r"""
Interface to KASH
Sage provides an interface to the KASH computer algebra system,
which is a *free* (as in beer!) but *closed source* program for
algebraic number theory that shares much common code with Magma. To
use KASH, you must install the appropriate optional Sage package by
typing something like "sage -i kash3-linux-2005.11.22" or "sage -i
kash3_osx-2005.11.22". For a list of optional packages type "sage
-optional". If you type one of the above commands, the (about 16MB)
package will be downloaded automatically (you don't have to do
that).
It is not enough to just have KASH installed on your computer. Note
that the KASH Sage package is currently only available for Linux
and OSX. If you need Windows, support contact me
([email protected]).
The KASH interface offers three pieces of functionality:
#. ``kash_console()`` - A function that dumps you into
an interactive command-line KASH session. Alternatively,
type ``!kash`` from the Sage prompt.
#. ``kash(expr)`` - Creation of a Sage object that
wraps a KASH object. This provides a Pythonic interface to KASH.
For example, if ``f=kash.new(10)``, then
``f.Factors()`` returns the prime factorization of
`10` computed using KASH.
#. ``kash.function_name(args ...)`` - Call the
indicated KASH function with the given arguments are return the
result as a KASH object.
#. ``kash.eval(expr)`` - Evaluation of arbitrary KASH
expressions, with the result returned as a string.
Issues
------
For some reason hitting Control-C to interrupt a calculation
doesn't work correctly. (TODO)
Tutorial
--------
The examples in this tutorial require that the optional kash
package be installed.
Basics
~~~~~~
Basic arithmetic is straightforward. First, we obtain the result as
a string.
::
sage: kash.eval('(9 - 7) * (5 + 6)') # optional -- kash
'22'
Next we obtain the result as a new KASH object.
::
sage: a = kash('(9 - 7) * (5 + 6)'); a # optional -- kash
22
sage: a.parent() # optional -- kash
Kash
We can do arithmetic and call functions on KASH objects::
sage: a*a # optional -- kash
484
sage: a.Factorial() # optional -- kash
1124000727777607680000
Integrated Help
~~~~~~~~~~~~~~~
Use the ``kash.help(name)`` command to get help about a
given command. This returns a list of help for each of the
definitions of ``name``. Use ``print
kash.help(name)`` to nicely print out all signatures.
Arithmetic
~~~~~~~~~~
Using the ``kash.new`` command we create Kash objects
on which one can do arithmetic.
::
sage: a = kash(12345) # optional -- kash
sage: b = kash(25) # optional -- kash
sage: a/b # optional -- kash
2469/5
sage: a**b # optional -- kash
1937659030411463935651167391656422626577614411586152317674869233464019922771432158872187137603759765625
Variable assignment
~~~~~~~~~~~~~~~~~~~
Variable assignment using ``kash`` is takes place in
Sage.
::
sage: a = kash('32233') # optional -- kash
sage: a # optional -- kash
32233
In particular, ``a`` is not defined as part of the KASH
session itself.
::
sage: kash.eval('a') # optional -- kash
"Error, the variable 'a' must have a value"
Use ``a.name()`` to get the name of the KASH variable::
sage: a.name() # somewhat random and optional - kash
'sage0'
sage: kash(a.name()) # optional -- kash
32233
Integers and Rationals
~~~~~~~~~~~~~~~~~~~~~~
We illustrate arithmetic with integers and rationals in KASH.
::
sage: F = kash.Factorization(4352) # optional -- kash
sage: F[1] # optional -- kash
<2, 8>
sage: F[2] # optional -- kash
<17, 1>
sage: F # optional -- kash
[ <2, 8>, <17, 1> ], extended by:
ext1 := 1,
ext2 := Unassign
.. note::
For some very large numbers KASH's integer factorization seems much
faster than PARI's (which is the default in Sage).
::
sage: kash.GCD(15,25) # optional -- kash
5
sage: kash.LCM(15,25) # optional -- kash
75
sage: kash.Div(25,15) # optional -- kash
1
sage: kash(17) % kash(5) # optional -- kash
2
sage: kash.IsPrime(10007) # optional -- kash
TRUE
sage: kash.IsPrime(2005) # optional -- kash
FALSE
sage: kash.NextPrime(10007) # optional -- kash
10009
Real and Complex Numbers
~~~~~~~~~~~~~~~~~~~~~~~~
::
sage: kash.Precision() # optional -- kash
30
sage: kash('R') # optional -- kash
Real field of precision 30
sage: kash.Precision(40) # optional -- kash
40
sage: kash('R') # optional -- kash
Real field of precision 40
sage: z = kash('1 + 2*I') # optional -- kash
sage: z # optional -- kash
1.000000000000000000000000000000000000000 + 2.000000000000000000000000000000000000000*I
sage: z*z # optional -- kash
-3.000000000000000000000000000000000000000 + 4.000000000000000000000000000000000000000*I
sage: kash.Cos('1.24') # optional -- kash
0.3247962844387762365776934156973803996992
sage: kash('1.24').Cos() # optional -- kash
0.3247962844387762365776934156973803996992
sage: kash.Exp('1.24') # optional -- kash
3.455613464762675598057615494121998175400
sage: kash.Precision(30) # optional -- kash
30
sage: kash.Log('3+4*I') # optional -- kash
1.60943791243410037460075933323 + 0.927295218001612232428512462922*I
sage: kash.Log('I') # optional -- kash
1.57079632679489661923132169164*I
sage: kash.Sqrt(4) # optional -- kash
2.00000000000000000000000000000
sage: kash.Sqrt(2) # optional -- kash
1.41421356237309504880168872421
sage: kash.Floor('9/5') # optional -- kash
1
sage: kash.Floor('3/5') # optional -- kash
0
sage: x_c = kash('3+I') # optional -- kash
sage: x_c.Argument() # optional -- kash
0.321750554396642193401404614359
sage: x_c.Imaginary() # optional -- kash
1.00000000000000000000000000000
Lists
~~~~~
Note that list appends are completely different in KASH than in
Python. Use underscore after the function name for the mutation
version.
::
sage: v = kash([1,2,3]); v # optional -- kash
[ 1, 2, 3 ]
sage: v[1] # optional -- kash
1
sage: v[3] # optional -- kash
3
sage: v.Append([5]) # optional -- kash
[ 1, 2, 3, 5 ]
sage: v # optional -- kash
[ 1, 2, 3 ]
sage: v.Append_([5, 6]) # optional -- kash
SUCCESS
sage: v # optional -- kash
[ 1, 2, 3, 5, 6 ]
sage: v.Add(5) # optional -- kash
[ 1, 2, 3, 5, 6, 5 ]
sage: v # optional -- kash
[ 1, 2, 3, 5, 6 ]
sage: v.Add_(5) # optional -- kash
SUCCESS
sage: v # optional -- kash
[ 1, 2, 3, 5, 6, 5 ]
The ``Apply`` command applies a function to each
element of a list.
::
sage: L = kash([1,2,3,4]) # optional -- kash
sage: L.Apply('i -> 3*i') # optional -- kash
[ 3, 6, 9, 12 ]
sage: L # optional -- kash
[ 1, 2, 3, 4 ]
sage: L.Apply('IsEven') # optional -- kash
[ FALSE, TRUE, FALSE, TRUE ]
sage: L # optional -- kash
[ 1, 2, 3, 4 ]
Ranges
~~~~~~
the following are examples of ranges.
::
sage: L = kash('[1..10]') # optional -- kash
sage: L # optional -- kash
[ 1 .. 10 ]
sage: L = kash('[2,4..100]') # optional -- kash
sage: L # optional -- kash
[ 2, 4 .. 100 ]
Sequences
~~~~~~~~~
Tuples
~~~~~~
Polynomials
~~~~~~~~~~~
::
sage: f = kash('X^3 + X + 1') # optional -- kash
sage: f + f # optional -- kash
2*X^3 + 2*X + 2
sage: f * f # optional -- kash
X^6 + 2*X^4 + 2*X^3 + X^2 + 2*X + 1
sage: f.Evaluate(10) # optional -- kash
1011
sage: Qx = kash.PolynomialAlgebra('Q') # optional -- kash
sage: Qx.gen(1)**5 + kash('7/3') # sage1 below somewhat random; optional -- kash
sage1.1^5 + 7/3
Number Fields
~~~~~~~~~~~~~
We create an equation order.
::
sage: f = kash('X^5 + 4*X^4 - 56*X^2 -16*X + 192') # optional -- kash
sage: OK = f.EquationOrder() # optional -- kash
sage: OK # optional -- kash
Equation Order with defining polynomial X^5 + 4*X^4 - 56*X^2 - 16*X + 192 over Z
::
sage: f = kash('X^5 + 4*X^4 - 56*X^2 -16*X + 192') # optional -- kash
sage: O = f.EquationOrder() # optional -- kash
sage: a = O.gen(2) # optional -- kash
sage: a # optional -- kash
[0, 1, 0, 0, 0]
sage: O.Basis() # output somewhat random; optional -- kash
[
_NG.1,
_NG.2,
_NG.3,
_NG.4,
_NG.5
]
sage: O.Discriminant() # optional -- kash
1364202618880
sage: O.MaximalOrder() # name sage2 below somewhat random; optional -- kash
Maximal Order of sage2
sage: O = kash.MaximalOrder('X^3 - 77') # optional -- kash
sage: I = O.Ideal(5,[2, 1, 0]) # optional -- kash
sage: I # name sage14 below random; optional -- kash
Ideal of sage14
Two element generators:
[5, 0, 0]
[2, 1, 0]
sage: F = I.Factorisation() # optional -- kash
sage: F # name sage14 random; optional -- kash
[
<Prime Ideal of sage14
Two element generators:
[5, 0, 0]
[2, 1, 0], 1>
]
Determining whether an ideal is principal.
::
sage: I.IsPrincipal() # optional -- kash
FALSE, extended by:
ext1 := Unassign
Computation of class groups and unit groups::
sage: f = kash('X^5 + 4*X^4 - 56*X^2 -16*X + 192') # optional -- kash
sage: O = kash.EquationOrder(f) # optional -- kash
sage: OK = O.MaximalOrder() # optional -- kash
sage: OK.ClassGroup() # name sage32 below random; optional -- kash
Abelian Group isomorphic to Z/6
Defined on 1 generator
Relations:
6*sage32.1 = 0, extended by:
ext1 := Mapping from: grp^abl: sage32 to ids/ord^num: _AA
::
sage: U = OK.UnitGroup() # optional -- kash
sage: U # name sage34 below random; optional -- kash
Abelian Group isomorphic to Z/2 + Z + Z
Defined on 3 generators
Relations:
2*sage34.1 = 0, extended by:
ext1 := Mapping from: grp^abl: sage34 to ord^num: sage30
sage: kash.Apply('x->%s.ext1(x)'%U.name(), U.Generators().List()) # optional -- kash
[ [1, -1, 0, 0, 0], [1, 1, 0, 0, 0], [-1, 0, 0, 0, 0] ]
Function Fields
~~~~~~~~~~~~~~~
::
sage: k = kash.FiniteField(25) # optional -- kash
sage: kT = k.RationalFunctionField() # optional -- kash
sage: kTy = kT.PolynomialAlgebra() # optional -- kash
sage: T = kT.gen(1) # optional -- kash
sage: y = kTy.gen(1) # optional -- kash
sage: f = y**3 + T**4 + 1 # optional -- kash
Long Input
----------
The KASH interface reads in even very long input (using files) in a
robust manner, as long as you are creating a new object.
.. note::
Using ``kash.eval`` for long input is much less robust, and is not
recommended.
::
sage: a = kash(range(10000)) # optional -- kash
Note that KASH seems to not support string or integer literals with
more than 1024 digits, which is why the above example uses a list
unlike for the other interfaces.
"""
from expect import Expect, ExpectElement
import os
class Kash(Expect):
r"""
Interface to the Kash interpreter.
AUTHORS:
- William Stein and David Joyner
"""
def __init__(self,
max_workspace_size=None,
maxread=100000,
script_subdirectory=None,
restart_on_ctrlc = True,
logfile=None,
server=None,
server_tmpdir=None):
"""
INPUT:
max_workspace_size -- (default: None)
set maximal workspace memory usage to <mem>
<mem> stands for byte-wise allocation
<mem>k stands for kilobyte-wise allocation
<mem>m stands for megabyte-wise allocation
"""
cmd = "kash3 -b -c -d "
if max_workspace_size != None:
cmd += " -a %s"%int(max_workspace)
Expect.__init__(self,
name = 'kash',
prompt = 'kash% ',
command = cmd,
maxread = maxread,
server = server,
server_tmpdir = server_tmpdir,
script_subdirectory = script_subdirectory,
restart_on_ctrlc = True,
verbose_start = False,
logfile = logfile,
eval_using_file_cutoff=100,
init_code = ['X:=ZX.1;']
)
self.__seq = 0
def _next_var_name(self):
if self.__seq == 0:
self.eval('_s_ := [ ];')
self.__seq += 1
return '_s_[%s]'%self.__seq
def _read_in_file_command(self,filename):
return 'Read("%s");'%filename
def _eval_line_using_file(self, line):
F = open(self._local_tmpfile(), 'w')
F.write(line)
F.close()
tmp_to_use = self._local_tmpfile()
if self.is_remote():
self._send_tmpfile_to_server()
tmp_to_use = self._remote_tmpfile()
return self._eval_line(self._read_in_file_command(tmp_to_use),
allow_use_file=False)
def _eval_line(self, line, allow_use_file=False, wait_for_prompt=True, restart_if_needed=False):
return Expect._eval_line(self, line, allow_use_file=allow_use_file,
wait_for_prompt=wait_for_prompt)
def __reduce__(self):
return reduce_load_Kash, tuple([])
def _quit_string(self):
return 'quit;'
def _start(self):
try:
Expect._start(self)
except RuntimeError:
raise RuntimeError, "You must install the optional Kash package to use Kash from Sage."
self.eval('Time(false);')
def _object_class(self):
return KashElement
def eval(self, x, newlines=False, strip=True, **kwds):
r"""
Send the code in the string s to the Kash interpreter and return
the output as a string.
INPUT:
- ``s`` - string containing Kash code.
- ``newlines`` - bool (default: True); if False,
remove all backslash-newlines inserted by the Kash output
formatter.
- ``strip`` - ignored
"""
x = str(x)
x = x.rstrip()
if len(x) == 0 or x[len(x) - 1] != ';':
x += ';'
s = Expect.eval(self, x, **kwds)
i = s.find('\r\n')
if i != -1:
s = s[i+2:]
if newlines:
return s
else:
return s.replace("\\\n","")
def help(self, name=None):
"""
Return help on KASH commands.
Returns help on all commands with a given name. If name is None,
return the location of the installed Kash HTML documentation.
EXAMPLES::
sage: X = kash.help('IntegerRing') # optional -- kash
There is one entry in X for each item found in the documentation
for this function: If you type ``print X[0]`` you will
get help on about the first one, printed nicely to the screen.
AUTHORS:
- Sebastion Pauli (2006-02-04): during Sage coding sprint
"""
if name is None:
print '\nTo use KASH help enter kash.help(s). '
print 'The syntax of the string s is given below.\n'
print self.eval('?')
return
name = str(name)
if name[0] == '?':
print self.eval(name)
else:
print self.eval('?%s'%name)
def _doc(self, V):
if V.lstrip()[:11] == 'No matches.':
return KashDocumentation([])
V = V.split('\n')[1:-1]
X = []
for C in V:
i = C.find('m')
j = C.find(':')
try:
n = int(C[i+1:j])
except ValueError:
full = C
else:
full = self.eval('?%s'%n)
X.append(full)
return KashDocumentation(X)
def help_search(self, name):
return self._doc(self.eval('?*%s'%name))
def set(self, var, value):
"""
Set the variable var to the given value.
"""
cmd = '%s:=%s;;'%(var,value)
out = self._eval_line(cmd, allow_use_file=True)
if out.lower().find('error') != -1:
raise TypeError, "Error executing code in Kash\nCODE:\n\t%s\nKash ERROR:\n\t%s"%(cmd, out)
def get(self, var):
"""
Get the value of the variable var.
"""
return self.eval('%s;'%var, newlines=False)
def _contains(self, v1, v2):
return self.eval('%s in %s'%(v1,v2)) == "true"
def _assign_symbol(self):
return ":="
def _equality_symbol(self):
return "="
def _true_symbol(self):
return "TRUE"
def _false_symbol(self):
return "FALSE"
def console(self):
kash_console()
def version(self):
return kash_version()
class KashElement(ExpectElement):
def __mod__(self, other):
self._check_valid()
if not isinstance(other, KashElement):
other = self.parent()(other)
other._check_valid()
return self.parent()('%s mod %s'%(self._name,other._name))
def __len__(self):
self._check_valid()
return int(self.parent().eval('Length(%s)'%self.name()))
class KashDocumentation(list):
def __repr__(self):
if len(self) == 0:
return "No matches."
return '\n'.join(self)
def is_KashElement(x):
return isinstance(x, KashElement)
kash = Kash()
def reduce_load_Kash():
return kash
import os
def kash_console():
os.system("kash3 ")
def kash_version():
return kash.eval('VERSION')
def __doctest_cleanup():
import sage.interfaces.quit
sage.interfaces.quit.expect_quitall()
