Magma versus Sage
The goal of Magma is to provides a mathematically rigorous environment for solving computationally hard problems in algebra, number theory, geometry and combinatorics. The core goal of Sage is to provide a free open source alternative to Magma. This includes being able to do everything Magma does and to do it better. This page is meant to track our progress in that direction. We intend to accomplish this by some combination of: (1) extremely hard work, (2) better technology, (3) getting Magma open sourced and incorporated into Sage, (4) sharing of effort (e.g., people who work for Magma sharing the code and algorithms they produce with the Sage community).
The point of this page is to list functionality that Magma has and whether Sage has it or not, and if Sage has said functionality, how does the speed compare. It is basically to answer the question "can Magma still do anything Sage can't do".
The main reference for what Magma does is the Magma reference manual.
Contents
Functionality in Magma not in Sage
There are tons of things that Magma does that Sage also obviously does, e.g., "compute with univariate polynomials". The goal is to list here only things that Magma does that Sage doesn't do. Also, if Magma can do something much more efficiently than Sage, it should be listed here.
Platform Support
Magma officially support the following hardware/OS platforms that Sage does not officially support:
Alpha (Linux), Alpha (OSF/Tru64), IBM PowerPC64 (AIX), IBM PowerPC64 (Linux), Macintosh 64-bit Intel (OS X 10.5 [Leopard]), Sparc (Solaris), Sparc64 (Solaris)
Notes:
- Nobody cares about Alpha support anymore.
- We do not have access to any PPC linux boxes.
- We do not have access to any AIX boxes.
There is active work to port sage to 64-bit MacIntel, but it is not done. Libsingular, fortran, and pexpect issues remain. Sage-3.4.1.alpha0 builds with the spkg at http://trac.sagemath.org/sage_trac/ticket/5057, but does not start.
Univariate Polynomials
Interpolation
Magma has the following function, which Sage doesn't have.
Interpolation(I, V) : [ RngElt ], [ RngElt ] -> RngUPolElt
This function finds a univariate polynomial that evaluates to the values V in the interpolation points I. Let K be a field and n>0 an integer; given sequences I and V, both consisting of n elements of K, return the unique univariate polynomial p over K of degree less than n such that p(I[i]) = V[i] for each 1≤i≤n. The corresponding Sage function would likely work like this:
sage: R.<x> = QQ[] sage: f = R.interpolation([1,2/3,3], [-1,2,3/5])
Then f would be the monic polynomial over QQ s.t. f(1)=-1, f(2/3)=2, f(3)=3/5.
Reductum
Magma has this and Sage doesn't:
Reductum(f) : RngUPolElt -> RngUPolElt
The reductum of a polynomial f, which is the polynomial obtained by removing the leading term of f.
PowerPolynomial
Magma has this and Sage doesn't:
PowerPolynomial(f,n) : RngUPolElt, RngIntElt -> RngUPolElt
The polynomial whose roots are the nth powers of the roots of the given polynomial (which should have coefficients in some field).
PrimitivePart, MaxNorm, SumNorm
For polynomials over ZZ, Magma has these and Sage doesn't:
PrimitivePart(p) : RngUPolElt -> RngUPolElt
The primitive part of p, being p divided by the content of p.
MaxNorm(p) : RngUPolElt -> RngIntElt
The maximum of the absolute values of the coefficients of p.
SumNorm(p) : RngUPolElt -> RngIntElt
The sum of the coefficients of p.
DedekindTest(p, m) : RngUPolElt, RngIntElt -> Boolelt
Given a monic polynomial p (univariate or multivariate in one variable) and a
prime number m, this returns true if p satisfies the Dedekind criterion at m,
and false otherwise. The Dedekind criterion is satisfied at m if and only if
the equation order corresponding to p is locally maximal at m [PZ89, p. 295]. In Sage we would have:
sage: R.<x> = ZZ[] sage: f = 6*(x^3 + 2*x + 3) sage: f.primitive_part() x^3 + 2*x + 3 sage: f.max_norm() sage: f.sum_norm() sage: f.dedekind_test(3)
Normalize
Magma has this and Sage doesn't, and it looks like it could be useful.
Normalize(f) : RngUPolElt -> RngUPolElt
Given a univariate polynomial f over the ring R, this function returns the unique
normalized polynomial g which is associate to f (so g=uf for some unit in R). This is
chosen so that if R is a field then g is monic, if R is Z then the leading coefficient
of g is positive, if R is a polynomial ring itself, then the leading coefficient of g
is recursively normalized, and so on for other rings.
Factorization
Sage is missing these:
DistinctDegreeFactorization(f) : RngUPolElt -> [ <RngIntElt, RngUPolElt> ]
Degree: RngIntElt Default: 0
Given a squarefree univariate polynomial f∈F[x] with F a finite field, this
function returns the distinct-degree factorization of f as a sequence of pairs,
each consisting of a degree d, together with the product of the degree-d
irreducible factors of f.
If the optional parameter Degree is given a value L > 0, then only (products of)
factors up to degree L are returned. EqualDegreeFactorization(f, d, g) : RngUPolElt, RngIntElt, RngUPolElt -> [ RngUPolElt ]
Given a squarefree univariate polynomial f∈F[x] with F a finite field, and
integer d and another polynomial g∈F[x] such that F is known to be the product of
distinct degree-d irreducible polynomials alone, and g is xq mod f, where q is the
cardinality of F, this function returns the irreducible factors of f as a sequence
of polynomials (no multiplicities are needed).
If the conditions are not satisfied, the result is unpredictable. This function
allows one to split f, assuming that one has computed f in some special way. Sage should definitely have this, but doesn't.
IsSeparable(f) : RngUPolElt -> BoolElt
Given a polynomial f∈K[x] such that f is a polynomial of degree ≥1 and K
is a field allowing polynomial factorization, this function returns true
iff f is separable.
Misc poly functions
I have no idea what QMatrix is...
QMatrix(f) : RngUPolElt -> AlgMatElt
Given a univariate polynomial f of degree d over a finite field F this
function returns the Berlekamp Q-matrix associated with f, which is an
element of the degree d - 1 matrix algebra over F. Sage doesn't have this easy function:
CompanionMatrix(f) : RngUPolElt -> AlgMatElt
Given a monic univariate polynomial f of degree d over some ring R, return the
companion matrix of f as an element of the full matrix algebra of degree d - 1 over R.
The companion matrix for f=a0 + a1x + ... + a1x1 + xd is given by
[ 0 1 0 ... 0 ]
[ 0 0 1 ... 0 ]
[ . . . . . ]
[ . . . . . ]
[ . . . . . ]
[ -a_0 -a_1 -a_2 ... -a_(d-1) ]
Hensel Lifting
HenselLift(f, s, P) : RngUPolElt, [ RngUPolElt ], RngUPol -> [ RngUPolElt ]
Given the sequence of irreducible factors s modulo some prime p of the univariate
integer polynomial f, return the Hensel lifting into the polynomial ring P, which must
be the univariate polynomial ring over a residue class ring modulo some power of p.
Thus given f = ∏i si mod p, this returns f = ∏i ti mod pk for some k ≥1, as a
sequence of polynomials in Z/pkZ. The factorization of f modulo p must be
squarefree, that is, s should not contain repeated factors. See http://magma.maths.usyd.edu.au/magma/htmlhelp/text304.htm#1869 for an example of Hensel lifting in Magma.
Specialized Functionality in Magma also in Sage
Here we list specialized things Magma does that Sage also does. For example, both Magma and Sage have extensive support for computing with modular symbols (far beyond all other math software).
Specialized Functionality in Sage not in Magma
Here we list functionality in Sage that Magma doesn't have, but is functionality that is part of Magma's core goals, i.e., solving computationally hard problems in algebra, number theory, geometry and combinatorics. Note that because Sage does calculus, graphics, statics and numerical computation, there are thousands of functions and features in Sage that are not in Magma and never will be in Magma, and this won't be mentioned here.