r"""
Interface to Magma
.. note::
You must have ``magma`` installed on your
computer for this interface to work. Magma is not free, so it is
not included with Sage, but you can obtain it from
http://magma.maths.usyd.edu.au/.
Type ``magma.[tab]`` for a list of all the functions
available from your Magma install. Type
``magma.[tab]?`` for Magma's help about a given
function. Type ``magma(...)`` to create a new Magma
object, and ``magma.eval(...)`` to run a string using
Magma (and get the result back as a string).
Sage provides an interface to the Magma computational algebra
system. This system provides extensive functionality for number
theory, group theory, combinatorics and algebra.
The Magma interface offers three pieces of functionality:
#. ``magma_console()`` - A function that dumps you
into an interactive command-line Magma session.
#. ``magma(expr)`` - Evaluation of arbitrary Magma
expressions, with the result returned as a string.
#. ``magma.new(expr)`` - Creation of a Sage object that
wraps a Magma object. This provides a Pythonic interface to Magma.
For example, if ``f=magma.new(10)``, then
``f.Factors()`` returns the prime factorization of
`10` computed using Magma.
Parameters
----------
Some Magma functions have optional "parameters", which are
arguments that in Magma go after a colon. In Sage, you pass these
using named function arguments. For example,
::
sage: E = magma('EllipticCurve([0,1,1,-1,0])') # optional - magma
sage: E.Rank(Bound = 5) # optional - magma
0
Multiple Return Values
----------------------
Some Magma functions return more than one value. You can control
how many you get using the ``nvals`` named parameter to
a function call::
sage: n = magma(100) # optional - magma
sage: n.IsSquare(nvals = 1) # optional - magma
true
sage: n.IsSquare(nvals = 2) # optional - magma
(true, 10)
sage: n = magma(-2006) # optional - magma
sage: n.Factorization() # optional - magma
[ <2, 1>, <17, 1>, <59, 1> ]
sage: n.Factorization(nvals=2) # optional - magma
([ <2, 1>, <17, 1>, <59, 1> ], -1)
We verify that an obviously principal ideal is principal::
sage: _ = magma.eval('R<x> := PolynomialRing(RationalField())') # optional - magma
sage: O = magma.NumberField('x^2+23').MaximalOrder() # optional - magma
sage: I = magma('ideal<%s|%s.1>'%(O.name(),O.name())) # optional - magma
sage: I.IsPrincipal(nvals=2) # optional - magma
(true, [1, 0])
Long Input
----------
The Magma interface reads in even very long input (using files) in
a robust manner.
::
sage: t = '"%s"'%10^10000 # ten thousand character string. # optional - magma
sage: a = magma.eval(t) # optional - magma
sage: a = magma(t) # optional - magma
Garbage Collection
------------------
There is a subtle point with the Magma interface, which arises from
how garbage collection works. Consider the following session:
First, create a matrix m in Sage::
sage: m=matrix(ZZ,2,[1,2,3,4]) # optional - magma
Then I create a corresponding matrix A in Magma::
sage: A = magma(m) # optional - magma
It is called _sage_[...] in Magma::
sage: s = A.name(); s # optional - magma
'_sage_[...]'
It's there::
sage: magma.eval(s) # optional - magma
'[1 2]\n[3 4]'
Now I delete the reference to that matrix::
sage: del A # optional - magma
Now _sage_[...] is "zeroed out" in the Magma session::
sage: magma.eval(s) # optional - magma
'0'
If Sage did not do this garbage collection, then every single time you
ever create any magma object from a sage object, e.g., by doing
magma(m), you would use up a lot of memory in that Magma session.
This would lead to a horrible memory leak situation, which would make
the Magma interface nearly useless for serious work.
Other Examples
--------------
We compute a space of modular forms with character.
::
sage: N = 20
sage: D = 20
sage: eps_top = fundamental_discriminant(D)
sage: eps = magma.KroneckerCharacter(eps_top, RationalField()) # optional - magma
sage: M2 = magma.ModularForms(eps) # optional - magma
sage: print M2 # optional - magma
Space of modular forms on Gamma_1(5) ...
sage: print M2.Basis() # optional - magma
[
1 + 10*q^2 + 20*q^3 + 20*q^5 + 60*q^7 + ...
q + q^2 + 2*q^3 + 3*q^4 + 5*q^5 + 2*q^6 + ...
]
In Sage/Python (and sort of C++) coercion of an element x into a
structure S is denoted by S(x). This also works for the Magma
interface::
sage: G = magma.DirichletGroup(20) # optional - magma
sage: G.AssignNames(['a', 'b']) # optional - magma
sage: (G.1).Modulus() # optional - magma
20
sage: e = magma.DirichletGroup(40)(G.1) # optional - magma
sage: print e # optional - magma
$.1
sage: print e.Modulus() # optional - magma
40
We coerce some polynomial rings into Magma::
sage: R.<y> = PolynomialRing(QQ)
sage: S = magma(R) # optional - magma
sage: print S # optional - magma
Univariate Polynomial Ring in y over Rational Field
sage: S.1 # optional - magma
y
This example illustrates that Sage doesn't magically extend how
Magma implicit coercion (what there is, at least) works. The errors
below are the result of Magma having a rather limited automatic
coercion system compared to Sage's::
sage: R.<x> = ZZ[]
sage: x * 5
5*x
sage: x * 1.0
x
sage: x * (2/3)
2/3*x
sage: y = magma(x) # optional - magma
sage: y * 5 # optional - magma
5*x
sage: y * 1.0 # optional - magma
$.1
sage: y * (2/3) # optional - magma
Traceback (most recent call last):
...
TypeError: Error evaluating Magma code.
...
Runtime error in '*': Bad argument types
Argument types given: RngUPolElt[RngInt], FldRatElt
AUTHORS:
- William Stein (2005): initial version
- William Stein (2006-02-28): added extensive tab completion and
interactive IPython documentation support.
- William Stein (2006-03-09): added nvals argument for
magma.functions...
"""
import re, sys
from sage.structure.parent import Parent
from expect import console, Expect, ExpectElement, ExpectFunction, FunctionElement
PROMPT = ">>>"
SAGE_REF = "_sage_ref"
SAGE_REF_RE = re.compile('%s\d+'%SAGE_REF)
import sage.misc.misc
import sage.misc.sage_eval
INTRINSIC_CACHE = '%s/magma_intrinsic_cache.sobj'%sage.misc.misc.DOT_SAGE
EXTCODE_DIR = None
def extcode_dir():
"""
Return directory that contains all the Magma extcode. This is put
in a writable directory owned by the user, since when attached,
Magma has to write sig and lck files.
EXAMPLES:
sage: sage.interfaces.magma.extcode_dir()
'...dir_.../data/'
"""
global EXTCODE_DIR
if not EXTCODE_DIR:
import shutil
tmp = sage.misc.misc.tmp_dir()
shutil.copytree('%s/magma/'%sage.misc.misc.SAGE_EXTCODE, tmp + '/data')
EXTCODE_DIR = "%s/data/"%tmp
return EXTCODE_DIR
class Magma(Expect):
r"""
Interface to the Magma interpreter.
Type ``magma.[tab]`` for a list of all the functions
available from your Magma install. Type
``magma.[tab]?`` for Magma's help about a given
function. Type ``magma(...)`` to create a new Magma
object, and ``magma.eval(...)`` to run a string using
Magma (and get the result back as a string).
.. note::
If you do not own a local copy of Magma, try using the
``magma_free`` command instead, which uses the free demo web
interface to Magma.
EXAMPLES:
You must use nvals = 0 to call a function that doesn't return
anything, otherwise you'll get an error. (nvals is the number of
return values.)
::
sage: magma.SetDefaultRealFieldPrecision(200, nvals=0) # magma >= v2.12; optional - magma
sage: magma.eval('1.1') # optional - magma
'1.1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000'
sage: magma.SetDefaultRealFieldPrecision(30, nvals=0) # optional - magma
"""
def __init__(self, maxread=10000, script_subdirectory=None,
logfile=None, server=None, server_tmpdir=None, user_config=False):
"""
INPUT:
- ``maxread`` - affects buffering
- ``script_subdirectory`` - directory where scripts
are read from
- ``logfile`` - output logged to this file
- ``server`` - address of remote server
- ``user_config`` - if True, then local user
configuration files will be read by Magma. If False (the default),
then Magma is started with the -n option which suppresses user
configuration files.
EXAMPLES::
sage: Magma(maxread=1000, logfile=tmp_filename())
Magma
"""
command = 'sage-native-execute magma'
if not user_config:
command += ' -n'
Expect.__init__(self,
name = "magma",
prompt = ">>SAGE>>",
command = command,
maxread = maxread,
server = server,
server_tmpdir = server_tmpdir,
script_subdirectory = script_subdirectory,
restart_on_ctrlc = False,
logfile = logfile,
eval_using_file_cutoff=100)
self.__seq = 0
self.__ref = 0
self.__available_var = []
self.__cache = {}
self._preparse_colon_equals = False
def __reduce__(self):
"""
Used to pickle a magma interface instance.
Unpickling results in the default magma interpreter; this is a
choice, and perhaps not the most logical one! It means that if you
make two distinct magma interfaces, pickle both, then unpickle
them, you get back exactly the same one. We illustrate this
behavior below.
OUTPUT: function, empty tuple
EXAMPLES::
sage: loads(dumps(magma)) is magma
True
Unpickling always gives the default global magma interpreter::
sage: m1 = Magma(); m2 = Magma()
sage: m1 is m2
False
sage: loads(dumps(m1)) is loads(dumps(m2))
True
sage: loads(dumps(m1)) is magma
True
"""
return reduce_load_Magma, tuple([])
def _read_in_file_command(self, filename):
"""
Return the command in Magma that reads in the contents of the given
file.
INPUT:
- ``filename`` - string
OUTPUT:
- ``string`` - a magma command
EXAMPLES::
sage: magma._read_in_file_command('file.m')
'load "file.m";'
"""
return 'load "%s";'%filename
def _post_process_from_file(self, s):
r"""
Used internally in the Magma interface to post-process the result
of evaluating a string using a file. For Magma what this does is
delete the first output line, since that is a verbose output line
that Magma displays about loading a file.
INPUT:
- ``s`` - a string
OUTPUT: a string
EXAMPLES::
sage: magma._post_process_from_file("Loading ...\nHello")
'Hello'
sage: magma._post_process_from_file("Hello")
''
"""
if not isinstance(s, str):
raise RuntimeError, "Error evaluating object in %s:\n%s"%(self,s)
i = s.find('\n')
if i == -1:
return ''
return s[i+1:]
def __getattr__(self, attrname):
"""
Return a formal wrapper around a Magma function, or raise an
AttributeError if attrname starts with an underscore.
INPUT:
- ``attrname`` - a string
OUTPUT: MagmaFunction instance
EXAMPLES::
sage: g = magma.__getattr__('EllipticCurve')
sage: g
EllipticCurve
sage: type(g)
<class 'sage.interfaces.magma.MagmaFunction'>
In fact, __getattr__ is called implicitly in the following
case::
sage: f = magma.EllipticCurve
sage: type(f)
<class 'sage.interfaces.magma.MagmaFunction'>
sage: f
EllipticCurve
"""
if attrname[:1] == "_":
raise AttributeError
return MagmaFunction(self, attrname)
def chdir(self, dir):
"""
Change the Magma interpreters current working directory.
INPUT:
- ``dir`` - a string
EXAMPLES::
sage: magma.eval('System("pwd")') # random, optional - magma
'/Users/was/s/devel/sage-main/sage'
sage: magma.chdir('..') # optional - magma
sage: magma.eval('System("pwd")') # random, optional - magma
'/Users/was/s/devel/sage-main'
"""
self.eval('ChangeDirectory("%s")'%dir, strip=False)
def eval(self, x, strip=True, **kwds):
"""
Evaluate the given block x of code in Magma and return the output
as a string.
INPUT:
- ``x`` - string of code
- ``strip`` - ignored
OUTPUT: string
EXAMPLES:
We evaluate a string that involves assigning to a
variable and printing.
::
sage: magma.eval("a := 10;print 2+a;") # optional - magma
'12'
We evaluate a large input line (note that no weird output appears
and that this works quickly).
::
sage: magma.eval("a := %s;"%(10^10000)) # optional - magma
''
Verify that trac 9705 is fixed::
sage: nl=chr(10) # newline character
sage: magma.eval( # optional - magma
... "_<x>:=PolynomialRing(Rationals());"+nl+
... "repeat"+nl+
... " g:=3*b*x^4+18*c*x^3-6*b^2*x^2-6*b*c*x-b^3-9*c^2 where b:=Random([-10..10]) where c:=Random([-10..10]);"+nl+
... "until g ne 0 and Roots(g) ne [];"+nl+
... "print \"success\";")
'success'
Verify that trac 11401 is fixed::
sage: nl=chr(10) # newline character
sage: magma.eval("a:=3;"+nl+"b:=5;") == nl # optional - magma
True
sage: magma.eval("[a,b];") # optional - magma
'[ 3, 5 ]'
"""
x = self._preparse(x)
x = str(x).rstrip()
if len(x) == 0 or x[len(x) - 1] != ';':
x += ';'
ans = Expect.eval(self, x, **kwds).replace('\\\n','')
if 'Runtime error' in ans or 'User error' in ans:
raise RuntimeError, "Error evaluating Magma code.\nIN:%s\nOUT:%s"%(x, ans)
return ans
def _preparse(self, s):
"""
All input gets preparsed by calling this function before it gets evaluated.
EXAMPLES::
sage: magma._preparse_colon_equals = False
sage: magma._preparse('a = 5')
'a = 5'
sage: magma._preparse_colon_equals = True
sage: magma._preparse('a = 5')
'a := 5'
sage: magma._preparse('a = 5; b := 7; c =a+b;')
'a := 5; b := 7; c :=a+b;'
"""
try:
if self._preparse_colon_equals:
s = s.replace(':=','=').replace('=',':=')
except AttributeError: pass
return s
def _start(self):
"""
Initialize a Magma interface instance. This involves (1) setting up
an obfuscated prompt, and (2) attaching the MAGMA_SPEC file (see
EXTCODE_DIR/spec file (see sage.interfaces.magma.EXTCODE_DIR/spec).
EXAMPLES: This is not too exciting::
sage: magma._start() # optional - magma
"""
self._change_prompt('>')
Expect._start(self)
self.eval('SetPrompt("%s"); SetLineEditor(false); SetColumns(0);'%PROMPT)
self._change_prompt(PROMPT)
self.expect().expect(PROMPT)
self.expect().expect(PROMPT)
self.expect().expect(PROMPT)
self.attach_spec(extcode_dir() + '/spec')
def set(self, var, value):
"""
Set the variable var to the given value in the Magma interpreter.
INPUT:
- ``var`` - string; a variable name
- ``value`` - string; what to set var equal to
EXAMPLES::
sage: magma.set('abc', '2 + 3/5') # optional - magma
sage: magma('abc') # optional - magma
13/5
"""
out = self.eval("%s:=%s"%(var, value))
if out.lower().find("error") != -1:
raise TypeError, "Error executing Magma code:\n%s"%out
def get(self, var):
"""
Get the value of the variable var.
INPUT:
- ``var`` - string; name of a variable defined in the
Magma session
OUTPUT:
- ``string`` - string representation of the value of
the variable.
EXAMPLES::
sage: magma.set('abc', '2 + 3/5') # optional - magma
sage: magma.get('abc') # optional - magma
'13/5'
"""
return self.eval("%s"%var)
def objgens(self, value, gens):
"""
Create a new object with given value and gens.
INPUT:
- ``value`` - something coercible to an element of this Magma
interface
- ``gens`` - string; comma separated list of variable names
OUTPUT: new Magma element that is equal to value with given gens
EXAMPLES::
sage: R = magma.objgens('PolynomialRing(Rationals(),2)', 'alpha,beta') # optional - magma
sage: R.gens() # optional - magma
[alpha, beta]
Because of how Magma works you can use this to change the variable
names of the generators of an object::
sage: S = magma.objgens(R, 'X,Y') # optional - magma
sage: R # optional - magma
Polynomial ring of rank 2 over Rational Field
Order: Lexicographical
Variables: X, Y
sage: S # optional - magma
Polynomial ring of rank 2 over Rational Field
Order: Lexicographical
Variables: X, Y
"""
var = self._next_var_name()
value = self(value)
out = self.eval("_zsage_<%s> := %s; %s := _zsage_"%(gens, value.name(), var))
if out.lower().find("error") != -1:
raise TypeError, "Error executing Magma code:\n%s"%out
return self(var)
def __call__(self, x, gens=None):
"""
Coerce x into this Magma interpreter interface.
INPUT:
- ``x`` - object
- ``gens`` - string; names of generators of self,
separated by commas
OUTPUT: MagmaElement
EXAMPLES::
sage: magma(EllipticCurve('37a')) # optional - magma
Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
sage: magma('EllipticCurve([GF(5)|1,2,3,4,1])') # optional - magma
Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 1 over GF(5)
sage: magma('PowerSeriesRing(Rationals())', 't') # optional - magma
Power series ring in t over Rational Field
sage: magma('PolynomialRing(RationalField(), 3)', 'x,y,z') # optional - magma
Polynomial ring of rank 3 over Rational Field
Order: Lexicographical
Variables: x, y, z
We test a coercion between different Magma instances::
sage: m = Magma()
sage: n = Magma()
sage: a = n(m(2)) # optional - magma
sage: a.parent() is n # optional - magma
True
sage: a.parent() is m # optional - magma
False
We test caching::
sage: R.<x> = ZZ[] # optional - magma
sage: magma(R) is magma(R) # optional - magma
True
sage: m = Magma() # optional - magma
sage: m(R) # optional - magma
Univariate Polynomial Ring in x over Integer Ring
sage: m(R) is magma(R) # optional - magma
False
sage: R._magma_cache # optional - magma
{Magma: Univariate Polynomial Ring in x over Integer Ring,
Magma: Univariate Polynomial Ring in x over Integer Ring}
"""
if isinstance(x, bool):
return Expect.__call__(self, 'true' if x else 'false')
if gens is not None:
return self.objgens(x, gens)
has_cache = hasattr(x, '_magma_cache')
try:
if has_cache and x._magma_cache.has_key(self):
A = x._magma_cache[self]
if A._session_number == self._session_number:
return A
except AttributeError:
x._magma_cache = {}
try:
if self.__cache.has_key(x):
A = self.__cache[x]
if A._session_number == self._session_number:
return A
except TypeError:
pass
A = Expect.__call__(self, x)
if has_cache:
x._magma_cache[self] = A
else:
try:
x._magma_cache = {self:A}
except AttributeError, msg:
if isinstance(x, Parent):
self.__cache[x] = A
return A
def _coerce_from_special_method(self, x):
"""
Tries to coerce to self by calling a special underscore method.
If no such method is defined, raises an AttributeError instead of a
TypeError.
EXAMPLES::
sage: magma._coerce_from_special_method(-3/5) # optional - magma
-3/5
Note that AttributeError::
sage: magma._coerce_from_special_method('2 + 3') # optional - magma
Traceback (most recent call last):
...
AttributeError: 'str' object has no attribute '_magma_init_'
"""
s = x._magma_init_(self)
a = self(s)
while True:
z = SAGE_REF_RE.search(s)
if not z: break
self.eval('delete %s;'%s[z.start():z.end()])
s = s[z.end()+1:]
return a
def _with_names(self, s, names):
"""
Return s but wrapped by a call to SageCreateWithNames. This is just
a very simple convenience function so that code is cleaner.
INPUT:
- ``s`` - string
- ``names`` - list of strings
OUTPUT: string
EXAMPLES::
sage: magma._with_names('PolynomialRing(RationalField())', ['y']) # optional - magma
'SageCreateWithNames(PolynomialRing(RationalField()),["y"])'
"""
return 'SageCreateWithNames(%s,[%s])'%(s, ','.join('"%s"'%x for x in names))
def clear(self, var):
"""
Clear the variable named var and make it available to be used
again.
INPUT:
- ``var`` - a string
EXAMPLES::
sage: magma = Magma() # optional - magma
sage: magma.clear('foo') # sets foo to 0 in magma; optional - magma
sage: magma.eval('foo') # optional - magma
'0'
Because we cleared foo, it is set to be used as a variable name in
the future::
sage: a = magma('10') # optional - magma
sage: a.name() # optional - magma
'foo'
The following tests that the whole variable clearing and freeing
system is working correctly.
::
sage: magma = Magma() # optional - magma
sage: a = magma('100') # optional - magma
sage: a.name() # optional - magma
'_sage_[1]'
sage: del a # optional - magma
sage: b = magma('257') # optional - magma
sage: b.name() # optional - magma
'_sage_[1]'
sage: del b # optional - magma
sage: magma('_sage_[1]') # optional - magma
0
"""
self.__available_var.insert(0,var)
self.eval("%s:=0"%var)
def cputime(self, t=None):
"""
Return the CPU time in seconds that has elapsed since this Magma
session started. This is a floating point number, computed by
Magma.
If t is given, then instead return the floating point time from
when t seconds had elapsed. This is useful for computing elapsed
times between two points in a running program.
INPUT:
- ``t`` - float (default: None); if not None, return
cputime since t
OUTPUT:
- ``float`` - seconds
EXAMPLES::
sage: type(magma.cputime()) # optional - magma
<type 'float'>
sage: magma.cputime() # random, optional - magma
1.9399999999999999
sage: t = magma.cputime() # optional - magma
sage: magma.cputime(t) # random, optional - magma
0.02
"""
if t:
return float(self.eval('Cputime(%s)'%t))
else:
return float(self.eval('Cputime()'))
def chdir(self, dir):
"""
Change to the given directory.
INPUT:
- ``dir`` - string; name of a directory
EXAMPLES::
sage: magma.chdir('/') # optional - magma
sage: magma.eval('System("pwd")') # optional - magma
'/'
"""
magma.eval('ChangeDirectory("%s")'%dir)
def attach(self, filename):
r"""
Attach the given file to the running instance of Magma.
Attaching a file in Magma makes all intrinsics defined in the file
available to the shell. Moreover, if the file doesn't start with
the ``freeze;`` command, then the file is reloaded
whenever it is changed. Note that functions and procedures defined
in the file are *not* available. For only those, use
``magma.load(filename)``.
INPUT:
- ``filename`` - a string
EXAMPLES: Attaching a file that exists is fine::
sage: magma.attach('%s/data/extcode/magma/sage/basic.m'%SAGE_ROOT) # optional - magma
Attaching a file that doesn't exist raises an exception::
sage: magma.attach('%s/data/extcode/magma/sage/basic2.m'%SAGE_ROOT) # optional - magma
Traceback (most recent call last):
...
RuntimeError: Error evaluating Magma code...
"""
self.eval('Attach("%s")'%filename)
Attach = attach
def attach_spec(self, filename):
r"""
Attach the given spec file to the running instance of Magma.
This can attach numerous other files to the running Magma (see the
Magma documentation for more details).
INPUT:
- ``filename`` - a string
EXAMPLES::
sage: magma.attach_spec('%s/data/extcode/magma/spec'%SAGE_ROOT) # optional - magma
sage: magma.attach_spec('%s/data/extcode/magma/spec2'%SAGE_ROOT) # optional - magma
Traceback (most recent call last):
...
RuntimeError: Can't open package spec file .../data/extcode/magma/spec2 for reading (No such file or directory)
"""
s = self.eval('AttachSpec("%s")'%filename)
if s:
raise RuntimeError, s.strip()
AttachSpec = attach_spec
def load(self, filename):
r"""
Load the file with given filename using the 'load' command in the
Magma shell.
Loading a file in Magma makes all the functions and procedures in
the file available. The file should not contain any intrinsics (or
you'll get errors). It also runs code in the file, which can
produce output.
INPUT:
- ``filename`` - string
OUTPUT: output printed when loading the file
EXAMPLES::
sage: open(SAGE_TMP + 'a.m','w').write('function f(n) return n^2; end function;\nprint "hi";')
sage: print magma.load(SAGE_TMP + 'a.m') # optional - magma
Loading ".../.sage//temp/.../a.m"
hi
sage: magma('f(12)') # optional - magma
144
"""
return self.eval('load "%s"'%filename)
def _next_var_name(self):
"""
Return the next available variable name in Magma.
OUTPUT: string
EXAMPLES::
sage: m = Magma()
sage: m._next_var_name() # optional - magma
'_sage_[1]'
sage: m._next_var_name() # optional - magma
'_sage_[2]'
sage: a = m(3/8); a # optional - magma
3/8
sage: a.name() # optional - magma
'_sage_[3]'
sage: m._next_var_name() # optional - magma
'_sage_[4]'
"""
if self.__seq == 0:
self.eval('_sage_ := [* *];')
else:
try:
self.eval('Append(~_sage_, 0);')
except:
self.eval('_sage_ := [* 0 : i in [1..%s] *];'%self.__seq)
try:
return self.__available_var.pop()
except IndexError:
self.__seq += 1
return '_sage_[%s]'%self.__seq
def _next_ref_name(self):
"""
Return the next reference name. This is used internally to deal
with Magma objects that would be deallocated before they are used
in constructing another object.
OUTPUT: string
EXAMPLES::
sage: magma._next_ref_name()
'_sage_ref...'
"""
self.__ref += 1
return '%s%s'%(SAGE_REF, self.__ref)
def function_call(self, function, args=[], params={}, nvals=1):
"""
Return result of evaluating a Magma function with given input,
parameters, and asking for nvals as output.
INPUT:
- ``function`` - string, a Magma function name
- ``args`` - list of objects coercible into this magma
interface
- ``params`` - Magma parameters, passed in after a
colon
- ``nvals`` - number of return values from the
function to ask Magma for
OUTPUT: MagmaElement or tuple of nvals MagmaElement's
EXAMPLES::
sage: magma.function_call('Factorization', 100) # optional - magma
[ <2, 2>, <5, 2> ]
sage: magma.function_call('NextPrime', 100, {'Proof':False}) # optional - magma
101
sage: magma.function_call('PolynomialRing', [QQ,2]) # optional - magma
Polynomial ring of rank 2 over Rational Field
Order: Lexicographical
Variables: $.1, $.2
Next, we illustrate multiple return values::
sage: magma.function_call('IsSquare', 100) # optional - magma
true
sage: magma.function_call('IsSquare', 100, nvals=2) # optional - magma
(true, 10)
sage: magma.function_call('IsSquare', 100, nvals=3) # optional - magma
Traceback (most recent call last):
...
RuntimeError: Error evaluating Magma code...
Runtime error in :=: Expected to assign 3 value(s) but only computed 2 value(s)
"""
args, params = self._convert_args_kwds(args, params)
nvals = int(nvals)
if len(params) == 0:
par = ''
else:
par = ' : ' + ','.join(['%s:=%s'%(a,b.name()) for a,b in params.items()])
fun = "%s(%s%s)"%(function, ",".join([s.name() for s in args]), par)
return self._do_call(fun, nvals)
def _do_call(self, code, nvals):
"""
Evaluate the given code expression assuming that it outputs nvals
distinct values. Return the resulting values as a tuple if nvals =
2.
INPUT:
- ``code`` - a string; code to evaluate
- ``nvals`` - an integer; number of return values
OUTPUT: nvals distinct values
EXAMPLES::
sage: magma._do_call('SetVerbose("Groebner",2)', 0) # optional - magma
sage: magma._do_call('Factorization(-5)', 1) # optional - magma
[ <5, 1> ]
Here we get two outputs, as a tuple.
::
sage: magma._do_call('Factorization(-5)', 2) # optional - magma
([ <5, 1> ], -1)
You can also do this::
sage: F, sign = magma._do_call('Factorization(-5)', 2) # optional - magma
sage: F # optional - magma
[ <5, 1> ]
sage: sign # optional - magma
-1
An expression that has one value.
::
sage: magma._do_call('3^5', 1) # optional - magma
243
"""
if nvals <= 0:
out = self.eval(code)
ans = None
elif nvals == 1:
return self(code)
else:
v = [self._next_var_name() for _ in range(nvals)]
vars = ", ".join(v)
cmd = "%s := %s;"%(vars, code)
out = self.eval(cmd)
ans = tuple([MagmaElement(self, x, is_name = True) for x in v])
if out.lower().find("error") != -1:
raise TypeError, "Error executing Magma code:\n%s"%out
return ans
def bar_call(self, left, name, gens, nvals=1):
"""
This is a wrapper around the Magma constructor
nameleft gens
returning nvals.
INPUT:
- ``left`` - something coerceable to a magma object
- ``name`` - name of the constructor, e.g., sub, quo,
ideal, etc.
- ``gens`` - if a list/tuple, each item is coerced to
magma; otherwise gens itself is converted to magma
- ``nvals`` - positive integer; number of return
values
OUTPUT: a single magma object if nvals == 1; otherwise a tuple of
nvals magma objects.
EXAMPLES: The bar_call function is used by the sub, quo, and ideal
methods of Magma elements. Here we illustrate directly using
bar_call to create quotients::
sage: V = magma.RModule(ZZ,3) # optional - magma
sage: V # optional - magma
RModule(IntegerRing(), 3)
sage: magma.bar_call(V, 'quo', [[1,2,3]], nvals=1) # optional - magma
RModule(IntegerRing(), 2)
sage: magma.bar_call(V, 'quo', [[1,2,3]], nvals=2) # optional - magma
(RModule(IntegerRing(), 2),
Mapping from: RModule(IntegerRing(), 3) to RModule(IntegerRing(), 2))
sage: magma.bar_call(V, 'quo', V, nvals=2) # optional - magma
(RModule(IntegerRing(), 0),
Mapping from: RModule(IntegerRing(), 3) to RModule(IntegerRing(), 0))
"""
magma = self
if isinstance(gens, (list, tuple)):
gens = [magma(z) for z in gens]
v = ', '.join([w.name() for w in gens])
else:
gens = magma(gens)
v = gens.name()
s = '%s< %s | %s >'%(name, left.name(), v)
return self._do_call(s, nvals)
def _object_class(self):
"""
Return the Python class of elements of the Magma interface.
OUTPUT: a Python class
EXAMPLES::
sage: magma._object_class()
<class 'sage.interfaces.magma.MagmaElement'>
"""
return MagmaElement
def _left_list_delim(self):
"""
Return the left sequence delimiter in Magma. Despite the name in
this function, this is really the least painful choice.
EXAMPLES::
sage: magma._left_list_delim()
'['
"""
return "["
def _right_list_delim(self):
"""
Return the right sequence delimiter in Magma. Despite the name in
this function, this is really the least painful choice.
EXAMPLES::
sage: magma._right_list_delim()
']'
"""
return "]"
def _assign_symbol(self):
"""
Returns the assignment symbol in Magma.
EXAMPLES::
sage: magma._assign_symbol()
':='
"""
return ":="
def _equality_symbol(self):
"""
Returns the equality testing logical symbol in Magma.
EXAMPLES::
sage: magma._equality_symbol()
'eq'
"""
return 'eq'
def _lessthan_symbol(self):
"""
Returns the less than testing logical symbol in Magma.
EXAMPLES::
sage: magma._lessthan_symbol()
' lt '
"""
return ' lt '
def _greaterthan_symbol(self):
"""
Returns the greater than testing logical symbol in Magma.
EXAMPLES::
sage: magma._greaterthan_symbol()
' gt '
"""
return ' gt '
def _true_symbol(self):
"""
Returns the string representation of "truth" in Magma.
EXAMPLES::
sage: magma._true_symbol()
'true'
"""
return 'true'
def _false_symbol(self):
"""
Returns the string representation of "false" in Magma.
EXAMPLES::
sage: magma._false_symbol()
'false'
"""
return 'false'
def console(self):
"""
Run a command line Magma session. This session is completely
separate from this Magma interface.
EXAMPLES::
sage: magma.console() # not tested
Magma V2.14-9 Sat Oct 11 2008 06:36:41 on one [Seed = 1157408761]
Type ? for help. Type <Ctrl>-D to quit.
>
Total time: 2.820 seconds, Total memory usage: 3.95MB
"""
magma_console()
def version(self):
"""
Return the version of Magma that you have in your PATH on your
computer.
OUTPUT:
- ``numbers`` - 3-tuple: major, minor, etc.
- ``string`` - version as a string
EXAMPLES::
sage: magma.version() # random, optional - magma
((2, 14, 9), 'V2.14-9')
"""
return magma_version()
def help(self, s):
"""
Return Magma help on string s.
This returns what typing ?s would return in Magma.
INPUT:
- ``s`` - string
OUTPUT: string
EXAMPLES::
sage: magma.help("NextPrime") # optional - magma
===============================================================================
PATH: /magma/ring-field-algebra/integer/prime/next-previous/NextPrime
KIND: Intrinsic
===============================================================================
NextPrime(n) : RngIntElt -> RngIntElt
NextPrime(n: parameter) : RngIntElt -> RngIntElt
...
"""
print self.eval('? %s'%s)
def trait_names(self, verbose=True, use_disk_cache=True):
"""
Return a list of all Magma commands.
This is used as a hook to enable custom command completion.
Magma doesn't provide any fast way to make a list of all commands,
which is why caching is done by default. Note that an adverse
impact of caching is that *new* commands are not picked up, e.g.,
user defined variables or functions.
INPUT:
- ``verbose`` - bool (default: True); whether to
verbosely output status info the first time the command list is
built
- ``use_disk_cache`` - bool (default: True); use
cached command list, which is saved to disk.
OUTPUT: list of strings
EXAMPLES::
sage: len(magma.trait_names(verbose=False)) # random, optional - magma
7261
"""
try:
return self.__trait_names
except AttributeError:
import sage.misc.persist
if use_disk_cache:
try:
self.__trait_names = sage.misc.persist.load(INTRINSIC_CACHE)
return self.__trait_names
except IOError:
pass
if verbose:
print "\nCreating list of all Magma intrinsics for use in tab completion."
print "This takes a few minutes the first time, but is saved to the"
print "file '%s' for future instant use."%INTRINSIC_CACHE
print "Magma may produce errors during this process, which are safe to ignore."
print "Delete that file to force recreation of this cache."
print "Scanning Magma types ..."
tm = sage.misc.misc.cputime()
T = self.eval('ListTypes()').split()
N = []
for t in T:
if verbose:
print t, " ",
sys.stdout.flush()
try:
s = self.eval('ListSignatures(%s)'%t)
for x in s.split('\n'):
i = x.find('(')
N.append(x[:i])
except RuntimeError, msg:
print 'Error -- %s'%msg
pass
if verbose:
print "Done! (%s seconds)"%sage.misc.misc.cputime(tm)
N = list(set(N))
N.sort()
print "Saving cache to '%s' for future instant use."%INTRINSIC_CACHE
print "Delete the above file to force re-creation of the cache."
sage.misc.persist.save(N, INTRINSIC_CACHE)
self.__trait_names = N
return N
def ideal(self, L):
"""
Return the Magma ideal defined by L.
INPUT:
- ``L`` - a list of elements of a Sage multivariate
polynomial ring.
OUTPUT: The magma ideal generated by the elements of L.
EXAMPLES::
sage: R.<x,y> = QQ[]
sage: magma.ideal([x^2, y^3*x]) # optional - magma
Ideal of Polynomial ring of rank 2 over Rational Field
Order: Graded Reverse Lexicographical
Variables: x, y
Homogeneous
Basis:
[
x^2,
x*y^3
]
"""
P = iter(L).next().parent()
Pn = self(P).name()
k = P.base_ring()
if k.degree() > 1:
i = str(k.gen())
o = self("BaseRing(%s).1"%Pn).name()
self.eval("%s := %s"%(i,o))
mlist = self(L)
return self("ideal<%s|%s>"%(Pn,mlist.name()))
def set_verbose(self, type, level):
"""
Set the verbosity level for a given algorithm, class, etc. in
Magma.
INPUT:
- ``type`` - string (e.g. 'Groebner')
- ``level`` - integer = 0
EXAMPLES::
sage: magma.set_verbose("Groebner", 2) # optional - magma
sage: magma.get_verbose("Groebner") # optional - magma
2
"""
self.SetVerbose(type,level)
def SetVerbose(self, type, level):
"""
Set the verbosity level for a given algorithm class etc. in Magma.
INPUT:
- ``type`` - string (e.g. 'Groebner'), see Magma
documentation
- ``level`` - integer = 0
.. note::
This method is provided to be consistent with the Magma
naming convention.
::
sage: magma.SetVerbose("Groebner", 2) # optional - magma
sage: magma.GetVerbose("Groebner") # optional - magma
2
"""
if level < 0:
raise TypeError, "level must be >= 0"
self.eval('SetVerbose("%s",%d)'%(type,level))
def get_verbose(self, type):
"""
Get the verbosity level of a given algorithm class etc. in Magma.
INPUT:
- ``type`` - string (e.g. 'Groebner'), see Magma
documentation
EXAMPLES::
sage: magma.set_verbose("Groebner", 2) # optional - magma
sage: magma.get_verbose("Groebner") # optional - magma
2
"""
return self.GetVerbose(type)
def GetVerbose(self, type):
"""
Get the verbosity level of a given algorithm class etc. in Magma.
INPUT:
- ``type`` - string (e.g. 'Groebner'), see Magma
documentation
.. note::
This method is provided to be consistent with the Magma
naming convention.
EXAMPLES::
sage: magma.SetVerbose("Groebner", 2) # optional - magma
sage: magma.GetVerbose("Groebner") # optional - magma
2
"""
return int(self.eval('GetVerbose("%s")'%type))
class MagmaFunctionElement(FunctionElement):
def __call__(self, *args, **kwds):
"""
Return the result of calling this Magma function at given inputs.
Use the optional nvals keyword argument to specify that there are
multiple return values.
EXAMPLES: We create a MagmaFunctionElement::
sage: n = magma(-15) # optional - magma
sage: f = n.Factorisation # optional - magma
sage: type(f) # optional - magma
<class 'sage.interfaces.magma.MagmaFunctionElement'>
sage: f() # optional - magma
[ <3, 1>, <5, 1> ]
We verify that the nvals argument works.
::
sage: f(nvals=2) # optional - magma
([ <3, 1>, <5, 1> ], -1)
This illustrates the more conventional way of calling a method on
an object. It's equivalent to the above, but done in all in one
step.
::
sage: n.Factorization(nvals = 2) # optional - magma
([ <3, 1>, <5, 1> ], -1)
"""
nvals = 1
if len(kwds) > 0:
if kwds.has_key('nvals'):
nvals = kwds['nvals']
del kwds['nvals']
M = self._obj.parent()
return M.function_call(self._name,
[self._obj.name()] + list(args),
params = kwds,
nvals = nvals)
def _sage_doc_(self):
"""
Return the docstring for this function of an element.
OUTPUT: string
EXAMPLES::
sage: n = magma(-15) # optional - magma
sage: f = n.Factorisation # optional - magma
sage: print f._sage_doc_() # optional - magma
(<RngIntElt> n) -> RngIntEltFact, RngIntElt, SeqEnum
...
sage: print n.Factorisation._sage_doc_() # optional - magma
(<RngIntElt> n) -> RngIntEltFact, RngIntElt, SeqEnum
...
"""
M = self._obj.parent()
t = str(self._obj.Type())
s = M.eval(self._name)
Z = s.split('(<')[1:]
W = []
tt = '(<%s'%t
for X in Z:
X = '(<' + X
if '(<All>' in X or tt in X:
W.append(X)
s = '\n'.join(W)
s = sage.misc.misc.word_wrap(s)
return s
def __repr__(self):
"""
Return string representation of this partially evaluated function.
This is basically the docstring (as returned by _sage_doc_)
unless self._name is the name of an attribute of the object, in
which case this returns the value of that attribute.
EXAMPLES::
sage: magma(-15).Factorisation # optional - magma
Partially evaluated Magma function or intrinsic 'Factorisation'
...
We create a vector space, set its M attribute to a number, then
display/get the attribute as a string.
::
sage: V = magma('VectorSpace(RationalField(),2)') # optional - magma
sage: V.set_magma_attribute('M', 290398) # optional - magma
sage: V.M # optional - magma
290398
sage: type(V.M) # optional - magma
<class 'sage.interfaces.magma.MagmaFunctionElement'>
sage: type(V.M.__repr__()) # optional - magma
<type 'str'>
Displaying a non-attribute function works as above.
::
sage: V.Dimension # optional - magma
Partially evaluated Magma function or intrinsic 'Dimension'
...
"""
M = self._obj.parent()
try:
return M.eval('%s`%s'%(self._obj.name(), self._name))
except RuntimeError:
return "Partially evaluated Magma function or intrinsic '%s'\n\nSignature:\n\n%s"%(
self._name, self._sage_doc_())
class MagmaFunction(ExpectFunction):
def __call__(self, *args, **kwds):
"""
Return the result of calling this Magma function at given inputs.
Use the optional nvals keyword argument to specify that there are
multiple return values.
EXAMPLES: We create a MagmaFunction::
sage: f = magma.Factorisation # optional - magma
sage: type(f) # optional - magma
<class 'sage.interfaces.magma.MagmaFunction'>
sage: f(-15) # optional - magma
[ <3, 1>, <5, 1> ]
We verify that the nvals argument works.
::
sage: f(-15, nvals=2) # optional - magma
([ <3, 1>, <5, 1> ], -1)
sage: f.__call__(-15, nvals=2) # optional - magma
([ <3, 1>, <5, 1> ], -1)
"""
nvals = 1
if len(kwds) > 0:
if kwds.has_key('nvals'):
nvals = kwds['nvals']
del kwds['nvals']
M = self._parent
return M.function_call(self._name,
list(args),
params = kwds,
nvals = nvals)
def _sage_doc_(self):
"""
Return docstring about this function.
OUTPUT: string
EXAMPLES::
sage: f = magma.Factorisation
sage: type(f)
<class 'sage.interfaces.magma.MagmaFunction'>
sage: print f._sage_doc_() # optional - magma
Intrinsic 'Factorisation'
...
"""
M = self._parent
s = M.eval(self._name)
s = sage.misc.misc.word_wrap(s, 80)
return s
def is_MagmaElement(x):
"""
Return True if x is of type MagmaElement, and False otherwise.
INPUT:
- ``x`` - any object
OUTPUT: bool
EXAMPLES::
sage: from sage.interfaces.magma import is_MagmaElement
sage: is_MagmaElement(2)
False
sage: is_MagmaElement(magma(2)) # optional - magma
True
"""
return isinstance(x, MagmaElement)
class MagmaElement(ExpectElement):
def _ref(self):
"""
Return a variable name that is a new reference to this particular
MagmaElement in Magma. This keeps this object from being garbage
collected by Magma, even if all the Sage references to it are
freed.
Important special behavior: When _ref is used during an implicit
call to _magma_init_, then the reference disappears after the
coercion is done. More precisely, if the output of _ref() appears
as part of the output of a call to _magma_init_ that is then
going to be input to magma(...), then it is deleted in the Magma
interface. The main use for this behavior is that in
_magma_init_ it allows you to get a reference to one object, and
use it exactly once in constructing a string to evaluate in Magma,
without having to worry about that object being deallocated. There
are more sophisticated ways that the same problem (with
_magma_init_ and references) could have been solved, but this
solution is much simpler and easier to understand than all others I
came up with. If this doesn't make sense, read the source code to
_coerce_from_special_method, which is much shorter than this
paragraph.
.. warning::
Use _ref sparingly, since it involves a full call to Magma,
which can be slow.
OUTPUT: string
EXAMPLES::
sage: a = magma('-2/3') # optional - magma
sage: s = a._ref(); s # optional - magma
'_sage_ref...'
sage: magma(s) # optional - magma
-2/3
"""
P = self._check_valid()
n = P._next_ref_name()
P.set(n, self.name())
return n
def __getattr__(self, attrname):
"""
INPUT:
- ``attrname`` - string
OUTPUT: a Magma function partially evaluated with self as the first
input.
.. note::
If the input ``attrname`` starts with an underscore, an
AttributeError is raised so that the actual Python _ method/value
can be accessed.
EXAMPLES::
sage: n = magma(-15) # optional - magma
sage: type(n) # optional - magma
<class 'sage.interfaces.magma.MagmaElement'>
sage: f = n.__getattr__('Factorization') # optional - magma
sage: type(f) # optional - magma
<class 'sage.interfaces.magma.MagmaFunctionElement'>
sage: f # optional - magma
Partially evaluated Magma function or intrinsic 'Factorization'
...
"""
if attrname[:1] == "_":
raise AttributeError
return MagmaFunctionElement(self, attrname)
def _sage_(self):
"""
Return Sage version of this object. Use self.sage() to get the Sage
version.
Edit extcode/magma/sage.m to add functionality.
EXAMPLES: Enumerated Sets::
sage: a = magma('{1,2/3,-5/9}') # optional - magma
sage: a.sage() # optional - magma
{1, -5/9, 2/3}
sage: a._sage_() # optional - magma
{1, -5/9, 2/3}
sage: type(a.sage()) # optional - magma
<class 'sage.sets.set.Set_object_enumerated_with_category'>
sage: a = magma('{1,2/3,-5/9}'); a # optional - magma
{ -5/9, 2/3, 1 }
sage: a.Type() # optional - magma
SetEnum
sage: b = a.sage(); b # optional - magma
{1, -5/9, 2/3}
sage: type(b) # optional - magma
<class 'sage.sets.set.Set_object_enumerated_with_category'>
sage: c = magma(b); c # optional - magma
{ -5/9, 2/3, 1 }
sage: c.Type() # optional - magma
SetEnum
Multisets are converted to lists::
sage: m = magma('{* 1,2,2,2,4^^2,3 *}') # optional - magma
sage: z = m.sage(); z # optional - magma
[1, 2, 2, 2, 3, 4, 4]
sage: type(z) # optional - magma
<type 'list'>
Tuples get converted to tuples::
sage: m = magma('<1,2,<3>>') # optional - magma
sage: z = m.sage(); z # optional - magma
(1, 2, (3,))
sage: type(z) # optional - magma
<type 'tuple'>
Sequences get converted to lists::
sage: m = magma('[<1>,<2>]') # optional - magma
sage: z = m.sage(); z # optional - magma
[(1,), (2,)]
sage: type(z) # optional - magma
<type 'list'>
Matrices::
sage: a = matrix(ZZ,3,3,[1..9])
sage: m = magma(a) # optional - magma
sage: b = m.sage(); b # optional - magma
[1 2 3]
[4 5 6]
[7 8 9]
sage: b == a # optional - magma
True
A nonsquare matrix::
sage: a = matrix(ZZ,2,3,[1..6])
sage: m = magma(a) # optional - magma
sage: m.sage() # optional - magma
[1 2 3]
[4 5 6]
Multivariate polynomials::
sage: R.<x,y,z> = QQ[] # optional - magma
sage: f = x^2+3*y # optional - magma
sage: g = magma(f).sage(); g # optional - magma
x^2 + 3*y
sage: parent(f) == parent(g) # optional - magma
True
Number fields and their elements::
sage: x = var('x')
sage: L.<alpha> = NumberField(x^3+2*x+2)
sage: K = magma(L) # optional - magma
sage: K.sage() # optional - magma
Number Field in alpha with defining polynomial x^3 + 2*x + 2
sage: K.sage() is L # optional - magma
True
sage: magma(alpha).sage() # optional - magma
alpha
Relative number field elements can be converted from Magma
to Sage, but the other direction has not yet been implemented.::
sage: P.<y> = L[]
sage: N.<b> = NumberField(y^2-alpha)
sage: M = magma(N) # optional - magma
sage: M.1.sage() # optional - magma
b
sage: _^2 # optional - magma
alpha
sage: magma(b) # optional - magma
Traceback (most recent call last):
...
TypeError: coercion of relative number field elements to Magma is not implemented
Sage does not have absolute number fields defined by
two polynomials, like Magma does. They are converted
to relative number fields. Conversion of their elements
has not yet been implemented.::
sage: magma.eval('P<x> := PolynomialRing(Rationals());') # optional - magma
''
sage: K = magma('NumberField([x^2-2,x^2-3]:Abs);') # optional - magma
sage: L = K.sage(); L # optional - magma
Number Field in K1 with defining polynomial x^2 - 2 over its base field
sage: L.base_field() # optional - magma
Number Field in K2 with defining polynomial x^2 - 3
sage: K.GeneratorsSequence()[1].sage() # optional - magma
Traceback (most recent call last):
...
NameError: name 'K' is not defined
"""
z, preparse = self.Sage(nvals = 2)
s = str(z)
preparse = str(preparse) == 'true'
return sage.misc.sage_eval.sage_eval(s, preparse=preparse)
def AssignNames(self, names):
"""
EXAMPLES::
sage: G = magma.DirichletGroup(20) # optional - magma
sage: G.AssignNames(['a','b']) # optional - magma
sage: G.1 # optional - magma
a
::
sage: G.Elements() # optional - magma
[
1,
a,
b,
a*b
]
"""
P = self._check_valid()
cmd = 'AssignNames(~%s, [%s])'%(self.name(),
','.join('"%s"'%x for x in names))
P.eval(cmd)
assign_names = AssignNames
def gen(self, n):
"""
Return the n-th generator of this Magma element. Note that
generators are 1-based in Magma rather than 0 based!
INPUT:
- ``n`` - a *positive* integer
OUTPUT: MagmaElement
EXAMPLES::
sage: k.<a> = GF(9)
sage: magma(k).gen(1) # optional -- magma
a
sage: R.<s,t,w> = k[]
sage: m = magma(R) # optional -- magma
sage: m.gen(1) # optional -- magma
s
sage: m.gen(2) # optional -- magma
t
sage: m.gen(3) # optional -- magma
w
sage: m.gen(0) # optional -- magma
Traceback (most recent call last):
...
IndexError: index must be positive since Magma indexes are 1-based
sage: m.gen(4) # optional -- magma
Traceback (most recent call last):
...
IndexError: list index out of range
"""
if n <= 0:
raise IndexError, "index must be positive since Magma indexes are 1-based"
return self.gens()[n-1]
def gens(self):
"""
Return generators for self.
If self is named X is Magma, this function evaluates X.1, X.2,
etc., in Magma until an error occurs. It then returns a Sage list
of the resulting X.i. Note - I don't think there is a Magma command
that returns the list of valid X.i. There are numerous ad hoc
functions for various classes but nothing systematic. This function
gets around that problem. Again, this is something that should
probably be reported to the Magma group and fixed there.
AUTHORS:
- William Stein (2006-07-02)
EXAMPLES::
sage: magma("VectorSpace(RationalField(),3)").gens() # optional - magma
[(1 0 0), (0 1 0), (0 0 1)]
sage: magma("AbelianGroup(EllipticCurve([1..5]))").gens() # optional - magma
[$.1]
"""
try:
return self._magma_gens
except AttributeError:
pass
G = []
i = 1
P = self._check_valid()
n = self.name()
while True:
try:
G.append(P('%s.%s'%(n,i)))
except (RuntimeError, TypeError):
break
i += 1
self._magma_gens = G
return G
def gen_names(self):
"""
Return list of Magma variable names of the generators of self.
.. note::
As illustrated below, these are not the print names of the
the generators of the Magma object, but special variable
names in the Magma session that reference the generators.
EXAMPLES::
sage: R.<x,zw> = QQ[]
sage: S = magma(R) # optional - magma
sage: S.gen_names() # optional - magma
('_sage_[...]', '_sage_[...]')
sage: magma(S.gen_names()[1]) # optional - magma
zw
"""
try:
return self.__gen_names
except AttributeError:
self.__gen_names = tuple([x.name() for x in self.gens()])
return self.__gen_names
def evaluate(self, *args):
"""
Evaluate self at the inputs.
INPUT:
- ``*args`` - import arguments
OUTPUT: self(\*args)
EXAMPLES::
sage: f = magma('Factorization') # optional - magma
sage: f.evaluate(15) # optional - magma
[ <3, 1>, <5, 1> ]
sage: f(15) # optional - magma
[ <3, 1>, <5, 1> ]
sage: f = magma('GCD') # optional - magma
sage: f.evaluate(15,20) # optional - magma
5
"""
P = self._check_valid()
v = [P(a) for a in args]
names = ','.join([str(x) for x in v])
return P('%s(%s)'%(self.name(), names))
eval = evaluate
def __call__(self, *args):
"""
Coerce something into the object (using the Magma ! notation).
For function calls, use self.eval(...).
EXAMPLES::
sage: M = magma.RMatrixSpace(magma.IntegerRing(), 2, 2) # optional - magma
sage: A = M([1,2,3,4]); A # optional - magma
[1 2]
[3 4]
sage: type(A) # optional - magma
<class 'sage.interfaces.magma.MagmaElement'>
sage: A.Type() # optional - magma
ModMatRngElt
"""
if len(args) > 1:
return self.evaluate(*args)
P = self._check_valid()
x = P(args[0])
try:
return P('%s!%s'%(self.name(), x.name()))
except (RuntimeError, TypeError):
return self.evaluate(*args)
def __iter__(self):
"""
Return iterator over this Magma element.
OUTPUT: generator object
.. warning::
Internally this constructs the list of elements in self in
Magma, which is not a lazy operation. This is because Magma
doesn't have a notion of lazy iterators, unfortunately.
EXAMPLES::
sage: V = magma('VectorSpace(GF(3),2)') # optional - magma
sage: V # optional - magma
Full Vector space of degree 2 over GF(3)
sage: w = V.__iter__(); w # optional - magma
<generator object __iter__ at ...>
sage: w.next() # optional - magma
(0 0)
sage: w.next() # optional - magma
(1 0)
sage: list(w) # optional - magma
[(2 0), (0 1), (1 1), (2 1), (0 2), (1 2), (2 2)]
"""
P = self._check_valid()
z = P('[_a : _a in %s]'%self.name())
for i in range(1,len(z)+1):
yield z[i]
def __len__(self):
r"""
Return cardinality of this Magma element.
This is the same as ``#self`` in Magma.
EXAMPLES::
sage: V = magma('VectorSpace(GF(3),2)') # optional - magma
sage: V # optional - magma
Full Vector space of degree 2 over GF(3)
sage: len(V) # optional - magma
9
sage: V.__len__() # optional - magma
9
"""
P = self._check_valid()
return int(P.eval('#%s'%self.name()))
def _polynomial_(self, R):
"""
Try to convert self into a polynomial in the univariate polynomial
ring R.
EXAMPLES::
sage: R.<x> = QQ[]
sage: f = magma(x^2 + 2/3*x + 5) # optional - magma
sage: f # optional - magma
x^2 + 2/3*x + 5
sage: f.Type() # optional - magma
RngUPolElt
sage: f._polynomial_(R) # optional - magma
x^2 + 2/3*x + 5
"""
return R(list(self.Eltseq()))
def _latex_(self):
r"""
Return latex representation of self.
AUTHORS:
- Jennifer Balakrishnan
Types that are nicely latex include:
- rationals
- matrices
- polynomials
- binary quadratic forms
- elements of quadratic, cyclotomic number fields, and general
number fields
- points
- elliptic curves
- power series
IMPLEMENTATION: Calls latex.m, which is in
Sage_ROOT/data/extcode/magma/latex.m
EXAMPLES::
sage: latex(magma('-2/3')) # optional - magma
\frac{-2}{3}
sage: magma('-2/3')._latex_() # optional - magma
'\\frac{-2}{3}'
::
sage: magma.eval('R<x> := PolynomialRing(RationalField()); f := (x-17/2)^3;') # optional - magma
''
sage: latex(magma('f')) # optional - magma
x^{3}-\frac{51}{2}x^{2}+\frac{867}{4}x-\frac{4913}{8}
::
sage: latex(magma('(MatrixAlgebra(RationalField(),3)![0,2,3,4,5,6,7,8,9])^(-1)')) # optional - magma
\left(\begin{array}{ccc}-1&2&-1\\2&-7&4\\-1&\frac{14}{3}&\frac{-8}{3}\end{array}\right)
::
sage: magma.eval('K<a> := CyclotomicField(11)') # optional - magma
''
sage: latex(magma('a^3 + a - 17/3')) # optional - magma
\frac{-17}{3}+\zeta_{11}+\zeta_{11}^{3}
::
sage: latex(magma('EllipticCurve([1,2/3,3/4,4/5,-5/6])')) # optional - magma
y^2+xy+\frac{3}{4}y=x^3+\frac{2}{3}x^2+\frac{4}{5}x-\frac{5}{6}
::
sage: _=magma.eval('R<x> := PolynomialRing(RationalField())') # optional - magma
sage: _=magma.eval('K<a> := NumberField(x^3+17*x+2)') # optional - magma
sage: latex(magma('(1/3)*a^2 - 17/3*a + 2')) # optional - magma
2-\frac{17}{3}a+\frac{1}{3}a^{2}
Sage auto-detects the greek letters and puts backslashes in::
sage: _=magma.eval('R<x> := PolynomialRing(RationalField())') # optional - magma
sage: _=magma.eval('K<alpha> := NumberField(x^3+17*x+2)') # optional - magma
sage: latex(magma('(1/3)*alpha^2 - 17/3*alpha + 2')) # optional - magma
2-\frac{17}{3}\alpha+\frac{1}{3}\alpha^{2}
::
sage: _=magma.eval('R<alpha> := PolynomialRing(RationalField())') # optional - magma
sage: latex(magma('alpha^3-1/7*alpha + 3')) # optional - magma
\alpha^{3}-\frac{1}{7}\alpha+3
Finite field elements::
sage: _=magma.eval('K<a> := GF(27)') # optional - magma
sage: latex(magma('a^2+2')) # optional - magma
2+a^{2}
Printing of unnamed (dollar sign) generators works correctly::
sage: latex(magma('FiniteField(81).1^2+1')) # optional - magma
1+\$.1^{2}
Finite fields::
sage: latex(magma('FiniteField(3)')) # optional - magma
\mathbf{F}_{{3}}
sage: latex(magma('FiniteField(27)')) # optional - magma
\mathbf{F}_{{3}^{3}}
Power Series::
sage: _=magma.eval('R<x> := PowerSeriesRing(RationalField())') # optional - magma
sage: latex(magma('(1/(1+x))')) # optional - magma
1-x+x^{2}-x^{3}+x^{4}-x^{5}+x^{6}-x^{7}+x^{8}-x^{9}+x^{10}-x^{11}+x^{12}-x^{13}+x^{14}-x^{15}+x^{16}-x^{17}+x^{18}-x^{19}+O(x^{20})
sage: _=magma.eval('R<x> := PowerSeriesRing(RationalField())') # optional - magma
sage: latex(magma('(-1/(2+x + O(x^3)))')) # optional - magma
\frac{-1}{2}+\frac{1}{4}x-\frac{1}{8}x^{2}+O(x^{3})
p-adic Numbers::
sage: latex(magma('pAdicField(7,4)!9333294394/49')) # optional - magma
4\cdot{}7^{-2} + 5\cdot{}7^{-1} + 5+ 6\cdot{}7^{1} + O(7^{2})
"""
P = self._check_valid()
s = str(P.eval('Latex(%s)'%self.name()))
v = '\\mathrm{'
if s[:len(v)] == v:
raise AttributeError
return s
def set_magma_attribute(self, attrname, value):
"""
INPUTS: attrname - string value - something coercible to a
MagmaElement
EXAMPLES::
sage: V = magma("VectorSpace(RationalField(),2)") # optional - magma
sage: V.set_magma_attribute('M',10) # optional - magma
sage: V.get_magma_attribute('M') # optional - magma
10
sage: V.M # optional - magma
10
"""
P = self.parent()
if not (isinstance(value, MagmaElement) and value.parent() is P):
value = P(value)
P.eval('%s`%s := %s'%(self.name(), attrname, value.name()))
def get_magma_attribute(self, attrname):
"""
Return value of a given Magma attribute. This is like selfattrname
in Magma.
OUTPUT: MagmaElement
EXAMPLES::
sage: V = magma("VectorSpace(RationalField(),10)") # optional - magma
sage: V.set_magma_attribute('M','"hello"') # optional - magma
sage: V.get_magma_attribute('M') # optional - magma
hello
sage: V.M # optional - magma
hello
"""
P = self.parent()
return P('%s`%s'%(self.name(), attrname))
def list_attributes(self):
"""
Return the attributes of self, obtained by calling the
ListAttributes function in Magma.
OUTPUT: list of strings
EXAMPLES: We observe that vector spaces in Magma have numerous
funny and mysterious attributes.
::
sage: V = magma("VectorSpace(RationalField(),2)") # optional - magma
sage: v = V.list_attributes(); v.sort(); v # optional - magma
['Coroots', 'Involution', 'M', 'RootDatum', 'Roots', 'StrLocalData', 'T', 'decomp', 'eisen', 'exthom', 'hSplit', 'ip_form', 'p', 'ssbasis', 'weights']
"""
return magma.eval('ListAttributes(Type(%s))'%\
self.name()).split()
def trait_names(self):
"""
Return all Magma functions that have this Magma element as first
input. This is used for tab completion.
.. note::
This function can unfortunately be slow if there are a very
large number of functions, e.g., when self is an
integer. (This could be fixed by the addition of an
appropriate function to the Magma kernel, which is
something that can only be done by the Magma developers.)
OUTPUT:
- ``list`` - sorted list of distinct strings
EXAMPLES::
sage: R.<x> = ZZ[] # optional - magma
sage: v = magma(R).trait_names() # optional - magma
sage: v # optional - magma
["'*'", "'+'", "'.'", "'/'", "'eq'", "'in'", "'meet'", "'subset'", ...]
"""
M = self.methods()
N = []
for x in M:
i = x.find('(')
N.append(x[:i])
v = list(set(N + self.list_attributes()))
v.sort()
return v
def methods(self, any=False):
"""
Return signatures of all Magma intrinsics that can take self as the
first argument, as strings.
INPUT:
- ``any`` - (bool: default is False) if True, also
include signatures with Any as first argument.
OUTPUT: list of strings
EXAMPLES::
sage: v = magma('2/3').methods() # optional - magma
sage: v[0] # optional - magma
"'*'..."
"""
t = str(self.Type())
X = self.parent().eval('ListSignatures(%s)'%self.Type()).split('\n')
tt = "(<"+t
if any:
Y = [x for x in X if tt in x or "(<Any>" in x]
else:
Y = [x for x in X if tt in x]
return Y
def __floordiv__(self, x):
"""
Quotient of division of self by other. This is denoted // ("div" in
magma).
EXAMPLE::
sage: R.<x,y,z> = QQ[]
sage: magma(5)//magma(2) # optional - magma
2
sage: m = magma(x*z + x*y) # optional - magma
sage: n = magma(x) # optional - magma
sage: m//n # optional - magma
y + z
"""
return self.parent()('%s div %s'%(self.name(), x.name()))
def __nonzero__(self):
"""
Return True if self is nonzero according to Magma. If Magma can't
decide, i.e., raising an error then also return True.
EXAMPLES: We define a Magma vector space.
::
sage: V = magma('VectorSpace(GF(3),2)'); V # optional - magma
Full Vector space of degree 2 over GF(3)
The first generator is nonzero.
::
sage: V.gen(1).__nonzero__() # optional - magma
True
The zero element is zero.
::
sage: V(0).__nonzero__() # optional - magma
False
The space itself is nonzero (the default - in Magma no comparison
to 0 is possible).
::
sage: V.__nonzero__() # optional - magma
True
We can also call bool which is the opposite of __nonzero__.
::
sage: bool(V(0)) # optional - magma
False
sage: bool(V.gen(1)) # optional - magma
True
sage: bool(V) # optional - magma
True
Test use in bool conversions of bools::
sage: bool(magma(False)) # optional - magma
False
sage: bool(magma(True)) # optional - magma
True
sage: bool(magma(1)) # optional - magma
True
sage: bool(magma(0)) # optional - magma
False
"""
try:
return not self.parent()("%s eq 0"%self.name()).bool()
except TypeError:
try:
return not self.parent()("%s eq false"%self.name()).bool()
except TypeError:
pass
return True
def sub(self, gens):
"""
Return the sub-object of self with given gens.
INPUT:
- ``gens`` - object or list/tuple of generators
EXAMPLES::
sage: V = magma('VectorSpace(RationalField(),3)') # optional - magma
sage: W = V.sub([ [1,2,3], [1,1,2] ]); W # optional - magma
Vector space of degree 3, dimension 2 over Rational Field
Generators:
(1 2 3)
(1 1 2)
Echelonized basis:
(1 0 1)
(0 1 1)
"""
return self.parent().bar_call(self, 'sub', gens)
def quo(self, gens):
"""
Return the quotient of self by the given object or list of
generators.
INPUT:
- ``gens`` - object or list/tuple of generators
OUTPUT:
- ``magma element`` - the quotient object
- ``magma element`` - mapping from self to the
quotient object
EXAMPLES::
sage: V = magma('VectorSpace(RationalField(),3)') # optional - magma
sage: V.quo([[1,2,3], [1,1,2]]) # optional - magma
(Full Vector space of degree 1 over Rational Field, Mapping from: Full Vector space of degree 3 over Rational Field to Full Vector space of degree 1 over Rational Field)
We illustrate quotienting out by an object instead of a list of
generators::
sage: W = V.sub([ [1,2,3], [1,1,2] ]) # optional - magma
sage: V.quo(W) # optional - magma
(Full Vector space of degree 1 over Rational Field, Mapping from: Full Vector space of degree 3 over Rational Field to Full Vector space of degree 1 over Rational Field)
We quotient a ZZ module out by a submodule.
::
sage: V = magma.RModule(ZZ,3); V # optional - magma
RModule(IntegerRing(), 3)
sage: W, phi = V.quo([[1,2,3]]) # optional - magma
sage: W # optional - magma
RModule(IntegerRing(), 2)
sage: phi # optional - magma
Mapping from: RModule(IntegerRing(), 3) to RModule(IntegerRing(), 2)
"""
return self.parent().bar_call(self, 'quo', gens, nvals=2)
def ideal(self, gens):
"""
Return the ideal of self with given list of generators.
INPUT:
- ``gens`` - object or list/tuple of generators
OUTPUT:
- ``magma element`` - a Magma ideal
EXAMPLES::
sage: R = magma('PolynomialRing(RationalField())') # optional - magma
sage: R.assign_names(['x']) # optional - magma
sage: x = R.1 # optional - magma
sage: R.ideal([x^2 - 1, x^3 - 1]) # optional - magma
Ideal of Univariate Polynomial Ring in x over Rational Field generated by x - 1
"""
return self.parent().bar_call(self, 'ideal', gens, nvals=1)
magma = Magma()
def reduce_load_Magma():
"""
Used in unpickling a Magma interface.
This functions just returns the global default Magma interface.
EXAMPLES::
sage: sage.interfaces.magma.reduce_load_Magma()
Magma
"""
return magma
def magma_console():
"""
Run a command line Magma session.
EXAMPLES::
sage: magma_console() # not tested
Magma V2.14-9 Sat Oct 11 2008 06:36:41 on one [Seed = 1157408761]
Type ? for help. Type <Ctrl>-D to quit.
>
Total time: 2.820 seconds, Total memory usage: 3.95MB
"""
console('sage-native-execute magma')
def magma_version():
"""
Return the version of Magma that you have in your PATH on your
computer.
OUTPUT:
- ``numbers`` - 3-tuple: major, minor, etc.
- ``string`` - version as a string
EXAMPLES::
sage: magma_version() # random, optional - magma
((2, 14, 9), 'V2.14-9')
"""
t = tuple([int(n) for n in magma.eval('GetVersion()').split()])
return t, 'V%s.%s-%s'%t
class MagmaGBLogPrettyPrinter:
"""
A device which filters Magma Groebner basis computation logs.
"""
cmd_inpt = re.compile("^>>>$")
app_inpt = re.compile("^Append\(~_sage_, 0\);$")
deg_curr = re.compile("^Basis length\: (\d+), queue length\: (\d+), step degree\: (\d+), num pairs\: (\d+)$")
pol_curr = re.compile("^Number of pair polynomials\: (\d+), at (\d+) column\(s\), .*")
def __init__(self, verbosity=1, style='magma'):
"""
Construct a new Magma Groebner Basis log pretty printer.
INPUT:
- ``verbosity`` - how much information should be printed
(between 0 and 1)
- ``style`` - if "magma" the full Magma log is printed; if
'sage' only the current degree and the number of pairs in
the queue is printed (default: "magma").
EXAMPLE::
sage: P.<x,y,z> = GF(32003)[]
sage: I = sage.rings.ideal.Cyclic(P)
sage: _ = I.groebner_basis('magma',prot='sage') # indirect doctest, optional - magma, not tested
Leading term degree: 2. Critical pairs: 2.
Leading term degree: 3. Critical pairs: 1.
Highest degree reached during computation: 3.
sage: P.<x,y,z> = GF(32003)[]
sage: I = sage.rings.ideal.Cyclic(P)
sage: _ = I.groebner_basis('magma',prot=True) # indirect doctest, optional - magma, not tested
Homogeneous weights search
Number of variables: 3, nullity: 1
Exact search time: 0.000
********************
FAUGERE F4 ALGORITHM
********************
Coefficient ring: GF(32003)
Rank: 3
Order: Graded Reverse Lexicographical
NEW hash table
Matrix kind: Modular FP
Datum size: 4
No queue sort
Initial length: 3
Inhomogeneous
Initial queue setup time: 0.000
Initial queue length: 2
*******
STEP 1
Basis length: 3, queue length: 2, step degree: 2, num pairs: 1
Basis total mons: 8, average length: 2.667
Number of pair polynomials: 1, at 4 column(s), 0.000
...
Total Faugere F4 time: 0.000, real time: 0.000
sage: set_random_seed(1)
sage: sr = mq.SR(1,1,2,4)
sage: F,s = sr.polynomial_system()
sage: I = F.ideal()
sage: _ = I.groebner_basis('magma',prot='sage') # indirect doctest, optional - magma, not tested
Leading term degree: 1. Critical pairs: 40.
Leading term degree: 2. Critical pairs: 40.
Leading term degree: 3. Critical pairs: 38.
Leading term degree: 2. Critical pairs: 327.
Leading term degree: 2. Critical pairs: 450.
Leading term degree: 2. Critical pairs: 416.
Leading term degree: 3. Critical pairs: 415.
Leading term degree: 4. Critical pairs: 98 (all pairs of current degree eliminated by criteria).
Leading term degree: 5. Critical pairs: 3 (all pairs of current degree eliminated by criteria).
Highest degree reached during computation: 3.
"""
self.verbosity = verbosity
self.style = style
self.curr_deg = 0
self.curr_npairs = 0
self.max_deg = 0
self.storage = ""
self.sync = None
def write(self, s):
"""
EXAMPLE::
sage: P.<x,y,z> = GF(32003)[]
sage: I = sage.rings.ideal.Katsura(P)
sage: _ = I.groebner_basis('magma',prot=True) # indirect doctest, optional - magma
<BLANKLINE>
Homogeneous weights search
...
Total Faugere F4 time: ..., real time: ...
"""
verbosity,style = self.verbosity,self.style
if self.storage:
s = self.storage + s
self.storage = ""
for line in s.splitlines():
match = re.match(MagmaGBLogPrettyPrinter.cmd_inpt,line)
if match:
self.sync = 1
continue
if self.sync:
if self.sync == 1:
self.sync = line
continue
else:
if line == '':
continue
self.sync = None
continue
if re.match(MagmaGBLogPrettyPrinter.app_inpt,line):
continue
if re.match(MagmaGBLogPrettyPrinter.deg_curr,line):
match = re.match(MagmaGBLogPrettyPrinter.deg_curr,line)
nbasis,npairs,deg,npairs_deg = map(int,match.groups())
self.curr_deg = deg
self.curr_npairs = npairs
if re.match(MagmaGBLogPrettyPrinter.pol_curr,line):
match = re.match(MagmaGBLogPrettyPrinter.pol_curr,line)
pol_curr,col_curr = map(int,match.groups())
if pol_curr != 0:
if self.max_deg < self.curr_deg:
self.max_deg = self.curr_deg
if style == "sage" and verbosity >= 1:
print "Leading term degree: %2d. Critical pairs: %d."%(self.curr_deg,self.curr_npairs)
else:
if style == "sage" and verbosity >= 1:
print "Leading term degree: %2d. Critical pairs: %d (all pairs of current degree eliminated by criteria)."%(self.curr_deg,self.curr_npairs)
if style == "magma" and verbosity >= 1:
print line
def flush(self):
"""
EXAMPLE::
sage: from sage.interfaces.magma import MagmaGBLogPrettyPrinter
sage: logs = MagmaGBLogPrettyPrinter()
sage: logs.flush()
"""
import sys
sys.stdout.flush()
