _do_sqrt() got an unexpected keyword argument 'hold'
"""
return new_Expression_from_GEx(self._parent,
g_hold2_wrapper(g_power_construct, self._gobj, g_ex1_2, hold))
def sin(self, hold=False):
"""
EXAMPLES::
sage: var('x, y')
(x, y)
sage: sin(x^2 + y^2)
sin(x^2 + y^2)
sage: sin(sage.symbolic.constants.pi)
0
sage: sin(SR(1))
sin(1)
sage: sin(SR(RealField(150)(1)))
0.84147098480789650665250232163029899962256306
Using the ``hold`` parameter it is possible to prevent automatic
evaluation::
sage: SR(0).sin()
0
sage: SR(0).sin(hold=True)
sin(0)
This also works using functional notation::
sage: sin(0,hold=True)
sin(0)
sage: sin(0)
0
To then evaluate again, we currently must use Maxima via
:meth:`simplify`::
sage: a = SR(0).sin(hold=True); a.simplify()
0
TESTS::
sage: SR(oo).sin()
Traceback (most recent call last):
...
RuntimeError: sin_eval(): sin(infinity) encountered
sage: SR(-oo).sin()
Traceback (most recent call last):
...
RuntimeError: sin_eval(): sin(infinity) encountered
sage: SR(unsigned_infinity).sin()
Traceback (most recent call last):
...
RuntimeError: sin_eval(): sin(infinity) encountered
"""
return new_Expression_from_GEx(self._parent,
g_hold_wrapper(g_sin, self._gobj, hold))
def cos(self, hold=False):
"""
Return the cosine of self.
EXAMPLES::
sage: var('x, y')
(x, y)
sage: cos(x^2 + y^2)
cos(x^2 + y^2)
sage: cos(sage.symbolic.constants.pi)
-1
sage: cos(SR(1))
cos(1)
sage: cos(SR(RealField(150)(1)))
0.54030230586813971740093660744297660373231042
In order to get a numeric approximation use .n()::
sage: SR(RR(1)).cos().n()
0.540302305868140
sage: SR(float(1)).cos().n()
0.540302305868140
To prevent automatic evaluation use the ``hold`` argument::
sage: pi.cos()
-1
sage: pi.cos(hold=True)
cos(pi)
This also works using functional notation::
sage: cos(pi,hold=True)
cos(pi)
sage: cos(pi)
-1
To then evaluate again, we currently must use Maxima via
:meth:`simplify`::
sage: a = pi.cos(hold=True); a.simplify()
-1
TESTS::
sage: SR(oo).cos()
Traceback (most recent call last):
...
RuntimeError: cos_eval(): cos(infinity) encountered
sage: SR(-oo).cos()
Traceback (most recent call last):
...
RuntimeError: cos_eval(): cos(infinity) encountered
sage: SR(unsigned_infinity).cos()
Traceback (most recent call last):
...
RuntimeError: cos_eval(): cos(infinity) encountered
"""
return new_Expression_from_GEx(self._parent,
g_hold_wrapper(g_cos, self._gobj, hold))
def tan(self, hold=False):
"""
EXAMPLES::
sage: var('x, y')
(x, y)
sage: tan(x^2 + y^2)
tan(x^2 + y^2)
sage: tan(sage.symbolic.constants.pi/2)
Infinity
sage: tan(SR(1))
tan(1)
sage: tan(SR(RealField(150)(1)))
1.5574077246549022305069748074583601730872508
To prevent automatic evaluation use the ``hold`` argument::
sage: (pi/12).tan()
-sqrt(3) + 2
sage: (pi/12).tan(hold=True)
tan(1/12*pi)
This also works using functional notation::
sage: tan(pi/12,hold=True)
tan(1/12*pi)
sage: tan(pi/12)
-sqrt(3) + 2
To then evaluate again, we currently must use Maxima via
:meth:`simplify`::
sage: a = (pi/12).tan(hold=True); a.simplify()
-sqrt(3) + 2
TESTS::
sage: SR(oo).tan()
Traceback (most recent call last):
...
RuntimeError: tan_eval(): tan(infinity) encountered
sage: SR(-oo).tan()
Traceback (most recent call last):
...
RuntimeError: tan_eval(): tan(infinity) encountered
sage: SR(unsigned_infinity).tan()
Traceback (most recent call last):
...
RuntimeError: tan_eval(): tan(infinity) encountered
"""
return new_Expression_from_GEx(self._parent,
g_hold_wrapper(g_tan, self._gobj, hold))
def arcsin(self, hold=False):
"""
Return the arcsin of x, i.e., the number y between -pi and pi
such that sin(y) == x.
EXAMPLES::
sage: x.arcsin()
arcsin(x)
sage: SR(0.5).arcsin()
0.523598775598299
sage: SR(0.999).arcsin()
1.52607123962616
sage: SR(1/3).arcsin()
arcsin(1/3)
sage: SR(-1/3).arcsin()
-arcsin(1/3)
To prevent automatic evaluation use the ``hold`` argument::
sage: SR(0).arcsin()
0
sage: SR(0).arcsin(hold=True)
arcsin(0)
This also works using functional notation::
sage: arcsin(0,hold=True)
arcsin(0)
sage: arcsin(0)
0
To then evaluate again, we currently must use Maxima via
:meth:`simplify`::
sage: a = SR(0).arcsin(hold=True); a.simplify()
0
TESTS::
sage: SR(oo).arcsin()
Traceback (most recent call last):
...
RuntimeError: arcsin_eval(): arcsin(infinity) encountered
sage: SR(-oo).arcsin()
Traceback (most recent call last):
...
RuntimeError: arcsin_eval(): arcsin(infinity) encountered
sage: SR(unsigned_infinity).arcsin()
Infinity
"""
return new_Expression_from_GEx(self._parent,
g_hold_wrapper(g_asin, self._gobj, hold))
def arccos(self, hold=False):
"""
Return the arc cosine of self.
EXAMPLES::
sage: x.arccos()
arccos(x)
sage: SR(1).arccos()
0
sage: SR(1/2).arccos()
1/3*pi
sage: SR(0.4).arccos()
1.15927948072741
sage: plot(lambda x: SR(x).arccos(), -1,1)
To prevent automatic evaluation use the ``hold`` argument::
sage: SR(1).arccos(hold=True)
arccos(1)
This also works using functional notation::
sage: arccos(1,hold=True)
arccos(1)
sage: arccos(1)
0
To then evaluate again, we currently must use Maxima via
:meth:`simplify`::
sage: a = SR(1).arccos(hold=True); a.simplify()
0
TESTS::
sage: SR(oo).arccos()
Traceback (most recent call last):
...
RuntimeError: arccos_eval(): arccos(infinity) encountered
sage: SR(-oo).arccos()
Traceback (most recent call last):
...
RuntimeError: arccos_eval(): arccos(infinity) encountered
sage: SR(unsigned_infinity).arccos()
Infinity
"""
return new_Expression_from_GEx(self._parent,
g_hold_wrapper(g_acos, self._gobj, hold))
def arctan(self, hold=False):
"""
Return the arc tangent of self.
EXAMPLES::
sage: x = var('x')
sage: x.arctan()
arctan(x)
sage: SR(1).arctan()
1/4*pi
sage: SR(1/2).arctan()
arctan(1/2)
sage: SR(0.5).arctan()
0.463647609000806
sage: plot(lambda x: SR(x).arctan(), -20,20)
To prevent automatic evaluation use the ``hold`` argument::
sage: SR(1).arctan(hold=True)
arctan(1)
This also works using functional notation::
sage: arctan(1,hold=True)
arctan(1)
sage: arctan(1)
1/4*pi
To then evaluate again, we currently must use Maxima via
:meth:`simplify`::
sage: a = SR(1).arctan(hold=True); a.simplify()
1/4*pi
TESTS::
sage: SR(oo).arctan()
1/2*pi
sage: SR(-oo).arctan()
-1/2*pi
sage: SR(unsigned_infinity).arctan()
Traceback (most recent call last):
...
RuntimeError: arctan_eval(): arctan(unsigned_infinity) encountered
"""
return new_Expression_from_GEx(self._parent,
g_hold_wrapper(g_atan, self._gobj, hold))
def arctan2(self, x, hold=False):
"""
Return the inverse of the 2-variable tan function on self and x.
EXAMPLES::
sage: var('x,y')
(x, y)
sage: x.arctan2(y)
arctan2(x, y)
sage: SR(1/2).arctan2(1/2)
1/4*pi
sage: maxima.eval('atan2(1/2,1/2)')
'%pi/4'
sage: SR(-0.7).arctan2(SR(-0.6))
-2.27942259892257
To prevent automatic evaluation use the ``hold`` argument::
sage: SR(1/2).arctan2(1/2, hold=True)
arctan2(1/2, 1/2)
This also works using functional notation::
sage: arctan2(1,2,hold=True)
arctan2(1, 2)
sage: arctan2(1,2)
arctan(1/2)
To then evaluate again, we currently must use Maxima via
:meth:`simplify`::
sage: a = SR(1/2).arctan2(1/2, hold=True); a.simplify()
1/4*pi
TESTS:
We compare a bunch of different evaluation points between
Sage and Maxima::
sage: float(SR(0.7).arctan2(0.6))
0.8621700546672264
sage: maxima('atan2(0.7,0.6)')
.862170054667226...
sage: float(SR(0.7).arctan2(-0.6))
2.279422598922567
sage: maxima('atan2(0.7,-0.6)')
2.279422598922567
sage: float(SR(-0.7).arctan2(0.6))
-0.8621700546672264
sage: maxima('atan2(-0.7,0.6)')
-.862170054667226...
sage: float(SR(-0.7).arctan2(-0.6))
-2.279422598922567
sage: maxima('atan2(-0.7,-0.6)')
-2.279422598922567
sage: float(SR(0).arctan2(-0.6))
3.141592653589793
sage: maxima('atan2(0,-0.6)')
3.141592653589793
sage: float(SR(0).arctan2(0.6))
0.0
sage: maxima('atan2(0,0.6)')
0.0
sage: SR(0).arctan2(0) # see trac ticket #11423
Traceback (most recent call last):
...
RuntimeError: arctan2_eval(): arctan2(0,0) encountered
sage: SR(I).arctan2(1)
arctan2(I, 1)
sage: SR(CDF(0,1)).arctan2(1)
arctan2(1.0*I, 1)
sage: SR(1).arctan2(CDF(0,1))
arctan2(1, 1.0*I)
sage: arctan2(0,oo)
0
sage: SR(oo).arctan2(oo)
1/4*pi
sage: SR(oo).arctan2(0)
1/2*pi
sage: SR(-oo).arctan2(0)
-1/2*pi
sage: SR(-oo).arctan2(-2)
pi
sage: SR(unsigned_infinity).arctan2(2)
Traceback (most recent call last):
...
RuntimeError: arctan2_eval(): arctan2(x, unsigned_infinity) encountered
sage: SR(2).arctan2(oo)
1/2*pi
sage: SR(2).arctan2(-oo)
-1/2*pi
sage: SR(2).arctan2(SR(unsigned_infinity))
Traceback (most recent call last):
...
RuntimeError: arctan2_eval(): arctan2(unsigned_infinity, x) encountered
"""
cdef Expression nexp = self.coerce_in(x)
return new_Expression_from_GEx(self._parent,
g_hold2_wrapper(g_atan2, self._gobj, nexp._gobj, hold))
def sinh(self, hold=False):
r"""
Return sinh of self.
We have $\sinh(x) = (e^{x} - e^{-x})/2$.
EXAMPLES::
sage: x.sinh()
sinh(x)
sage: SR(1).sinh()
sinh(1)
sage: SR(0).sinh()
0
sage: SR(1.0).sinh()
1.17520119364380
sage: maxima('sinh(1.0)')
1.17520119364380...
sinh(1.0000000000000000000000000)
sage: SR(1).sinh().n(90)
1.1752011936438014568823819
sage: SR(RIF(1)).sinh()
1.175201193643802?
To prevent automatic evaluation use the ``hold`` argument::
sage: arccosh(x).sinh()
sqrt(x + 1)*sqrt(x - 1)
sage: arccosh(x).sinh(hold=True)
sinh(arccosh(x))
This also works using functional notation::
sage: sinh(arccosh(x),hold=True)
sinh(arccosh(x))
sage: sinh(arccosh(x))
sqrt(x + 1)*sqrt(x - 1)
To then evaluate again, we currently must use Maxima via
:meth:`simplify`::
sage: a = arccosh(x).sinh(hold=True); a.simplify()
sqrt(x + 1)*sqrt(x - 1)
TESTS::
sage: SR(oo).sinh()
+Infinity
sage: SR(-oo).sinh()
-Infinity
sage: SR(unsigned_infinity).sinh()
Traceback (most recent call last):
...
RuntimeError: sinh_eval(): sinh(unsigned_infinity) encountered
"""
return new_Expression_from_GEx(self._parent,
g_hold_wrapper(g_sinh, self._gobj, hold))
def cosh(self, hold=False):
r"""
Return cosh of self.
We have $\cosh(x) = (e^{x} + e^{-x})/2$.
EXAMPLES::
sage: x.cosh()
cosh(x)
sage: SR(1).cosh()
cosh(1)
sage: SR(0).cosh()
1
sage: SR(1.0).cosh()
1.54308063481524
sage: maxima('cosh(1.0)')
1.54308063481524...
sage: SR(1.00000000000000000000000000).cosh()
1.5430806348152437784779056
sage: SR(RIF(1)).cosh()
1.543080634815244?
To prevent automatic evaluation use the ``hold`` argument::
sage: arcsinh(x).cosh()
sqrt(x^2 + 1)
sage: arcsinh(x).cosh(hold=True)
cosh(arcsinh(x))
This also works using functional notation::
sage: cosh(arcsinh(x),hold=True)
cosh(arcsinh(x))
sage: cosh(arcsinh(x))
sqrt(x^2 + 1)
To then evaluate again, we currently must use Maxima via
:meth:`simplify`::
sage: a = arcsinh(x).cosh(hold=True); a.simplify()
sqrt(x^2 + 1)
TESTS::
sage: SR(oo).cosh()
+Infinity
sage: SR(-oo).cosh()
+Infinity
sage: SR(unsigned_infinity).cosh()
Traceback (most recent call last):
...
RuntimeError: cosh_eval(): cosh(unsigned_infinity) encountered
"""
return new_Expression_from_GEx(self._parent,
g_hold_wrapper(g_cosh, self._gobj, hold))
def tanh(self, hold=False):
r"""
Return tanh of self.
We have $\tanh(x) = \sinh(x) / \cosh(x)$.
EXAMPLES::
sage: x.tanh()
tanh(x)
sage: SR(1).tanh()
tanh(1)
sage: SR(0).tanh()
0
sage: SR(1.0).tanh()
0.761594155955765
sage: maxima('tanh(1.0)')
.7615941559557649
sage: plot(lambda x: SR(x).tanh(), -1, 1)
To prevent automatic evaluation use the ``hold`` argument::
sage: arcsinh(x).tanh()
x/sqrt(x^2 + 1)
sage: arcsinh(x).tanh(hold=True)
tanh(arcsinh(x))
This also works using functional notation::
sage: tanh(arcsinh(x),hold=True)
tanh(arcsinh(x))
sage: tanh(arcsinh(x))
x/sqrt(x^2 + 1)
To then evaluate again, we currently must use Maxima via
:meth:`simplify`::
sage: a = arcsinh(x).tanh(hold=True); a.simplify()
x/sqrt(x^2 + 1)
TESTS::
sage: SR(oo).tanh()
1
sage: SR(-oo).tanh()
-1
sage: SR(unsigned_infinity).tanh()
Traceback (most recent call last):
...
RuntimeError: tanh_eval(): tanh(unsigned_infinity) encountered
"""
return new_Expression_from_GEx(self._parent,
g_hold_wrapper(g_tanh, self._gobj, hold))
def arcsinh(self, hold=False):
"""
Return the inverse hyperbolic sine of self.
EXAMPLES::
sage: x.arcsinh()
arcsinh(x)
sage: SR(0).arcsinh()
0
sage: SR(1).arcsinh()
arcsinh(1)
sage: SR(1.0).arcsinh()
0.881373587019543
sage: maxima('asinh(2.0)')
1.4436354751788...
Sage automatically applies certain identities::
sage: SR(3/2).arcsinh().cosh()
1/2*sqrt(13)
To prevent automatic evaluation use the ``hold`` argument::
sage: SR(-2).arcsinh()
-arcsinh(2)
sage: SR(-2).arcsinh(hold=True)
arcsinh(-2)
This also works using functional notation::
sage: arcsinh(-2,hold=True)
arcsinh(-2)
sage: arcsinh(-2)
-arcsinh(2)
To then evaluate again, we currently must use Maxima via
:meth:`simplify`::
sage: a = SR(-2).arcsinh(hold=True); a.simplify()
-arcsinh(2)
TESTS::
sage: SR(oo).arcsinh()
+Infinity
sage: SR(-oo).arcsinh()
-Infinity
sage: SR(unsigned_infinity).arcsinh()
Infinity
"""
return new_Expression_from_GEx(self._parent,
g_hold_wrapper(g_asinh, self._gobj, hold))
def arccosh(self, hold=False):
"""
Return the inverse hyperbolic cosine of self.
EXAMPLES::
sage: x.arccosh()
arccosh(x)
sage: SR(0).arccosh()
1/2*I*pi
sage: SR(1/2).arccosh()
arccosh(1/2)
sage: SR(CDF(1/2)).arccosh()
1.0471975512*I
sage: maxima('acosh(0.5)')
1.04719755119659...*%i
To prevent automatic evaluation use the ``hold`` argument::
sage: SR(-1).arccosh()
I*pi
sage: SR(-1).arccosh(hold=True)
arccosh(-1)
This also works using functional notation::
sage: arccosh(-1,hold=True)
arccosh(-1)
sage: arccosh(-1)
I*pi
To then evaluate again, we currently must use Maxima via
:meth:`simplify`::
sage: a = SR(-1).arccosh(hold=True); a.simplify()
I*pi
TESTS::
sage: SR(oo).arccosh()
+Infinity
sage: SR(-oo).arccosh()
+Infinity
sage: SR(unsigned_infinity).arccosh()
+Infinity
"""
return new_Expression_from_GEx(self._parent,
g_hold_wrapper(g_acosh, self._gobj, hold))
def arctanh(self, hold=False):
"""
Return the inverse hyperbolic tangent of self.
EXAMPLES::
sage: x.arctanh()
arctanh(x)
sage: SR(0).arctanh()
0
sage: SR(1/2).arctanh()
arctanh(1/2)
sage: SR(0.5).arctanh()
0.549306144334055
sage: SR(0.5).arctanh().tanh()
0.500000000000000
sage: maxima('atanh(0.5)')
.5493061443340...
To prevent automatic evaluation use the ``hold`` argument::
sage: SR(-1/2).arctanh()
-arctanh(1/2)
sage: SR(-1/2).arctanh(hold=True)
arctanh(-1/2)
This also works using functional notation::
sage: arctanh(-1/2,hold=True)
arctanh(-1/2)
sage: arctanh(-1/2)
-arctanh(1/2)
To then evaluate again, we currently must use Maxima via
:meth:`simplify`::
sage: a = SR(-1/2).arctanh(hold=True); a.simplify()
-arctanh(1/2)
TESTS::
sage: SR(1).arctanh()
+Infinity
sage: SR(-1).arctanh()
-Infinity
sage: SR(oo).arctanh()
-1/2*I*pi
sage: SR(-oo).arctanh()
1/2*I*pi
sage: SR(unsigned_infinity).arctanh()
Traceback (most recent call last):
...
RuntimeError: arctanh_eval(): arctanh(unsigned_infinity) encountered
"""
return new_Expression_from_GEx(self._parent,
g_hold_wrapper(g_atanh, self._gobj, hold))
def exp(self, hold=False):
"""
Return exponential function of self, i.e., e to the
power of self.
EXAMPLES::
sage: x.exp()
e^x
sage: SR(0).exp()
1
sage: SR(1/2).exp()
e^(1/2)
sage: SR(0.5).exp()
1.64872127070013
sage: math.exp(0.5)
1.6487212707001282
sage: SR(0.5).exp().log()
0.500000000000000
sage: (pi*I).exp()
-1
To prevent automatic evaluation use the ``hold`` argument::
sage: (pi*I).exp(hold=True)
e^(I*pi)
This also works using functional notation::
sage: exp(I*pi,hold=True)
e^(I*pi)
sage: exp(I*pi)
-1
To then evaluate again, we currently must use Maxima via
:meth:`simplify`::
sage: a = (pi*I).exp(hold=True); a.simplify()
-1
TESTS:
Test if #6377 is fixed::
sage: SR(oo).exp()
+Infinity
sage: SR(-oo).exp()
0
sage: SR(unsigned_infinity).exp()
Traceback (most recent call last):
...
RuntimeError: exp_eval(): exp^(unsigned_infinity) encountered
"""
return new_Expression_from_GEx(self._parent,
g_hold_wrapper(g_exp, self._gobj, hold))
def log(self, b=None, hold=False):
"""
Return the logarithm of self.
EXAMPLES::
sage: x, y = var('x, y')
sage: x.log()
log(x)
sage: (x^y + y^x).log()
log(x^y + y^x)
sage: SR(0).log()
-Infinity
sage: SR(-1).log()
I*pi
sage: SR(1).log()
0
sage: SR(1/2).log()
log(1/2)
sage: SR(0.5).log()
-0.693147180559945
sage: SR(0.5).log().exp()
0.500000000000000
sage: math.log(0.5)
-0.6931471805599453
sage: plot(lambda x: SR(x).log(), 0.1,10)
To prevent automatic evaluation use the ``hold`` argument::
sage: I.log()
1/2*I*pi
sage: I.log(hold=True)
log(I)
To then evaluate again, we currently must use Maxima via
:meth:`simplify`::
sage: a = I.log(hold=True); a.simplify()
1/2*I*pi
We do not currently support a ``hold`` parameter in functional
notation::
sage: log(SR(-1),hold=True)
Traceback (most recent call last):
...
TypeError: log() got an unexpected keyword argument 'hold'
TESTS::
sage: SR(oo).log()
+Infinity
sage: SR(-oo).log()
+Infinity
sage: SR(unsigned_infinity).log()
+Infinity
"""
res = new_Expression_from_GEx(self._parent,
g_hold_wrapper(g_log, self._gobj, hold))
if b is None:
return res
else:
return res/self.coerce_in(b).log(hold=hold)
def zeta(self, hold=False):
"""
EXAMPLES::
sage: x, y = var('x, y')
sage: (x/y).zeta()
zeta(x/y)
sage: SR(2).zeta()
1/6*pi^2
sage: SR(3).zeta()
zeta(3)
sage: SR(CDF(0,1)).zeta()
0.00330022368532 - 0.418155449141*I
sage: CDF(0,1).zeta()
0.00330022368532 - 0.418155449141*I
sage: plot(lambda x: SR(x).zeta(), -10,10).show(ymin=-3,ymax=3)
To prevent automatic evaluation use the ``hold`` argument::
sage: SR(2).zeta(hold=True)
zeta(2)
This also works using functional notation::
sage: zeta(2,hold=True)
zeta(2)
sage: zeta(2)
1/6*pi^2
To then evaluate again, we currently must use Maxima via
:meth:`simplify`::
sage: a = SR(2).zeta(hold=True); a.simplify()
1/6*pi^2
TESTS::
sage: t = SR(1).zeta(); t
Infinity
"""
cdef GEx x
sig_on()
try:
x = g_hold_wrapper(g_zeta, self._gobj, hold)
finally:
sig_off()
return new_Expression_from_GEx(self._parent, x)
def factorial(self, hold=False):
"""
Return the factorial of self.
OUTPUT:
A symbolic expression.
EXAMPLES::
sage: var('x, y')
(x, y)
sage: SR(5).factorial()
120
sage: x.factorial()
factorial(x)
sage: (x^2+y^3).factorial()
factorial(y^3 + x^2)
To prevent automatic evaluation use the ``hold`` argument::
sage: SR(5).factorial(hold=True)
factorial(5)
This also works using functional notation::
sage: factorial(5,hold=True)
factorial(5)
sage: factorial(5)
120
To then evaluate again, we currently must use Maxima via
:meth:`simplify`::
sage: a = SR(5).factorial(hold=True); a.simplify()
120
"""
cdef GEx x
sig_on()
try:
x = g_hold_wrapper(g_factorial, self._gobj, hold)
finally:
sig_off()
return new_Expression_from_GEx(self._parent, x)
def binomial(self, k, hold=False):
"""
Return binomial coefficient "self choose k".
OUTPUT:
A symbolic expression.
EXAMPLES::
sage: var('x, y')
(x, y)
sage: SR(5).binomial(SR(3))
10
sage: x.binomial(SR(3))
1/6*x^3 - 1/2*x^2 + 1/3*x
sage: x.binomial(y)
binomial(x, y)
To prevent automatic evaluation use the ``hold`` argument::
sage: x.binomial(3, hold=True)
binomial(x, 3)
sage: SR(5).binomial(3, hold=True)
binomial(5, 3)
To then evaluate again, we currently must use Maxima via
:meth:`simplify`::
sage: a = SR(5).binomial(3, hold=True); a.simplify()
10
We do not currently support a ``hold`` parameter in functional
notation::
sage: binomial(5,3, hold=True)
Traceback (most recent call last):
...
TypeError: binomial() got an unexpected keyword argument 'hold'
TESTS:
Check if we handle zero correctly (#8561)::
sage: x.binomial(0)
1
sage: SR(0).binomial(0)
1
"""
cdef Expression nexp = self.coerce_in(k)
cdef GEx x
sig_on()
try:
x = g_hold2_wrapper(g_binomial, self._gobj, nexp._gobj, hold)
finally:
sig_off()
return new_Expression_from_GEx(self._parent, x)
def Order(self, hold=False):
"""
Order, as in big oh notation.
OUTPUT:
A symbolic expression.
EXAMPLES::
sage: n = var('n')
sage: t = (17*n^3).Order(); t
Order(n^3)
sage: t.derivative(n)
Order(n^2)
To prevent automatic evaluation use the ``hold`` argument::
sage: (17*n^3).Order(hold=True)
Order(17*n^3)
"""
return new_Expression_from_GEx(self._parent,
g_hold_wrapper(g_Order, self._gobj, hold))
def gamma(self, hold=False):
"""
Return the Gamma function evaluated at self.
EXAMPLES::
sage: x = var('x')
sage: x.gamma()
gamma(x)
sage: SR(2).gamma()
1
sage: SR(10).gamma()
362880
sage: SR(10.0r).gamma()
362880.0
sage: SR(CDF(1,1)).gamma()
0.498015668118 - 0.154949828302*I
::
sage: gp('gamma(1+I)')
0.4980156681183560427136911175 - 0.1549498283018106851249551305*I # 32-bit
0.49801566811835604271369111746219809195 - 0.15494982830181068512495513048388660520*I # 64-bit
We plot the familiar plot of this log-convex function::
sage: plot(gamma(x), -6,4).show(ymin=-3,ymax=3)
To prevent automatic evaluation use the ``hold`` argument::
sage: SR(1/2).gamma()
sqrt(pi)
sage: SR(1/2).gamma(hold=True)
gamma(1/2)
This also works using functional notation::
sage: gamma(1/2,hold=True)
gamma(1/2)
sage: gamma(1/2)
sqrt(pi)
To then evaluate again, we currently must use Maxima via
:meth:`simplify`::
sage: a = SR(1/2).gamma(hold=True); a.simplify()
sqrt(pi)
"""
cdef GEx x
sig_on()
try:
x = g_hold_wrapper(g_tgamma, self._gobj, hold)
finally:
sig_off()
return new_Expression_from_GEx(self._parent, x)
def lgamma(self, hold=False):
"""
This method is deprecated, please use the ``.log_gamma()`` function instead.
Log gamma function evaluated at self.
EXAMPLES::
sage: x.lgamma()
doctest:...: DeprecationWarning: The lgamma() function is deprecated. Use log_gamma() instead.
See http://trac.sagemath.org/6992 for details.
log_gamma(x)
"""
from sage.misc.superseded import deprecation
deprecation(6992, "The lgamma() function is deprecated. Use log_gamma() instead.")
return self.log_gamma(hold=hold)
def log_gamma(self, hold=False):
"""
Return the log gamma function evaluated at self.
This is the logarithm of gamma of self, where
gamma is a complex function such that `gamma(n)`
equals `factorial(n-1)`.
EXAMPLES::
sage: x = var('x')
sage: x.log_gamma()
log_gamma(x)
sage: SR(2).log_gamma()
0
sage: SR(5).log_gamma()
log(24)
sage: a = SR(5).log_gamma(); a.n()
3.17805383034795
sage: SR(5-1).factorial().log()
log(24)
sage: set_verbose(-1); plot(lambda x: SR(x).log_gamma(), -7,8, plot_points=1000).show()
sage: math.exp(0.5)
1.6487212707001282
sage: plot(lambda x: (SR(x).exp() - SR(-x).exp())/2 - SR(x).sinh(), -1, 1)
To prevent automatic evaluation use the ``hold`` argument::
sage: SR(5).log_gamma(hold=True)
log_gamma(5)
To evaluate again, currently we must use numerical evaluation
via :meth:`n`::
sage: a = SR(5).log_gamma(hold=True); a.n()
3.17805383034795
"""
cdef GEx x
sig_on()
try:
x = g_hold_wrapper(g_lgamma, self._gobj, hold)
finally:
sig_off()
return new_Expression_from_GEx(self._parent, x)
def default_variable(self):
"""
Return the default variable, which is by definition the first
variable in self, or `x` is there are no variables in self.
The result is cached.
EXAMPLES::
sage: sqrt(2).default_variable()
x
sage: x, theta, a = var('x, theta, a')
sage: f = x^2 + theta^3 - a^x
sage: f.default_variable()
a
Note that this is the first *variable*, not the first *argument*::
sage: f(theta, a, x) = a + theta^3
sage: f.default_variable()
a
sage: f.variables()
(a, theta)
sage: f.arguments()
(theta, a, x)
"""
v = self.variables()
if len(v) == 0:
return self.parent().var('x')
else:
return v[0]
def combine(self):
r"""
Returns a simplified version of this symbolic expression
by combining all terms with the same denominator into a single
term.
EXAMPLES::
sage: var('x, y, a, b, c')
(x, y, a, b, c)
sage: f = x*(x-1)/(x^2 - 7) + y^2/(x^2-7) + 1/(x+1) + b/a + c/a; f
(x - 1)*x/(x^2 - 7) + y^2/(x^2 - 7) + b/a + c/a + 1/(x + 1)
sage: f.combine()
((x - 1)*x + y^2)/(x^2 - 7) + (b + c)/a + 1/(x + 1)
"""
return self.parent()(self._maxima_().combine())
def normalize(self):
"""
Return this expression normalized as a fraction
.. SEEALSO:
:meth:`numerator`, :meth:`denominator`,
:meth:`numerator_denominator`, :meth:`combine`
EXAMPLES::
sage: var('x, y, a, b, c')
(x, y, a, b, c)
sage: g = x + y/(x + 2)
sage: g.normalize()
(x^2 + 2*x + y)/(x + 2)
sage: f = x*(x-1)/(x^2 - 7) + y^2/(x^2-7) + 1/(x+1) + b/a + c/a
sage: f.normalize()
(a*x^3 + b*x^3 + c*x^3 + a*x*y^2 + a*x^2 + b*x^2 + c*x^2 +
a*y^2 - a*x - 7*b*x - 7*c*x - 7*a - 7*b - 7*c)/((x^2 -
7)*a*(x + 1))
ALGORITHM: Uses GiNaC.
"""
return new_Expression_from_GEx(self._parent, self._gobj.normal())
def numerator(self, bint normalize = True):
"""
Returns the numerator of this symbolic expression
INPUT:
- ``normalize`` -- (default: ``True``) a boolean.
If ``normalize`` is ``True``, the expression is first normalized to
have it as a fraction before getting the numerator.
If ``normalize`` is ``False``, the expression is kept and if it is not
a quotient, then this will return the expression itself.
.. SEEALSO::
:meth:`normalize`, :meth:`denominator`,
:meth:`numerator_denominator`, :meth:`combine`
EXAMPLES::
sage: a, x, y = var('a,x,y')
sage: f = x*(x-a)/((x^2 - y)*(x-a)); f
x/(x^2 - y)
sage: f.numerator()
x
sage: f.denominator()
x^2 - y
sage: f.numerator(normalize=False)
x
sage: f.denominator(normalize=False)
x^2 - y
sage: y = var('y')
sage: g = x + y/(x + 2); g
x + y/(x + 2)
sage: g.numerator()
x^2 + 2*x + y
sage: g.denominator()
x + 2
sage: g.numerator(normalize=False)
x + y/(x + 2)
sage: g.denominator(normalize=False)
1
TESTS::
sage: ((x+y)^2/(x-y)^3*x^3).numerator(normalize=False)
(x + y)^2*x^3
sage: ((x+y)^2*x^3).numerator(normalize=False)
(x + y)^2*x^3
sage: (y/x^3).numerator(normalize=False)
y
sage: t = y/x^3/(x+y)^(1/2); t
y/(sqrt(x + y)*x^3)
sage: t.numerator(normalize=False)
y
sage: (1/x^3).numerator(normalize=False)
1
sage: (x^3).numerator(normalize=False)
x^3
sage: (y*x^sin(x)).numerator(normalize=False)
Traceback (most recent call last):
...
TypeError: self is not a rational expression
"""
cdef GExVector vec
cdef GEx oper, power
if normalize:
return new_Expression_from_GEx(self._parent, self._gobj.numer())
elif is_a_mul(self._gobj):
for i from 0 <= i < self._gobj.nops():
oper = self._gobj.op(i)
if not is_a_power(oper):
vec.push_back(oper)
else:
power = oper.op(1)
if not is_a_numeric(power):
raise TypeError, "self is not a rational expression"
elif ex_to_numeric(power).is_positive():
vec.push_back(oper)
return new_Expression_from_GEx(self._parent,
g_mul_construct(vec, True))
elif is_a_power(self._gobj):
power = self._gobj.op(1)
if is_a_numeric(power) and ex_to_numeric(power).is_negative():
return self._parent.one()
return self
def denominator(self, bint normalize=True):
"""
Returns the denominator of this symbolic expression
INPUT:
- ``normalize`` -- (default: ``True``) a boolean.
If ``normalize`` is ``True``, the expression is first normalized to
have it as a fraction before getting the denominator.
If ``normalize`` is ``False``, the expression is kept and if it is not
a quotient, then this will just return 1.
.. SEEALSO::
:meth:`normalize`, :meth:`numerator`,
:meth:`numerator_denominator`, :meth:`combine`
EXAMPLES::
sage: x, y, z, theta = var('x, y, z, theta')
sage: f = (sqrt(x) + sqrt(y) + sqrt(z))/(x^10 - y^10 - sqrt(theta))
sage: f.numerator()
sqrt(x) + sqrt(y) + sqrt(z)
sage: f.denominator()
x^10 - y^10 - sqrt(theta)
sage: f.numerator(normalize=False)
(sqrt(x) + sqrt(y) + sqrt(z))
sage: f.denominator(normalize=False)
x^10 - y^10 - sqrt(theta)
sage: y = var('y')
sage: g = x + y/(x + 2); g
x + y/(x + 2)
sage: g.numerator(normalize=False)
x + y/(x + 2)
sage: g.denominator(normalize=False)
1
TESTS::
sage: ((x+y)^2/(x-y)^3*x^3).denominator(normalize=False)
(x - y)^3
sage: ((x+y)^2*x^3).denominator(normalize=False)
1
sage: (y/x^3).denominator(normalize=False)
x^3
sage: t = y/x^3/(x+y)^(1/2); t
y/(sqrt(x + y)*x^3)
sage: t.denominator(normalize=False)
sqrt(x + y)*x^3
sage: (1/x^3).denominator(normalize=False)
x^3
sage: (x^3).denominator(normalize=False)
1
sage: (y*x^sin(x)).denominator(normalize=False)
Traceback (most recent call last):
...
TypeError: self is not a rational expression
"""
cdef GExVector vec
cdef GEx oper, ex, power
if normalize:
return new_Expression_from_GEx(self._parent, self._gobj.denom())
elif is_a_mul(self._gobj):
for i from 0 <= i < self._gobj.nops():
oper = self._gobj.op(i)
if is_a_power(oper):
ex = oper.op(0)
power = oper.op(1)
if not is_a_numeric(power):
raise TypeError, "self is not a rational expression"
elif ex_to_numeric(power).is_negative():
vec.push_back(g_pow(ex, g_abs(power)))
return new_Expression_from_GEx(self._parent,
g_mul_construct(vec, False))
elif is_a_power(self._gobj):
power = self._gobj.op(1)
if is_a_numeric(power) and ex_to_numeric(power).is_negative():
return new_Expression_from_GEx(self._parent,
g_pow(self._gobj.op(0), g_abs(power)))
return self._parent.one()
def numerator_denominator(self, bint normalize=True):
"""
Returns the numerator and the denominator of this symbolic expression
INPUT:
- ``normalize`` -- (default: ``True``) a boolean.
If ``normalize`` is ``True``, the expression is first normalized to
have it as a fraction before getting the numerator and denominator.
If ``normalize`` is ``False``, the expression is kept and if it is not
a quotient, then this will return the expression itself together with
1.
.. SEEALSO::
:meth:`normalize`, :meth:`numerator`, :meth:`denominator`,
:meth:`combine`
EXAMPLE::
sage: x, y, a = var("x y a")
sage: ((x+y)^2/(x-y)^3*x^3).numerator_denominator()
((x + y)^2*x^3, (x - y)^3)
sage: ((x+y)^2/(x-y)^3*x^3).numerator_denominator(False)
((x + y)^2*x^3, (x - y)^3)
sage: g = x + y/(x + 2)
sage: g.numerator_denominator()
(x^2 + 2*x + y, x + 2)
sage: g.numerator_denominator(normalize=False)
(x + y/(x + 2), 1)
sage: g = x^2*(x + 2)
sage: g.numerator_denominator()
((x + 2)*x^2, 1)
sage: g.numerator_denominator(normalize=False)
((x + 2)*x^2, 1)
TESTS::
sage: ((x+y)^2/(x-y)^3*x^3).numerator_denominator(normalize=False)
((x + y)^2*x^3, (x - y)^3)
sage: ((x+y)^2*x^3).numerator_denominator(normalize=False)
((x + y)^2*x^3, 1)
sage: (y/x^3).numerator_denominator(normalize=False)
(y, x^3)
sage: t = y/x^3/(x+y)^(1/2); t
y/(sqrt(x + y)*x^3)
sage: t.numerator_denominator(normalize=False)
(y, sqrt(x + y)*x^3)
sage: (1/x^3).numerator_denominator(normalize=False)
(1, x^3)
sage: (x^3).numerator_denominator(normalize=False)
(x^3, 1)
sage: (y*x^sin(x)).numerator_denominator(normalize=False)
Traceback (most recent call last):
...
TypeError: self is not a rational expression
"""
cdef GExVector vecnumer, vecdenom
cdef GEx oper, ex, power
cdef GNumeric power_num
if normalize:
ex = self._gobj.numer_denom()
return (new_Expression_from_GEx(self._parent, ex.op(0)),
new_Expression_from_GEx(self._parent, ex.op(1)))
elif is_a_mul(self._gobj):
for i from 0 <= i < self._gobj.nops():
oper = self._gobj.op(i)
if is_a_power(oper):
ex = oper.op(0)
power = oper.op(1)
if not is_a_numeric(power):
raise TypeError, "self is not a rational expression"
elif is_a_numeric(power):
power_num = ex_to_numeric(power)
if power_num.is_positive():
vecnumer.push_back(oper)
else:
vecdenom.push_back(g_pow(ex, g_abs(power)))
else:
vecnumer.push_back(oper)
return (new_Expression_from_GEx(self._parent,
g_mul_construct(vecnumer, False)),
new_Expression_from_GEx(self._parent,
g_mul_construct(vecdenom, False)))
elif is_a_power(self._gobj):
power = self._gobj.op(1)
if is_a_numeric(power) and ex_to_numeric(power).is_positive():
return (self, self._parent.one())
else:
return (self._parent.one(),
new_Expression_from_GEx(self._parent,
g_pow(self._gobj.op(0), g_abs(power))))
else:
return (self, self._parent.one())
def partial_fraction(self, var=None):
r"""
Return the partial fraction expansion of ``self`` with
respect to the given variable.
INPUT:
- ``var`` - variable name or string (default: first
variable)
OUTPUT:
A symbolic expression.
EXAMPLES::
sage: f = x^2/(x+1)^3
sage: f.partial_fraction()
1/(x + 1) - 2/(x + 1)^2 + 1/(x + 1)^3
sage: f.partial_fraction()
1/(x + 1) - 2/(x + 1)^2 + 1/(x + 1)^3
Notice that the first variable in the expression is used by
default::
sage: y = var('y')
sage: f = y^2/(y+1)^3
sage: f.partial_fraction()
1/(y + 1) - 2/(y + 1)^2 + 1/(y + 1)^3
sage: f = y^2/(y+1)^3 + x/(x-1)^3
sage: f.partial_fraction()
y^2/(y^3 + 3*y^2 + 3*y + 1) + 1/(x - 1)^2 + 1/(x - 1)^3
You can explicitly specify which variable is used::
sage: f.partial_fraction(y)
x/(x^3 - 3*x^2 + 3*x - 1) + 1/(y + 1) - 2/(y + 1)^2 + 1/(y + 1)^3
"""
if var is None:
var = self.default_variable()
return self.parent()(self._maxima_().partfrac(var))
def maxima_methods(self):
"""
Provides easy access to maxima methods, converting the result to a
Sage expression automatically.
EXAMPLES::
sage: t = log(sqrt(2) - 1) + log(sqrt(2) + 1); t
log(sqrt(2) + 1) + log(sqrt(2) - 1)
sage: res = t.maxima_methods().logcontract(); res
log((sqrt(2) + 1)*(sqrt(2) - 1))
sage: type(res)
<type 'sage.symbolic.expression.Expression'>
"""
from sage.symbolic.maxima_wrapper import MaximaWrapper
return MaximaWrapper(self)
def rectform(self):
r"""
Convert this symbolic expression to rectangular form; that
is, the form `a + bi` where `a` and `b` are real numbers and
`i` is the imaginary unit.
.. note::
The name \"rectangular\" comes from the fact that, in the
complex plane, `a` and `bi` are perpendicular.
INPUT:
- ``self`` -- the expression to convert.
OUTPUT:
A new expression, equivalent to the original, but expressed in
the form `a + bi`.
ALGORITHM:
We call Maxima's ``rectform()`` and return the result unmodified.
EXAMPLES:
The exponential form of `\sin(x)`::
sage: f = (e^(I*x) - e^(-I*x)) / (2*I)
sage: f.rectform()
sin(x)
And `\cos(x)`::
sage: f = (e^(I*x) + e^(-I*x)) / 2
sage: f.rectform()
cos(x)
In some cases, this will simplify the given expression. For
example, here, `e^{ik\pi}`, `\sin(k\pi)=0` should cancel
leaving only `\cos(k\pi)` which can then be simplified::
sage: k = var('k')
sage: assume(k, 'integer')
sage: f = e^(I*pi*k)
sage: f.rectform()
(-1)^k
However, in general, the resulting expression may be more
complicated than the original::
sage: f = e^(I*x)
sage: f.rectform()
cos(x) + I*sin(x)
TESTS:
If the expression is already in rectangular form, it should be
left alone::
sage: a,b = var('a,b')
sage: assume((a, 'real'), (b, 'real'))
sage: f = a + b*I
sage: f.rectform()
a + I*b
sage: forget()
We can check with specific real numbers::
sage: a = RR.random_element()
sage: b = RR.random_element()
sage: f = a + b*I
sage: bool(f.rectform() == a + b*I)
True
If we decompose a complex number into its real and imaginary
parts, they should correspond to the real and imaginary terms
of the rectangular form::
sage: z = CC.random_element()
sage: a = z.real_part()
sage: b = z.imag_part()
sage: bool(SR(z).rectform() == a + b*I)
True
"""
return self.maxima_methods().rectform()
def simplify(self):
"""
Returns a simplified version of this symbolic expression.
.. note::
Currently, this just sends the expression to Maxima
and converts it back to Sage.
.. seealso::
:meth:`simplify_full`, :meth:`simplify_trig`,
:meth:`simplify_rational`, :meth:`simplify_radical`,
:meth:`simplify_factorial`, :meth:`simplify_log`
EXAMPLES::
sage: a = var('a'); f = x*sin(2)/(x^a); f
x*sin(2)/x^a
sage: f.simplify()
x^(-a + 1)*sin(2)
"""
return self._parent(self._maxima_())
def simplify_full(self):
"""
Applies simplify_factorial, simplify_trig, simplify_rational,
simplify_log, and again simplify_rational to self (in that order).
ALIAS: simplify_full and full_simplify are the same.
EXAMPLES::
sage: f = sin(x)^2 + cos(x)^2
sage: f.simplify_full()
1
::
sage: f = sin(x/(x^2 + x))
sage: f.simplify_full()
sin(1/(x + 1))
::
sage: var('n,k')
(n, k)
sage: f = binomial(n,k)*factorial(k)*factorial(n-k)
sage: f.simplify_full()
factorial(n)
TESTS:
There are two square roots of `$(x + 1)^2$`, so this should
not be simplified to `$x + 1$`, :trac:`12737`::
sage: f = sqrt((x + 1)^2)
sage: f.simplify_full()
sqrt(x^2 + 2*x + 1)
The imaginary part of an expression should not change under
simplification; :trac:`11934`::
sage: f = sqrt(-8*(4*sqrt(2) - 7)*x^4 + 16*(3*sqrt(2) - 5)*x^3)
sage: original = f.imag_part()
sage: simplified = f.full_simplify().imag_part()
sage: original - simplified
0
The invalid simplification from :trac:`12322` should not occur
after :trac:`12737`::
sage: t = var('t')
sage: assume(t, 'complex')
sage: assumptions()
[t is complex]
sage: f = (1/2)*log(2*t) + (1/2)*log(1/t)
sage: f.simplify_full()
1/2*log(2*t) - 1/2*log(t)
"""
x = self
x = x.simplify_factorial()
x = x.simplify_trig()
x = x.simplify_rational()
x = x.simplify_log('one')
x = x.simplify_rational()
return x
full_simplify = simplify_full
def simplify_trig(self,expand=True):
r"""
Optionally expands and then employs identities such as
`\sin(x)^2 + \cos(x)^2 = 1`, `\cosh(x)^2 - \sinh(x)^2 = 1`,
`\sin(x)\csc(x) = 1`, or `\tanh(x)=\sinh(x)/\cosh(x)`
to simplify expressions containing tan, sec, etc., to sin,
cos, sinh, cosh.
INPUT:
- ``self`` - symbolic expression
- ``expand`` - (default:True) if True, expands trigonometric
and hyperbolic functions of sums of angles and of multiple
angles occurring in ``self`` first. For best results,
``self`` should be expanded. See also :meth:`expand_trig` to
get more controls on this expansion.
ALIAS: :meth:`trig_simplify` and :meth:`simplify_trig` are the same
EXAMPLES::
sage: f = sin(x)^2 + cos(x)^2; f
cos(x)^2 + sin(x)^2
sage: f.simplify()
cos(x)^2 + sin(x)^2
sage: f.simplify_trig()
1
sage: h = sin(x)*csc(x)
sage: h.simplify_trig()
1
sage: k = tanh(x)*cosh(2*x)
sage: k.simplify_trig()
(2*sinh(x)^3 + sinh(x))/cosh(x)
In some cases we do not want to expand::
sage: f=tan(3*x)
sage: f.simplify_trig()
(4*cos(x)^2 - 1)*sin(x)/(4*cos(x)^3 - 3*cos(x))
sage: f.simplify_trig(False)
sin(3*x)/cos(3*x)
"""
if expand:
return self.parent()(self._maxima_().trigexpand().trigsimp())
else:
return self.parent()(self._maxima_().trigsimp())
trig_simplify = simplify_trig
@rename_keyword(deprecation=6094, method="algorithm")
def simplify_rational(self,algorithm='full', map=False):
r"""
Simplify rational expressions.
INPUT:
- ``self`` - symbolic expression
- ``algorithm`` - (default: 'full') string which switches the
algorithm for simplifications. Possible values are
- 'simple' (simplify rational functions into quotient of two
polynomials),
- 'full' (apply repeatedly, if necessary)
- 'noexpand' (convert to commmon denominator and add)
- ``map`` - (default: False) if True, the result is an
expression whose leading operator is the same as that of the
expression ``self`` but whose subparts are the results of
applying simplification rules to the corresponding subparts
of the expressions.
ALIAS: :meth:`rational_simplify` and :meth:`simplify_rational`
are the same
DETAILS: We call Maxima functions ratsimp, fullratsimp and
xthru. If each part of the expression has to be simplified
separately, we use Maxima function map.
EXAMPLES::
sage: f = sin(x/(x^2 + x))
sage: f
sin(x/(x^2 + x))
sage: f.simplify_rational()
sin(1/(x + 1))
::
sage: f = ((x - 1)^(3/2) - (x + 1)*sqrt(x - 1))/sqrt((x - 1)*(x + 1)); f
-((x + 1)*sqrt(x - 1) - (x - 1)^(3/2))/sqrt((x + 1)*(x - 1))
sage: f.simplify_rational()
-2*sqrt(x - 1)/sqrt(x^2 - 1)
With ``map=True`` each term in a sum is simplified separately
and thus the resuls are shorter for functions which are
combination of rational and nonrational funtions. In the
following example, we use this option if we want not to
combine logarithm and the rational function into one
fraction::
sage: f=(x^2-1)/(x+1)-ln(x)/(x+2)
sage: f.simplify_rational()
(x^2 + x - log(x) - 2)/(x + 2)
sage: f.simplify_rational(map=True)
x - log(x)/(x + 2) - 1
Here is an example from the Maxima documentation of where
``algorithm='simple'`` produces an (possibly useful) intermediate
step::
sage: y = var('y')
sage: g = (x^(y/2) + 1)^2*(x^(y/2) - 1)^2/(x^y - 1)
sage: g.simplify_rational(algorithm='simple')
(x^(2*y) - 2*x^y + 1)/(x^y - 1)
sage: g.simplify_rational()
x^y - 1
With option ``algorithm='noexpand'`` we only convert to common
denominators and add. No expansion of products is performed::
sage: f=1/(x+1)+x/(x+2)^2
sage: f.simplify_rational()
(2*x^2 + 5*x + 4)/(x^3 + 5*x^2 + 8*x + 4)
sage: f.simplify_rational(algorithm='noexpand')
((x + 2)^2 + (x + 1)*x)/((x + 2)^2*(x + 1))
"""
self_m = self._maxima_()
if algorithm == 'full':
maxima_method = 'fullratsimp'
elif algorithm == 'simple':
maxima_method = 'ratsimp'
elif algorithm == 'noexpand':
maxima_method = 'xthru'
else:
raise NotImplementedError, "unknown algorithm, see the help for available algorithms"
P = self_m.parent()
self_str=self_m.str()
if map:
cmd = "if atom(%s) then %s(%s) else map(%s,%s)"%(self_str,maxima_method,self_str,maxima_method,self_str)
else:
cmd = "%s(%s)"%(maxima_method,self_m.str())
res = P(cmd)
return self.parent()(res)
rational_simplify = simplify_rational
def simplify_factorial(self):
"""
Simplify by combining expressions with factorials, and by
expanding binomials into factorials.
ALIAS: factorial_simplify and simplify_factorial are the same
EXAMPLES:
Some examples are relatively clear::
sage: var('n,k')
(n, k)
sage: f = factorial(n+1)/factorial(n); f
factorial(n + 1)/factorial(n)
sage: f.simplify_factorial()
n + 1
::
sage: f = factorial(n)*(n+1); f
(n + 1)*factorial(n)
sage: simplify(f)
(n + 1)*factorial(n)
sage: f.simplify_factorial()
factorial(n + 1)
::
sage: f = binomial(n, k)*factorial(k)*factorial(n-k); f
binomial(n, k)*factorial(k)*factorial(-k + n)
sage: f.simplify_factorial()
factorial(n)
A more complicated example, which needs further processing::
sage: f = factorial(x)/factorial(x-2)/2 + factorial(x+1)/factorial(x)/2; f
1/2*factorial(x + 1)/factorial(x) + 1/2*factorial(x)/factorial(x - 2)
sage: g = f.simplify_factorial(); g
1/2*(x - 1)*x + 1/2*x + 1/2
sage: g.simplify_rational()
1/2*x^2 + 1/2
TESTS:
Check that the problem with applying full_simplify() to gamma functions (Trac 9240)
has been fixed::
sage: gamma(1/3)
gamma(1/3)
sage: gamma(1/3).full_simplify()
gamma(1/3)
sage: gamma(4/3)
gamma(4/3)
sage: gamma(4/3).full_simplify()
1/3*gamma(1/3)
"""
return self.parent()(self._maxima_().makefact().factcomb().minfactorial())
factorial_simplify = simplify_factorial
def simplify_radical(self):
r"""
Simplifies this symbolic expression, which can contain logs,
exponentials, and radicals, by trying to convert it into a canonical
form over a large class of expressions and a given ordering of
variables.
.. WARNING::
As shown in the examples below, a canonical form is not always
returned, i.e., two mathematically identical expressions might
be simplified to different expressions.
ALGORITHM:
This uses the Maxima ``radcan()`` command. From the Maxima
documentation: "All functionally equivalent forms are mapped into a
unique form. For a somewhat larger class of expressions, produces a
regular form. Two equivalent expressions in this class do not
necessarily have the same appearance, but their difference can be
simplified by radcan to zero. For some expressions radcan is quite
time consuming. This is the cost of exploring certain relationships
among the components of the expression for simplifications based on
factoring and partial fraction expansions of exponents."
.. NOTE::
:meth:`radical_simplify`, :meth:`simplify_radical`,
:meth:`exp_simplify`, :meth:`simplify_exp` are all the same.
EXAMPLES::
sage: var('x,y,a')
(x, y, a)
::
sage: f = log(x*y)
sage: f.simplify_radical()
log(x) + log(y)
::
sage: f = log(8)/log(2)
sage: f.simplify_radical()
3
::
sage: f = (log(x+x^2)-log(x))^a/log(1+x)^(a/2)
sage: f.simplify_radical()
log(x + 1)^(1/2*a)
::
sage: f = (e^x-1)/(1+e^(x/2))
sage: f.simplify_exp()
e^(1/2*x) - 1
The example below shows two expressions e1 and e2 which are
"simplified" to different expressions, while their difference is
"simplified" to zero, thus ``simplify_radical`` does not return a
canonical form, except maybe for 0. ::
sage: e1 = 1/(sqrt(5)+sqrt(2))
sage: e2 = (sqrt(5)-sqrt(2))/3
sage: e1.simplify_radical()
1/(sqrt(5) + sqrt(2))
sage: e2.simplify_radical()
1/3*sqrt(5) - 1/3*sqrt(2)
sage: (e1-e2).simplify_radical()
0
TESTS:
This tests that :trac:`11668` has been fixed (by :trac:`12780`)::
sage: a,b = var('a b')
sage: A = abs((a+I*b))^2
sage: A.simplify_radical()
abs(a + I*b)^2
sage: imag(A)
0
sage: imag(A.simplify_radical())
0
"""
from sage.calculus.calculus import maxima
res = self.parent()(self._maxima_().radcan())
return res
radical_simplify = simplify_radical
simplify_exp = exp_simplify = simplify_radical
@rename_keyword(deprecation=6094, method="algorithm")
def simplify_log(self,algorithm=None):
r"""
Simplifies symbolic expression, which can contain logs.
Recursively scans the expression self, transforming
subexpressions of the form `a1 \log(b1) + a2 \log(b2) + c` into
`\log( b1^{a1} b2^{a2} ) + c` and simplifies inside logarithm. User
can specify, which conditions must satisfy `a1` and `a2` to use
this transformation in optional parameter ``algorithm``.
INPUT:
- ``self`` - expression to be simplified
- ``algorithm`` - (default: None) optional, governs the condition
on `a1` and `a2` which must be satisfied to contract expression
`a1 \log(b1) + a2 \log(b2)`. Values are
- None (use Maxima default, integers),
- 'one' (1 and -1),
- 'ratios' (integers and fractions of integers),
- 'constants' (constants),
- 'all' (all expressions).
See also examples below.
DETAILS: This uses the Maxima logcontract() command. From the
Maxima documentation: "Recursively scans the expression expr,
transforming subexpressions of the form a1*log(b1) +
a2*log(b2) + c into log(ratsimp(b1^a1 * b2^a2)) + c. The user
can control which coefficients are contracted by setting the
option logconcoeffp to the name of a predicate function of one
argument. E.g. if you like to generate SQRTs, you can do
logconcoeffp:'logconfun$ logconfun(m):=featurep(m,integer) or
ratnump(m)$ . Then logcontract(1/2*log(x)); will give
log(sqrt(x))."
ALIAS: :meth:`log_simplify` and :meth:`simplify_log` are the
same
EXAMPLES::
sage: x,y,t=var('x y t')
Only two first terms are contracted in the following example ,
the logarithm with coefficient 1/2 is not contracted::
sage: f = log(x)+2*log(y)+1/2*log(t)
sage: f.simplify_log()
log(x*y^2) + 1/2*log(t)
To contract all terms in previous example use option ``algorithm``::
sage: f.simplify_log(algorithm='ratios')
log(sqrt(t)*x*y^2)
This shows that the option ``algorithm`` from the previous call
has no influence to future calls (we changed some default
Maxima flag, and have to ensure that this flag has been
restored)::
sage: f.simplify_log('one')
1/2*log(t) + log(x) + 2*log(y)
sage: f.simplify_log('ratios')
log(sqrt(t)*x*y^2)
sage: f.simplify_log()
log(x*y^2) + 1/2*log(t)
To contract terms with no coefficient (more precisely, with
coefficients 1 and -1) use option ``algorithm``::
sage: f = log(x)+2*log(y)-log(t)
sage: f.simplify_log('one')
2*log(y) + log(x/t)
::
sage: f = log(x)+log(y)-1/3*log((x+1))
sage: f.simplify_log()
log(x*y) - 1/3*log(x + 1)
sage: f.simplify_log('ratios')
log(x*y/(x + 1)^(1/3))
`\pi` is irrational number, to contract logarithms in the following example
we have to put ``algorithm`` to ``constants`` or ``all``::
sage: f = log(x)+log(y)-pi*log((x+1))
sage: f.simplify_log('constants')
log(x*y/(x + 1)^pi)
x*log(9) is contracted only if ``algorithm`` is ``all``::
sage: (x*log(9)).simplify_log()
x*log(9)
sage: (x*log(9)).simplify_log('all')
log(9^x)
TESTS:
This shows that the issue at trac #7334 is fixed. Maxima intentionally
keeps the expression inside the log factored::
sage: log_expr = (log(sqrt(2)-1)+log(sqrt(2)+1))
sage: log_expr.simplify_log('all')
log((sqrt(2) + 1)*(sqrt(2) - 1))
sage: _.simplify_rational()
0
sage: log_expr.simplify_full() # applies both simplify_log and simplify_rational
0
We should use the current simplification domain rather than
set it to 'real' explicitly (:trac:`12780`)::
sage: f = sqrt(x^2)
sage: f.simplify_log()
sqrt(x^2)
sage: from sage.calculus.calculus import maxima
sage: maxima('domain: real;')
real
sage: f.simplify_log()
abs(x)
sage: maxima('domain: complex;')
complex
AUTHORS:
- Robert Marik (11-2009)
"""
from sage.calculus.calculus import maxima
maxima.eval('savelogexpand:logexpand$ logexpand:false$')
if algorithm is not None:
maxima.eval('logconcoeffp:\'logconfun$')
if algorithm == 'ratios':
maxima.eval('logconfun(m):= featurep(m,integer) or ratnump(m)$')
elif algorithm == 'one':
maxima.eval('logconfun(m):= is(m=1) or is(m=-1)$')
elif algorithm == 'constants':
maxima.eval('logconfun(m):= constantp(m)$')
elif algorithm == 'all':
maxima.eval('logconfun(m):= true$')
elif algorithm is not None:
raise NotImplementedError, "unknown algorithm, see the help for available algorithms"
res = self.parent()(self._maxima_().logcontract())
if algorithm is not None:
maxima.eval('logconcoeffp:false$')
maxima.eval('logexpand:savelogexpand$')
return res
log_simplify = simplify_log
@rename_keyword(deprecation=6094, method="algorithm")
def expand_log(self,algorithm='products'):
r"""
Simplifies symbolic expression, which can contain logs.
Expands logarithms of powers, logarithms of products and
logarithms of quotients. The option ``algorithm`` specifies
which expression types should be expanded.
INPUT:
- ``self`` - expression to be simplified
- ``algorithm`` - (default: 'products') optional, governs which
expression is expanded. Possible values are
- 'nothing' (no expansion),
- 'powers' (log(a^r) is expanded),
- 'products' (like 'powers' and also log(a*b) are expanded),
- 'all' (all possible expansion).
See also examples below.
DETAILS: This uses the Maxima simplifier and sets
``logexpand`` option for this simplifier. From the Maxima
documentation: "Logexpand:true causes log(a^b) to become
b*log(a). If it is set to all, log(a*b) will also simplify to
log(a)+log(b). If it is set to super, then log(a/b) will also
simplify to log(a)-log(b) for rational numbers a/b,
a#1. (log(1/b), for integer b, always simplifies.) If it is
set to false, all of these simplifications will be turned
off. "
ALIAS: :meth:`log_expand` and :meth:`expand_log` are the same
EXAMPLES:
By default powers and products (and quotients) are expanded,
but not quotients of integers::
sage: (log(3/4*x^pi)).log_expand()
pi*log(x) + log(3/4)
To expand also log(3/4) use ``algorithm='all'``::
sage: (log(3/4*x^pi)).log_expand('all')
pi*log(x) - log(4) + log(3)
To expand only the power use ``algorithm='powers'``.::
sage: (log(x^6)).log_expand('powers')
6*log(x)
The expression ``log((3*x)^6)`` is not expanded with
``algorithm='powers'``, since it is converted into product
first::
sage: (log((3*x)^6)).log_expand('powers')
log(729*x^6)
This shows that the option ``algorithm`` from the previous call
has no influence to future calls (we changed some default
Maxima flag, and have to ensure that this flag has been
restored)::
sage: (log(3/4*x^pi)).log_expand()
pi*log(x) + log(3/4)
sage: (log(3/4*x^pi)).log_expand('all')
pi*log(x) - log(4) + log(3)
sage: (log(3/4*x^pi)).log_expand()
pi*log(x) + log(3/4)
TESTS:
Most of these log expansions only make sense over the
reals. So, we should set the Maxima ``domain`` variable to
'real' before we call out to Maxima. When we return, however, we
should set the ``domain`` back to what it was, rather than
assuming that it was 'complex'. See :trac:`12780`::
sage: from sage.calculus.calculus import maxima
sage: maxima('domain: real;')
real
sage: x.expand_log()
x
sage: maxima('domain;')
real
sage: maxima('domain: complex;')
complex
AUTHORS:
- Robert Marik (11-2009)
"""
from sage.calculus.calculus import maxima
original_domain = maxima.eval('domain')
maxima.eval('domain: real$ savelogexpand:logexpand$')
if algorithm == 'nothing':
maxima_method='false'
elif algorithm == 'powers':
maxima_method='true'
elif algorithm == 'products':
maxima_method='all'
elif algorithm == 'all':
maxima_method='super'
else:
raise NotImplementedError, "unknown algorithm, see the help for available algorithms"
maxima.eval('logexpand:%s'%maxima_method)
res = self._maxima_()
res = res.sage()
maxima.eval('domain: %s$ logexpand:savelogexpand$' % original_domain)
return res
log_expand = expand_log
def factor(self, dontfactor=[]):
"""
Factors self, containing any number of variables or functions, into
factors irreducible over the integers.
INPUT:
- ``self`` - a symbolic expression
- ``dontfactor`` - list (default: []), a list of
variables with respect to which factoring is not to occur.
Factoring also will not take place with respect to any variables
which are less important (using the variable ordering assumed for
CRE form) than those on the 'dontfactor' list.
EXAMPLES::
sage: x,y,z = var('x, y, z')
sage: (x^3-y^3).factor()
(x^2 + x*y + y^2)*(x - y)
sage: factor(-8*y - 4*x + z^2*(2*y + x))
(x + 2*y)*(z + 2)*(z - 2)
sage: f = -1 - 2*x - x^2 + y^2 + 2*x*y^2 + x^2*y^2
sage: F = factor(f/(36*(1 + 2*y + y^2)), dontfactor=[x]); F
1/36*(x^2 + 2*x + 1)*(y - 1)/(y + 1)
If you are factoring a polynomial with rational coefficients (and
dontfactor is empty) the factorization is done using Singular
instead of Maxima, so the following is very fast instead of
dreadfully slow::
sage: var('x,y')
(x, y)
sage: (x^99 + y^99).factor()
(x^60 + x^57*y^3 - x^51*y^9 - x^48*y^12 + x^42*y^18 + x^39*y^21 -
x^33*y^27 - x^30*y^30 - x^27*y^33 + x^21*y^39 + x^18*y^42 -
x^12*y^48 - x^9*y^51 + x^3*y^57 + y^60)*(x^20 + x^19*y -
x^17*y^3 - x^16*y^4 + x^14*y^6 + x^13*y^7 - x^11*y^9 -
x^10*y^10 - x^9*y^11 + x^7*y^13 + x^6*y^14 - x^4*y^16 -
x^3*y^17 + x*y^19 + y^20)*(x^10 - x^9*y + x^8*y^2 - x^7*y^3 +
x^6*y^4 - x^5*y^5 + x^4*y^6 - x^3*y^7 + x^2*y^8 - x*y^9 +
y^10)*(x^6 - x^3*y^3 + y^6)*(x^2 - x*y + y^2)*(x + y)
"""
from sage.calculus.calculus import symbolic_expression_from_maxima_string, symbolic_expression_from_string
if len(dontfactor) > 0:
m = self._maxima_()
name = m.name()
cmd = 'block([dontfactor:%s],factor(%s))'%(dontfactor, name)
return symbolic_expression_from_maxima_string(cmd)
else:
try:
from sage.rings.all import QQ
f = self.polynomial(QQ)
w = repr(f.factor())
return symbolic_expression_from_string(w)
except TypeError:
pass
return self.parent()(self._maxima_().factor())
def factor_list(self, dontfactor=[]):
"""
Returns a list of the factors of self, as computed by the
factor command.
INPUT:
- ``self`` - a symbolic expression
- ``dontfactor`` - see docs for :meth:`factor`
.. note::
If you already have a factored expression and just want to
get at the individual factors, use the `_factor_list` method
instead.
EXAMPLES::
sage: var('x, y, z')
(x, y, z)
sage: f = x^3-y^3
sage: f.factor()
(x^2 + x*y + y^2)*(x - y)
Notice that the -1 factor is separated out::
sage: f.factor_list()
[(x^2 + x*y + y^2, 1), (x - y, 1)]
We factor a fairly straightforward expression::
sage: factor(-8*y - 4*x + z^2*(2*y + x)).factor_list()
[(x + 2*y, 1), (z + 2, 1), (z - 2, 1)]
A more complicated example::
sage: var('x, u, v')
(x, u, v)
sage: f = expand((2*u*v^2-v^2-4*u^3)^2 * (-u)^3 * (x-sin(x))^3)
sage: f.factor()
-(4*u^3 - 2*u*v^2 + v^2)^2*u^3*(x - sin(x))^3
sage: g = f.factor_list(); g
[(4*u^3 - 2*u*v^2 + v^2, 2), (u, 3), (x - sin(x), 3), (-1, 1)]
This function also works for quotients::
sage: f = -1 - 2*x - x^2 + y^2 + 2*x*y^2 + x^2*y^2
sage: g = f/(36*(1 + 2*y + y^2)); g
1/36*(x^2*y^2 + 2*x*y^2 - x^2 + y^2 - 2*x - 1)/(y^2 + 2*y + 1)
sage: g.factor(dontfactor=[x])
1/36*(x^2 + 2*x + 1)*(y - 1)/(y + 1)
sage: g.factor_list(dontfactor=[x])
[(x^2 + 2*x + 1, 1), (y + 1, -1), (y - 1, 1), (1/36, 1)]
This example also illustrates that the exponents do not have to be
integers::
sage: f = x^(2*sin(x)) * (x-1)^(sqrt(2)*x); f
(x - 1)^(sqrt(2)*x)*x^(2*sin(x))
sage: f.factor_list()
[(x - 1, sqrt(2)*x), (x, 2*sin(x))]
"""
return self.factor(dontfactor=dontfactor)._factor_list()
def _factor_list(self):
r"""
Turn an expression already in factored form into a list of (prime,
power) pairs.
This is used, e.g., internally by the :meth:`factor_list`
command.
EXAMPLES::
sage: g = factor(x^3 - 1); g
(x^2 + x + 1)*(x - 1)
sage: v = g._factor_list(); v
[(x^2 + x + 1, 1), (x - 1, 1)]
sage: type(v)
<type 'list'>
"""
op = self.operator()
if op is operator.mul:
return sum([f._factor_list() for f in self.operands()], [])
elif op is operator.pow:
return [tuple(self.operands())]
else:
return [(self, 1)]
def convert(self, target=None):
"""
Calls the convert function in the units package. For symbolic
variables that are not units, this function just returns the
variable.
INPUT:
- ``self`` -- the symbolic expression converting from
- ``target`` -- (default None) the symbolic expression
converting to
OUTPUT:
A symbolic expression.
EXAMPLES::
sage: units.length.foot.convert()
381/1250*meter
sage: units.mass.kilogram.convert(units.mass.pound)
100000000/45359237*pound
We don't get anything new by converting an ordinary symbolic variable::
sage: a = var('a')
sage: a - a.convert()
0
Raises ValueError if self and target are not convertible::
sage: units.mass.kilogram.convert(units.length.foot)
Traceback (most recent call last):
...
ValueError: Incompatible units
sage: (units.length.meter^2).convert(units.length.foot)
Traceback (most recent call last):
...
ValueError: Incompatible units
Recognizes derived unit relationships to base units and other
derived units::
sage: (units.length.foot/units.time.second^2).convert(units.acceleration.galileo)
762/25*galileo
sage: (units.mass.kilogram*units.length.meter/units.time.second^2).convert(units.force.newton)
newton
sage: (units.length.foot^3).convert(units.area.acre*units.length.inch)
1/3630*(acre*inch)
sage: (units.charge.coulomb).convert(units.current.ampere*units.time.second)
(ampere*second)
sage: (units.pressure.pascal*units.si_prefixes.kilo).convert(units.pressure.pounds_per_square_inch)
1290320000000/8896443230521*pounds_per_square_inch
For decimal answers multiply by 1.0::
sage: (units.pressure.pascal*units.si_prefixes.kilo).convert(units.pressure.pounds_per_square_inch)*1.0
0.145037737730209*pounds_per_square_inch
Converting temperatures works as well::
sage: s = 68*units.temperature.fahrenheit
sage: s.convert(units.temperature.celsius)
20*celsius
sage: s.convert()
293.150000000000*kelvin
Trying to multiply temperatures by another unit then converting
raises a ValueError::
sage: wrong = 50*units.temperature.celsius*units.length.foot
sage: wrong.convert()
Traceback (most recent call last):
...
ValueError: Cannot convert
"""
import units
return units.convert(self, target)
def roots(self, x=None, explicit_solutions=True, multiplicities=True, ring=None):
r"""
Returns roots of ``self`` that can be found exactly,
possibly with multiplicities. Not all roots are guaranteed to
be found.
.. warning::
This is *not* a numerical solver - use ``find_root`` to
solve for self == 0 numerically on an interval.
INPUT:
- ``x`` - variable to view the function in terms of
(use default variable if not given)
- ``explicit_solutions`` - bool (default True); require that
roots be explicit rather than implicit
- ``multiplicities`` - bool (default True); when True, return
multiplicities
- ``ring`` - a ring (default None): if not None, convert
self to a polynomial over ring and find roots over ring
OUTPUT:
A list of pairs ``(root, multiplicity)`` or list of roots.
If there are infinitely many roots, e.g., a function like
`\sin(x)`, only one is returned.
EXAMPLES::
sage: var('x, a')
(x, a)
A simple example::
sage: ((x^2-1)^2).roots()
[(-1, 2), (1, 2)]
sage: ((x^2-1)^2).roots(multiplicities=False)
[-1, 1]
A complicated example::
sage: f = expand((x^2 - 1)^3*(x^2 + 1)*(x-a)); f
-a*x^8 + x^9 + 2*a*x^6 - 2*x^7 - 2*a*x^2 + 2*x^3 + a - x
The default variable is `a`, since it is the first in
alphabetical order::
sage: f.roots()
[(x, 1)]
As a polynomial in `a`, `x` is indeed a root::
sage: f.poly(a)
x^9 - 2*x^7 + 2*x^3 - (x^8 - 2*x^6 + 2*x^2 - 1)*a - x
sage: f(a=x)
0
The roots in terms of `x` are what we expect::
sage: f.roots(x)
[(a, 1), (-I, 1), (I, 1), (1, 3), (-1, 3)]
Only one root of `\sin(x) = 0` is given::
sage: f = sin(x)
sage: f.roots(x)
[(0, 1)]
.. note::
It is possible to solve a greater variety of equations
using ``solve()`` and the keyword ``to_poly_solve``,
but only at the price of possibly encountering
approximate solutions. See documentation for f.solve
for more details.
We derive the roots of a general quadratic polynomial::
sage: var('a,b,c,x')
(a, b, c, x)
sage: (a*x^2 + b*x + c).roots(x)
[(-1/2*(b + sqrt(b^2 - 4*a*c))/a, 1), (-1/2*(b - sqrt(b^2 - 4*a*c))/a, 1)]
By default, all the roots are required to be explicit rather than
implicit. To get implicit roots, pass ``explicit_solutions=False``
to ``.roots()`` ::
sage: var('x')
x
sage: f = x^(1/9) + (2^(8/9) - 2^(1/9))*(x - 1) - x^(8/9)
sage: f.roots()
Traceback (most recent call last):
...
RuntimeError: no explicit roots found
sage: f.roots(explicit_solutions=False)
[((2^(8/9) + x^(8/9) - 2^(1/9) - x^(1/9))/(2^(8/9) - 2^(1/9)), 1)]
Another example, but involving a degree 5 poly whose roots don't
get computed explicitly::
sage: f = x^5 + x^3 + 17*x + 1
sage: f.roots()
Traceback (most recent call last):
...
RuntimeError: no explicit roots found
sage: f.roots(explicit_solutions=False)
[(x^5 + x^3 + 17*x + 1, 1)]
sage: f.roots(explicit_solutions=False, multiplicities=False)
[x^5 + x^3 + 17*x + 1]
Now let's find some roots over different rings::
sage: f.roots(ring=CC)
[(-0.0588115223184..., 1), (-1.331099917875... - 1.52241655183732*I, 1), (-1.331099917875... + 1.52241655183732*I, 1), (1.36050567903502 - 1.51880872209965*I, 1), (1.36050567903502 + 1.51880872209965*I, 1)]
sage: (2.5*f).roots(ring=RR)
[(-0.058811522318449..., 1)]
sage: f.roots(ring=CC, multiplicities=False)
[-0.05881152231844..., -1.331099917875... - 1.52241655183732*I, -1.331099917875... + 1.52241655183732*I, 1.36050567903502 - 1.51880872209965*I, 1.36050567903502 + 1.51880872209965*I]
sage: f.roots(ring=QQ)
[]
sage: f.roots(ring=QQbar, multiplicities=False)
[-0.05881152231844944?, -1.331099917875796? - 1.522416551837318?*I, -1.331099917875796? + 1.522416551837318?*I, 1.360505679035020? - 1.518808722099650?*I, 1.360505679035020? + 1.518808722099650?*I]
Root finding over finite fields::
sage: f.roots(ring=GF(7^2, 'a'))
[(3, 1), (4*a + 6, 2), (3*a + 3, 2)]
TESTS::
sage: (sqrt(3) * f).roots(ring=QQ)
Traceback (most recent call last):
...
TypeError: unable to convert sqrt(3) to a rational
Check if #9538 is fixed::
sage: var('f6,f5,f4,x')
(f6, f5, f4, x)
sage: e=15*f6*x^2 + 5*f5*x + f4
sage: res = e.roots(x); res
[(-1/30*(5*f5 + sqrt(25*f5^2 - 60*f4*f6))/f6, 1), (-1/30*(5*f5 - sqrt(25*f5^2 - 60*f4*f6))/f6, 1)]
sage: e.subs(x=res[0][0]).is_zero()
True
"""
if x is None:
x = self.default_variable()
if ring is not None:
p = self.polynomial(ring)
return p.roots(ring=ring, multiplicities=multiplicities)
S, mul = self.solve(x, multiplicities=True, explicit_solutions=explicit_solutions)
if len(mul) == 0 and explicit_solutions:
raise RuntimeError, "no explicit roots found"
else:
rt_muls = [(S[i].rhs(), mul[i]) for i in range(len(mul))]
if multiplicities:
return rt_muls
else:
return [ rt for rt, mul in rt_muls ]
def solve(self, x, multiplicities=False, solution_dict=False, explicit_solutions=False, to_poly_solve=False):
r"""
Analytically solve the equation ``self == 0`` or an univarite
inequality for the variable `x`.
.. warning::
This is not a numerical solver - use ``find_root`` to solve
for self == 0 numerically on an interval.
INPUT:
- ``x`` - variable to solve for
- ``multiplicities`` - bool (default: False); if True,
return corresponding multiplicities. This keyword is
incompatible with ``to_poly_solve=True`` and does not make
any sense when solving an inequality.
- ``solution_dict`` - bool (default: False); if True or non-zero,
return a list of dictionaries containing solutions. Not used
when solving an inequality.
- ``explicit_solutions`` - bool (default: False); require that
all roots be explicit rather than implicit. Not used
when solving an inequality.
- ``to_poly_solve`` - bool (default: False) or string; use
Maxima's ``to_poly_solver`` package to search for more possible
solutions, but possibly encounter approximate solutions.
This keyword is incompatible with ``multiplicities=True``
and is not used when solving an inequality. Setting ``to_poly_solve``
to 'force' (string) omits Maxima's solve command (usefull when
some solution of trigonometric equations are lost).
EXAMPLES::
sage: z = var('z')
sage: (z^5 - 1).solve(z)
[z == e^(2/5*I*pi), z == e^(4/5*I*pi), z == e^(-4/5*I*pi), z == e^(-2/5*I*pi), z == 1]
sage: solve((z^3-1)^3, z, multiplicities=True)
([z == 1/2*I*sqrt(3) - 1/2, z == -1/2*I*sqrt(3) - 1/2, z == 1], [3, 3, 3])
A simple example to show use of the keyword
``multiplicities``::
sage: ((x^2-1)^2).solve(x)
[x == -1, x == 1]
sage: ((x^2-1)^2).solve(x,multiplicities=True)
([x == -1, x == 1], [2, 2])
sage: ((x^2-1)^2).solve(x,multiplicities=True,to_poly_solve=True)
Traceback (most recent call last):
...
NotImplementedError: to_poly_solve does not return multiplicities
Here is how the ``explicit_solutions`` keyword functions::
sage: solve(sin(x)==x,x)
[x == sin(x)]
sage: solve(sin(x)==x,x,explicit_solutions=True)
[]
sage: solve(x*sin(x)==x^2,x)
[x == 0, x == sin(x)]
sage: solve(x*sin(x)==x^2,x,explicit_solutions=True)
[x == 0]
The following examples show use of the keyword ``to_poly_solve``::
sage: solve(abs(1-abs(1-x)) == 10, x)
[abs(abs(x - 1) - 1) == 10]
sage: solve(abs(1-abs(1-x)) == 10, x, to_poly_solve=True)
[x == -10, x == 12]
sage: var('Q')
Q
sage: solve(Q*sqrt(Q^2 + 2) - 1, Q)
[Q == 1/sqrt(Q^2 + 2)]
sage: solve(Q*sqrt(Q^2 + 2) - 1, Q, to_poly_solve=True)
[Q == 1/sqrt(-sqrt(2) + 1), Q == 1/sqrt(sqrt(2) + 1)]
In some cases there may be infinitely many solutions indexed
by a dummy variable. If it begins with ``z``, it is implicitly
assumed to be an integer, a real if with ``r``, and so on::
sage: solve( sin(x)==cos(x), x, to_poly_solve=True)
[x == 1/4*pi + pi*z...]
An effort is made to only return solutions that satisfy the current assumptions::
sage: solve(x^2==4, x)
[x == -2, x == 2]
sage: assume(x<0)
sage: solve(x^2==4, x)
[x == -2]
sage: solve((x^2-4)^2 == 0, x, multiplicities=True)
([x == -2], [2])
sage: solve(x^2==2, x)
[x == -sqrt(2)]
sage: assume(x, 'rational')
sage: solve(x^2 == 2, x)
[]
sage: solve(x^2==2-z, x)
[x == -sqrt(-z + 2)]
sage: solve((x-z)^2==2, x)
[x == z - sqrt(2), x == z + sqrt(2)]
There is still room for improvement::
sage: assume(x, 'integer')
sage: assume(z, 'integer')
sage: solve((x-z)^2==2, x)
[x == z - sqrt(2), x == z + sqrt(2)]
sage: forget()
In some cases it may be worthwhile to directly use to_poly_solve,
if one suspects some answers are being missed::
sage: solve(cos(x)==0,x)
[x == 1/2*pi]
sage: solve(cos(x)==0,x,to_poly_solve=True)
[x == 1/2*pi]
sage: from sage.calculus.calculus import maxima
sage: sol = maxima(cos(x)==0).to_poly_solve(x)
sage: sol.sage()
[[x == -1/2*pi + 2*pi*z...], [x == 1/2*pi + 2*pi*z...]]
If a returned unsolved expression has a denominator, but the
original one did not, this may also be true::
sage: solve(cos(x) * sin(x) == 1/2, x, to_poly_solve=True)
[sin(x) == 1/2/cos(x)]
sage: from sage.calculus.calculus import maxima
sage: sol = maxima(cos(x) * sin(x) == 1/2).to_poly_solve(x)
sage: sol.sage()
[[x == 1/4*pi + pi*z...]]
Some basic inequalities can be also solved::
sage: x,y=var('x,y'); (ln(x)-ln(y)>0).solve(x)
[[log(x) - log(y) > 0]]
::
sage: x,y=var('x,y'); (ln(x)>ln(y)).solve(x) # not tested - output depends on system
[[0 < y, y < x, 0 < x]]
[[y < x, 0 < y]]
TESTS:
:trac:`7325` (solving inequalities)::
sage: (x^2>1).solve(x)
[[x < -1], [x > 1]]
Catch error message from Maxima::
sage: solve(acot(x),x)
[]
::
sage: solve(acot(x),x,to_poly_solve=True)
[]
:trac:`7491` fixed::
sage: y=var('y')
sage: solve(y==y,y)
[y == r1]
sage: solve(y==y,y,multiplicities=True)
([y == r1], [])
sage: from sage.symbolic.assumptions import GenericDeclaration
sage: GenericDeclaration(x, 'rational').assume()
sage: solve(x^2 == 2, x)
[]
sage: forget()
:trac:`8390` fixed::
sage: solve(sin(x)==1/2,x)
[x == 1/6*pi]
::
sage: solve(sin(x)==1/2,x,to_poly_solve=True)
[x == 1/6*pi]
::
sage: solve(sin(x)==1/2,x,to_poly_solve='force')
[x == 5/6*pi + 2*pi*z..., x == 1/6*pi + 2*pi*z...]
:trac:`11618` fixed::
sage: g(x)=0
sage: solve(g(x)==0,x,solution_dict=True)
[{x: r1}]
:trac:`13286` fixed::
sage: solve([x-4], [x])
[x == 4]
:trac:`13645`: fixed::
sage: x.solve((1,2))
Traceback (most recent call last):
...
TypeError: 1 is not a valid variable.
"""
import operator
cdef Expression ex
if is_a_relational(self._gobj):
if self.operator() is not operator.eq:
from sage.symbolic.relation import solve_ineq
try:
return(solve_ineq(self))
except Exception:
pass
try:
return(solve_ineq([self]))
except Exception:
raise NotImplementedError, "solving only implemented for equalities and few special inequalities, see solve_ineq"
ex = self
else:
ex = (self == 0)
if multiplicities and to_poly_solve:
raise NotImplementedError, "to_poly_solve does not return multiplicities"
if isinstance(x, (list, tuple)):
x = x[0]
if x is None:
v = ex.variables()
if len(v) == 0:
if multiplicities:
return [], []
else:
return []
x = v[0]
if not isinstance(x, Expression):
raise TypeError("%s is not a valid variable."%repr(x))
m = ex._maxima_()
P = m.parent()
if explicit_solutions:
P.eval('solveexplicit: true')
try:
if to_poly_solve != 'force':
s = m.solve(x).str()
else:
s = str([])
except TypeError, mess:
if "Error executing code in Maxima" in str(mess):
s = str([])
else:
raise
if explicit_solutions:
P.eval('solveexplicit: false')
if s == 'all':
if solution_dict:
ans = [ {x: self.parent().var('r1')} ]
else:
ans = [x == self.parent().var('r1')]
if multiplicities:
return ans,[]
else:
return ans
from sage.symbolic.relation import string_to_list_of_solutions
X = string_to_list_of_solutions(s)
if multiplicities:
if len(X) == 0:
return X, []
else:
ret_multiplicities = [int(e) for e in str(P.get('multiplicities'))[1:-1].split(',')]
if to_poly_solve and not multiplicities:
if len(X)==0:
try:
s = m.to_poly_solve(x)
T = string_to_list_of_solutions(repr(s))
X = [t[0] for t in T]
except Exception:
X = []
for eq in X:
if repr(x) in map(repr, eq.rhs().variables()) or \
repr(x) in repr(eq.lhs()):
try:
Y = eq._maxima_().to_poly_solve(x).sage()
X.remove(eq)
X.extend([y[0] for y in Y])
except TypeError, mess:
if "Error executing code in Maxima" in str(mess) or \
"unable to make sense of Maxima expression" in\
str(mess):
if explicit_solutions:
X.remove(eq)
else:
pass
else:
raise
from sage.symbolic.assumptions import assumptions
to_check = assumptions()
if to_check:
for ix, soln in reversed(list(enumerate(X))):
if soln.lhs().is_symbol():
if any([a.contradicts(soln) for a in to_check]):
del X[ix]
if multiplicities:
del ret_multiplicities[ix]
continue
if solution_dict:
X=[dict([[sol.left(),sol.right()]]) for sol in X]
if multiplicities:
return X, ret_multiplicities
else:
return X
def find_root(self, a, b, var=None, xtol=10e-13, rtol=4.5e-16, maxiter=100, full_output=False):
"""
Numerically find a root of self on the closed interval [a,b] (or
[b,a]) if possible, where self is a function in the one variable.
Note: this function only works in fixed (machine) precision, it is not
possible to get arbitrary precision approximations with it.
INPUT:
- ``a, b`` - endpoints of the interval
- ``var`` - optional variable
- ``xtol, rtol`` - the routine converges when a root
is known to lie within xtol of the value return. Should be >= 0. The
routine modifies this to take into account the relative precision
of doubles.
- ``maxiter`` - integer; if convergence is not
achieved in maxiter iterations, an error is raised. Must be >= 0.
- ``full_output`` - bool (default: False), if True,
also return object that contains information about convergence.
EXAMPLES:
Note that in this example both f(-2) and f(3) are positive,
yet we still find a root in that interval::
sage: f = x^2 - 1
sage: f.find_root(-2, 3)
1.0
sage: f.find_root(-2, 3, x)
1.0
sage: z, result = f.find_root(-2, 3, full_output=True)
sage: result.converged
True
sage: result.flag
'converged'
sage: result.function_calls
11
sage: result.iterations
10
sage: result.root
1.0
More examples::
sage: (sin(x) + exp(x)).find_root(-10, 10)
-0.588532743981862...
sage: sin(x).find_root(-1,1)
0.0
sage: (1/tan(x)).find_root(3,3.5)
3.1415926535...
An example with a square root::
sage: f = 1 + x + sqrt(x+2); f.find_root(-2,10)
-1.618033988749895
Some examples that Ted Kosan came up with::
sage: t = var('t')
sage: v = 0.004*(9600*e^(-(1200*t)) - 2400*e^(-(300*t)))
sage: v.find_root(0, 0.002)
0.001540327067911417...
With this expression, we can see there is a
zero very close to the origin::
sage: a = .004*(8*e^(-(300*t)) - 8*e^(-(1200*t)))*(720000*e^(-(300*t)) - 11520000*e^(-(1200*t))) +.004*(9600*e^(-(1200*t)) - 2400*e^(-(300*t)))^2
sage: show(plot(a, 0, .002), xmin=0, xmax=.002)
It is easy to approximate with ``find_root``::
sage: a.find_root(0,0.002)
0.0004110514049349...
Using solve takes more effort, and even then gives
only a solution with free (integer) variables::
sage: a.solve(t)
[]
sage: b = a.simplify_radical(); b
-23040*(-2.0*e^(1800*t) + 25.0*e^(900*t) - 32.0)*e^(-2400*t)
sage: b.solve(t)
[]
sage: b.solve(t, to_poly_solve=True)
[t == 1/450*I*pi*z... + 1/900*log(3/4*sqrt(41) + 25/4), t == 1/450*I*pi*z... + 1/900*log(-3/4*sqrt(41) + 25/4)]
sage: n(1/900*log(-3/4*sqrt(41) + 25/4))
0.000411051404934985
We illustrate that root finding is only implemented in one
dimension::
sage: x, y = var('x,y')
sage: (x-y).find_root(-2,2)
Traceback (most recent call last):
...
NotImplementedError: root finding currently only implemented in 1 dimension.
TESTS:
Test the special case that failed for the first attempt to fix
#3980::
sage: t = var('t')
sage: find_root(1/t - x,0,2)
Traceback (most recent call last):
...
NotImplementedError: root finding currently only implemented in 1 dimension.
"""
if is_a_relational(self._gobj) and self.operator() is not operator.eq:
raise ValueError, "Symbolic equation must be an equality."
from sage.numerical.optimize import find_root
if self.number_of_arguments() == 0:
if bool(self == 0):
return a
else:
raise RuntimeError, "no zero in the interval, since constant expression is not 0."
elif self.number_of_arguments() == 1:
f = self._fast_float_(self.default_variable())
return find_root(f, a=a, b=b, xtol=xtol,
rtol=rtol,maxiter=maxiter,
full_output=full_output)
else:
raise NotImplementedError, "root finding currently only implemented in 1 dimension."
def find_local_maximum(self, a, b, var=None, tol=1.48e-08, maxfun=500):
r"""
Numerically find a local maximum of the expression ``self``
on the interval [a,b] (or [b,a]) along with the point at which the
maximum is attained.
See the documentation for
:func:`find_local_minimum` for more details.
EXAMPLES::
sage: f = x*cos(x)
sage: f.find_local_maximum(0,5)
(0.5610963381910451, 0.8603335890...)
sage: f.find_local_maximum(0,5, tol=0.1, maxfun=10)
(0.561090323458081..., 0.857926501456...)
"""
minval, x = (-self).find_local_minimum(a, b, var=var, tol=tol,
maxfun=maxfun)
return -minval, x
def find_local_minimum(self, a, b, var=None, tol=1.48e-08, maxfun=500):
r"""
Numerically find a local minimum of the expression ``self``
on the interval [a,b] (or [b,a]) and the point at which it attains
that minimum. Note that ``self`` must be a function of
(at most) one variable.
INPUT:
- ``var`` - variable (default: first variable in
self)
- ``a,b`` - endpoints of interval on which to minimize
self.
- ``tol`` - the convergence tolerance
- ``maxfun`` - maximum function evaluations
OUTPUT:
A tuple ``(minval, x)``, where
- ``minval`` -- float. The minimum value that self takes on in
the interval ``[a,b]``.
- ``x`` -- float. The point at which self takes on the minimum
value.
EXAMPLES::
sage: f = x*cos(x)
sage: f.find_local_minimum(1, 5)
(-3.288371395590..., 3.4256184695...)
sage: f.find_local_minimum(1, 5, tol=1e-3)
(-3.288371361890..., 3.4257507903...)
sage: f.find_local_minimum(1, 5, tol=1e-2, maxfun=10)
(-3.288370845983..., 3.4250840220...)
sage: show(f.plot(0, 20))
sage: f.find_local_minimum(1, 15)
(-9.477294259479..., 9.5293344109...)
ALGORITHM:
Uses :func:`sage.numerical.optimize.find_local_minimum`.
AUTHORS:
- William Stein (2007-12-07)
"""
from sage.numerical.optimize import find_local_minimum
if var is None:
var = self.default_variable()
return find_local_minimum(self._fast_float_(var),
a=a, b=b, tol=tol, maxfun=maxfun )
find_maximum_on_interval = deprecated_function_alias(2607, find_local_maximum)
find_minimum_on_interval = deprecated_function_alias(2607, find_local_minimum)
def _fast_float_(self, *vars):
"""
Returns an object which provides fast floating point
evaluation of this symbolic expression.
See :mod:`sage.ext.fast_eval` for more information.
EXAMPLES::
sage: f = sqrt(x+1)
sage: ff = f._fast_float_('x')
sage: ff(1.0)
1.4142135623730951
sage: type(_)
<type 'float'>
"""
from sage.symbolic.expression_conversions import fast_float
return fast_float(self, *vars)
def _fast_callable_(self, etb):
"""
Given an ExpressionTreeBuilder *etb*, return an Expression representing
this symbolic expression.
EXAMPLES::
sage: from sage.ext.fast_callable import ExpressionTreeBuilder
sage: etb = ExpressionTreeBuilder(vars=['x','y'])
sage: x,y = var('x,y')
sage: f = y+2*x^2
sage: f._fast_callable_(etb)
add(mul(ipow(v_0, 2), 2), v_1)
"""
from sage.symbolic.expression_conversions import fast_callable
return fast_callable(self, etb)
def show(self):
"""
Show this symbolic expression, i.e., typeset it nicely.
EXAMPLES::
sage: (x^2 + 1).show()
x^{2} + 1
"""
from sage.misc.functional import _do_show
return _do_show(self)
def plot(self, *args, **kwds):
"""
Plot a symbolic expression. All arguments are passed onto the standard plot command.
EXAMPLES:
This displays a straight line::
sage: sin(2).plot((x,0,3))
This draws a red oscillatory curve::
sage: sin(x^2).plot((x,0,2*pi), rgbcolor=(1,0,0))
Another plot using the variable theta::
sage: var('theta')
theta
sage: (cos(theta) - erf(theta)).plot((theta,-2*pi,2*pi))
A very thick green plot with a frame::
sage: sin(x).plot((x,-4*pi, 4*pi), thickness=20, rgbcolor=(0,0.7,0)).show(frame=True)
You can embed 2d plots in 3d space as follows::
sage: plot(sin(x^2), (x,-pi, pi), thickness=2).plot3d(z = 1)
A more complicated family::
sage: G = sum([plot(sin(n*x), (x,-2*pi, 2*pi)).plot3d(z=n) for n in [0,0.1,..1]])
sage: G.show(frame_aspect_ratio=[1,1,1/2]) # long time (5s on sage.math, 2012)
A plot involving the floor function::
sage: plot(1.0 - x * floor(1/x), (x,0.00001,1.0))
Sage used to allow symbolic functions with "no arguments";
this no longer works::
sage: plot(2*sin, -4, 4)
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for '*': 'Integer Ring' and '<class 'sage.functions.trig.Function_sin'>'
You should evaluate the function first::
sage: plot(2*sin(x), -4, 4)
TESTS::
sage: f(x) = x*(1 - x)
sage: plot(f,0,1)
"""
from sage.symbolic.callable import is_CallableSymbolicExpression
from sage.symbolic.ring import is_SymbolicVariable
from sage.plot.plot import plot
if kwds.has_key('param'):
param = kwds['param']
else:
param = None
for i, arg in enumerate(args):
if is_SymbolicVariable(arg):
param = arg
args = args[:i] + args[i+1:]
break
if param is None:
if is_CallableSymbolicExpression(self):
A = self.arguments()
if len(A) == 0:
raise ValueError, "function has no input arguments"
else:
param = A[0]
f = self._plot_fast_callable(param)
else:
A = self.variables()
if len(A) == 0:
n = self.number_of_arguments()
f = self._plot_fast_callable()
else:
param = A[0]
try:
f = self._plot_fast_callable(param)
except NotImplementedError:
return self.function(param)
else:
try:
f = self._plot_fast_callable(param)
except NotImplementedError:
return self.function(param)
return plot(f, *args, **kwds)
def _plot_fast_callable(self, *vars):
"""
Internal function used for creating a fast callable version of this
symbolic expression for plotting.
EXAMPLES::
sage: s = abs((1+I*x)^4); s
abs((I*x + 1)^4)
sage: s._plot_fast_callable(x)
<sage.ext.interpreters.wrapper_py.Wrapper_py object at ...>
sage: s._plot_fast_callable(x)(10)
10201
sage: abs((I*10+1)^4)
10201
sage: plot(s)
"""
try:
return self._fast_float_(*vars)
except (TypeError, NotImplementedError):
from sage.ext.fast_callable import fast_callable
return fast_callable(self, vars=vars)
def sum(self, *args, **kwds):
r"""
Returns the symbolic sum
`\sum_{v = a}^b self`
with respect to the variable `v` with endpoints
`a` and `b`.
INPUT:
- ``v`` - a variable or variable name
- ``a`` - lower endpoint of the sum
- ``b`` - upper endpoint of the sum
- ``algorithm`` - (default: 'maxima') one of
- 'maxima' - use Maxima (the default)
- 'maple' - (optional) use Maple
- 'mathematica' - (optional) use Mathematica
- 'giac' - (optional) use Giac
EXAMPLES::
sage: k, n = var('k,n')
sage: k.sum(k, 1, n).factor()
1/2*(n + 1)*n
::
sage: (1/k^4).sum(k, 1, oo)
1/90*pi^4
::
sage: (1/k^5).sum(k, 1, oo)
zeta(5)
A well known binomial identity::
sage: assume(n>=0)
sage: binomial(n,k).sum(k, 0, n)
2^n
And some truncations thereof::
sage: binomial(n,k).sum(k,1,n)
2^n - 1
sage: binomial(n,k).sum(k,2,n)
2^n - n - 1
sage: binomial(n,k).sum(k,0,n-1)
2^n - 1
sage: binomial(n,k).sum(k,1,n-1)
2^n - 2
The binomial theorem::
sage: x, y = var('x, y')
sage: (binomial(n,k) * x^k * y^(n-k)).sum(k, 0, n)
(x + y)^n
::
sage: (k * binomial(n, k)).sum(k, 1, n)
2^(n - 1)*n
::
sage: ((-1)^k*binomial(n,k)).sum(k, 0, n)
0
::
sage: (2^(-k)/(k*(k+1))).sum(k, 1, oo)
-log(2) + 1
Summing a hypergeometric term::
sage: (binomial(n, k) * factorial(k) / factorial(n+1+k)).sum(k, 0, n)
1/2*sqrt(pi)/factorial(n + 1/2)
We check a well known identity::
sage: bool((k^3).sum(k, 1, n) == k.sum(k, 1, n)^2)
True
A geometric sum::
sage: a, q = var('a, q')
sage: (a*q^k).sum(k, 0, n)
(a*q^(n + 1) - a)/(q - 1)
The geometric series::
sage: assume(abs(q) < 1)
sage: (a*q^k).sum(k, 0, oo)
-a/(q - 1)
A divergent geometric series. Don't forget
to forget your assumptions::
sage: forget()
sage: assume(q > 1)
sage: (a*q^k).sum(k, 0, oo)
Traceback (most recent call last):
...
ValueError: Sum is divergent.
This summation only Mathematica can perform::
sage: (1/(1+k^2)).sum(k, -oo, oo, algorithm = 'mathematica') # optional - mathematica
pi*coth(pi)
Use Giac to perform this summation::
sage: (sum(1/(1+k^2), k, -oo, oo, algorithm = 'giac')).factor() # optional - giac
(e^(2*pi) + 1)*pi/((e^pi - 1)*(e^pi + 1))
Use Maple as a backend for summation::
sage: (binomial(n,k)*x^k).sum(k, 0, n, algorithm = 'maple') # optional - maple
(x + 1)^n
Check that the sum in #10682 is done right::
sage: sum(binomial(n,k)*k^2, k, 2, n)
1/4*(n^2 + n)*2^n - n
.. note::
#. Sage can currently only understand a subset of the output of Maxima, Maple and
Mathematica, so even if the chosen backend can perform the summation the
result might not be convertable into a Sage expression.
"""
from sage.calculus.calculus import symbolic_sum
return symbolic_sum(self, *args, **kwds)
def integral(self, *args, **kwds):
"""
Compute the integral of self. Please see
:func:`sage.symbolic.integration.integral.integrate` for more details.
EXAMPLES::
sage: sin(x).integral(x,0,3)
-cos(3) + 1
sage: sin(x).integral(x)
-cos(x)
TESTS:
We check that :trac:`12438` is resolved::
sage: f(x) = x; f
x |--> x
sage: integral(f, x)
x |--> 1/2*x^2
sage: integral(f, x, 0, 1)
1/2
sage: f(x, y) = x + y
sage: f
(x, y) |--> x + y
sage: integral(f, y, 0, 1)
x |--> x + 1/2
sage: integral(f, x, 0, 1)
y |--> y + 1/2
sage: _(3)
7/2
sage: var("z")
z
sage: integral(f, z, 0, 2)
(x, y) |--> 2*x + 2*y
sage: integral(f, z)
(x, y) |--> (x + y)*z
"""
from sage.symbolic.integration.integral import \
integral, _normalize_integral_input
from sage.symbolic.callable import \
CallableSymbolicExpressionRing, is_CallableSymbolicExpressionRing
R = self._parent
if is_CallableSymbolicExpressionRing(R):
f = ring.SR(self)
f, v, a, b = _normalize_integral_input(f, *args)
if a is not None and v in R.arguments():
arguments = list(R.arguments())
arguments.remove(v)
if arguments:
arguments = tuple(arguments)
R = CallableSymbolicExpressionRing(arguments, check=False)
else:
R = ring.SR
return R(integral(f, v, a, b, **kwds))
return integral(self, *args, **kwds)
integrate = integral
def nintegral(self, *args, **kwds):
"""
Compute the numerical integral of self. Please see
:obj:`sage.calculus.calculus.nintegral` for more details.
EXAMPLES::
sage: sin(x).nintegral(x,0,3)
(1.989992496600..., 2.209335488557...e-14, 21, 0)
"""
from sage.calculus.calculus import nintegral
return nintegral(self, *args, **kwds)
nintegrate = nintegral
def minpoly(self, *args, **kwds):
"""
Return the minimal polynomial of this symbolic expression.
EXAMPLES::
sage: golden_ratio.minpoly()
x^2 - x - 1
"""
try:
obj = self.pyobject()
return obj.minpoly()
except AttributeError:
pass
except TypeError:
pass
from sage.calculus.calculus import minpoly
return minpoly(self, *args, **kwds)
def limit(self, *args, **kwds):
"""
Return a symbolic limit. See
:obj:`sage.calculus.calculus.limit`
EXAMPLES::
sage: (sin(x)/x).limit(x=0)
1
"""
from sage.calculus.calculus import limit
return limit(self, *args, **kwds)
def laplace(self, t, s):
"""
Return Laplace transform of self. See
:obj:`sage.calculus.calculus.laplace`
EXAMPLES::
sage: var('x,s,z')
(x, s, z)
sage: (z + exp(x)).laplace(x, s)
z/s + 1/(s - 1)
"""
from sage.calculus.calculus import laplace
return laplace(self, t, s)
def inverse_laplace(self, t, s):
"""
Return inverse Laplace transform of self. See
:obj:`sage.calculus.calculus.inverse_laplace`
EXAMPLES::
sage: var('w, m')
(w, m)
sage: f = (1/(w^2+10)).inverse_laplace(w, m); f
1/10*sqrt(10)*sin(sqrt(10)*m)
"""
from sage.calculus.calculus import inverse_laplace
return inverse_laplace(self, t, s)
def add_to_both_sides(self, x):
"""
Returns a relation obtained by adding *x* to both sides of
this relation.
EXAMPLES::
sage: var('x y z')
(x, y, z)
sage: eqn = x^2 + y^2 + z^2 <= 1
sage: eqn.add_to_both_sides(-z^2)
x^2 + y^2 <= -z^2 + 1
sage: eqn.add_to_both_sides(I)
x^2 + y^2 + z^2 + I <= (I + 1)
"""
if not is_a_relational(self._gobj):
raise TypeError, "this expression must be a relation"
return self + x
def subtract_from_both_sides(self, x):
"""
Returns a relation obtained by subtracting *x* from both sides
of this relation.
EXAMPLES::
sage: eqn = x*sin(x)*sqrt(3) + sqrt(2) > cos(sin(x))
sage: eqn.subtract_from_both_sides(sqrt(2))
sqrt(3)*x*sin(x) > -sqrt(2) + cos(sin(x))
sage: eqn.subtract_from_both_sides(cos(sin(x)))
sqrt(3)*x*sin(x) + sqrt(2) - cos(sin(x)) > 0
"""
if not is_a_relational(self._gobj):
raise TypeError, "this expression must be a relation"
return self - x
def multiply_both_sides(self, x, checksign=None):
"""
Returns a relation obtained by multiplying both sides of this
relation by *x*.
.. note::
The *checksign* keyword argument is currently ignored and
is included for backward compatibility reasons only.
EXAMPLES::
sage: var('x,y'); f = x + 3 < y - 2
(x, y)
sage: f.multiply_both_sides(7)
7*x + 21 < 7*y - 14
sage: f.multiply_both_sides(-1/2)
-1/2*x - 3/2 < -1/2*y + 1
sage: f*(-2/3)
-2/3*x - 2 < -2/3*y + 4/3
sage: f*(-pi)
-pi*(x + 3) < -pi*(y - 2)
Since the direction of the inequality never changes when doing
arithmetic with equations, you can multiply or divide the
equation by a quantity with unknown sign::
sage: f*(1+I)
(I + 1)*x + 3*I + 3 < (I + 1)*y - 2*I - 2
sage: f = sqrt(2) + x == y^3
sage: f.multiply_both_sides(I)
I*x + I*sqrt(2) == I*y^3
sage: f.multiply_both_sides(-1)
-x - sqrt(2) == -y^3
Note that the direction of the following inequalities is
not reversed::
sage: (x^3 + 1 > 2*sqrt(3)) * (-1)
-x^3 - 1 > -2*sqrt(3)
sage: (x^3 + 1 >= 2*sqrt(3)) * (-1)
-x^3 - 1 >= -2*sqrt(3)
sage: (x^3 + 1 <= 2*sqrt(3)) * (-1)
-x^3 - 1 <= -2*sqrt(3)
"""
if not is_a_relational(self._gobj):
raise TypeError, "this expression must be a relation"
return self * x
def divide_both_sides(self, x, checksign=None):
"""
Returns a relation obtained by dividing both sides of this
relation by *x*.
.. note::
The *checksign* keyword argument is currently ignored and
is included for backward compatibility reasons only.
EXAMPLES::
sage: theta = var('theta')
sage: eqn = (x^3 + theta < sin(x*theta))
sage: eqn.divide_both_sides(theta, checksign=False)
(x^3 + theta)/theta < sin(theta*x)/theta
sage: eqn.divide_both_sides(theta)
(x^3 + theta)/theta < sin(theta*x)/theta
sage: eqn/theta
(x^3 + theta)/theta < sin(theta*x)/theta
"""
if not is_a_relational(self._gobj):
raise TypeError, "this expression must be a relation"
return self / x
cdef dict dynamic_class_cache = {}
cdef get_dynamic_class_for_function(unsigned serial):
r"""
Create a dynamic class corresponding to the function with given
``serial`` that includes dynamic methods defined by the function.
Dynamic methods can be defined in a subclass ``EvaluationMethods`` in
the function body. These will be available in symbolic expressions
representing evaluations of the said function on some arguments.
EXAMPLES::
sage: from sage.symbolic.function import BuiltinFunction
sage: class TFunc(BuiltinFunction):
....: def __init__(self):
....: BuiltinFunction.__init__(self, 'tfunc', nargs=1)
....:
....: class EvaluationMethods:
....: def argp1(self, x):
....: '''
....: Some documentation about a bogus function.
....: '''
....: return x+1
....:
....: @property
....: def foo(self):
....: return 5
....:
sage: tfunc = TFunc()
sage: e = tfunc(x); e
tfunc(x)
sage: type(e)
<class '__main__.Expression_with_dynamic_methods'>
sage: e.argp1()
x + 1
sage: e.foo
5
sage: x.argp1()
Traceback (most recent call last):
...
AttributeError: 'sage.symbolic.expression.Expression' object has no
attribute 'argp1'
sage: t = (e + 1).op[0]; t
tfunc(x)
sage: t
tfunc(x)
sage: type(t)
<class '__main__.Expression_with_dynamic_methods'>
sage: t.argp1()
x + 1
sage: import sagenb.misc.support as s
sage: s.completions('t.argp', globals(), system='python')
['t.argp1']
sage: t.argp1.__doc__.strip()
'Some documentation about a bogus function.'
Now with two arguments::
sage: class TFunc2(BuiltinFunction):
....: def __init__(self):
....: BuiltinFunction.__init__(self, 'tfunc', nargs=2)
....:
....: class EvaluationMethods:
....: def argsum(self, x, y):
....: return x + y
....:
sage: tfunc2 = TFunc2()
sage: e = tfunc2(x, 1)
sage: e.argsum()
x + 1
"""
cls = dynamic_class_cache.get(serial)
if cls is None:
func_class = get_sfunction_from_serial(serial)
eval_methods = getattr(func_class, 'EvaluationMethods', None)
if eval_methods is not None:
from sage.symbolic.function_factory import eval_on_operands
from sage.structure.parent import getattr_from_other_class
for name in dir(eval_methods):
m = getattr_from_other_class(func_class, eval_methods, name)
if callable(m):
setattr(eval_methods, name, eval_on_operands(m))
cls = dynamic_class('Expression_with_dynamic_methods',
(eval_methods, Expression))
else:
cls = Expression
dynamic_class_cache[serial] = cls
return cls
cdef Expression new_Expression_from_GEx(parent, GEx juice):
cdef Expression nex
cdef unsigned serial
if is_exactly_a_function(juice):
cls = get_dynamic_class_for_function(ex_to_function(juice).get_serial())
else:
cls = Expression
nex = <Expression>PY_NEW(cls)
GEx_construct_ex(&nex._gobj, juice)
nex._parent = parent
return nex
cdef Expression new_Expression_from_pyobject(parent, x):
cdef GEx exp
GEx_construct_pyobject(exp, x)
return new_Expression_from_GEx(parent, exp)
cdef class ExpressionIterator:
cdef Expression _ex
cdef int _ind
cdef int _len
def __iter__(self):
"""
Return this iterator object itself.
EXAMPLES::
sage: x,y,z = var('x,y,z')
sage: i = (x+y).iterator()
sage: iter(i) is i
True
"""
return self
def __next__(self):
"""
Return the next component of the expression.
EXAMPLES::
sage: x,y,z = var('x,y,z')
sage: i = (x+y).iterator()
sage: i.next()
x
"""
cdef GEx ex
if self._ind == self._len:
raise StopIteration
ex = self._ex._gobj.op(self._ind)
self._ind+=1
return new_Expression_from_GEx(self._ex._parent, ex)
cdef inline ExpressionIterator new_ExpIter_from_Expression(Expression ex):
"""
Construct a new iterator over a symbolic expression.
EXAMPLES::
sage: x,y,z = var('x,y,z')
sage: i = (x+y).iterator() #indirect doctest
"""
cdef ExpressionIterator m = <ExpressionIterator>PY_NEW(ExpressionIterator)
m._ex = ex
m._ind = 0
m._len = ex._gobj.nops()
return m
cdef operators compatible_relation(operators lop, operators rop) except <operators>-1:
"""
TESTS::
sage: var('a,b,x,y')
(a, b, x, y)
sage: (x < a) + (y <= b) # indirect doctest
x + y < a + b
sage: (x >= 4) * (y > 7)
x*y > 28
"""
if lop == rop:
return lop
elif lop == not_equal or rop == not_equal:
raise TypeError, "incompatible relations"
elif lop == equal:
return rop
elif rop == equal:
return lop
elif lop in [less, less_or_equal] and rop in [less, less_or_equal]:
return less
elif lop in [greater, greater_or_equal] and rop in [greater, greater_or_equal]:
return greater
else:
raise TypeError, "incompatible relations"
