Add the Modulus option to RowReduce, NullSpace, MatrixRank, Inverse and LinearSolve

[tombouctou]

RowReduce, NullSpace, MatrixRank, Inverse and LinearSolve did not accept the Modulus option, so for example NullSpace[{{1, 1}, {1, 1}}, Modulus -> 2] stayed unevaluated. This adds it for a prime modulus, as described in the WMA docs ("If p is zero, ordinary arithmetic is used").

What it does. With Modulus -> p the matrix entries are reduced mod p and the computation is plain Gaussian elimination on Python ints (new module mathics/eval/numbers/linalg_modular.py). Entries may be integers, or rationals whose denominator is invertible mod p. Results are integers in 0..p-1:

RowReduce[{{1, 0, 4}, {2, 0, 3}, {2, 1, 2}}, Modulus -> 5]    (* {{1, 0, 4}, {0, 1, 4}, {0, 0, 0}} *)
MatrixRank[{{1, 0, 4}, {2, 0, 3}, {2, 1, 2}}, Modulus -> 5]   (* 2 *)
NullSpace[{{1, 1, 0}, {1, 0, 1}, {0, 1, 1}}, Modulus -> 2]     (* {{1, 1, 1}} *)
Inverse[{{12, -2}, {1, 3}}, Modulus -> 5]                     (* {{1, 4}, {3, 4}} *)
LinearSolve[{{1, 1, 1}, {1, 2, 3}, {1, 4, 9}}, {1, 2, 3}, Modulus -> 47]   (* {23, 2, 23} *)

Conventions:

• RowReduce gives the reduced row echelon form, as without Modulus.
• NullSpace gives one vector per non-pivot column, with 1 in that column and 0 in the other non-pivot columns, in decreasing order of that column. That order is what WMA gives: running Strickland's LambdaAlgebra.m (below) reproduces his WMA-computed homology bases exactly only with this order. The exact NullSpace (without Modulus) currently lists them in increasing order; I left that alone here.
• LinearSolve sets the free variables to 0.
• A singular matrix and an inconsistent system give the existing Inverse::sing and LinearSolve::nosol messages.

What it does not do. A modulus that is not a prime (WMA says the computation "may fail" then), a non-integer modulus, a symbolic entry, or a denominator divisible by p leave the expression unevaluated. Modulus -> 0 keeps the current code path.

Tests. Doc examples, plus test_linalg_modulus and test_linalg_modulus_random in test/builtin/numbers/test_linalg.py; the random test checks the defining properties (rank + nullity, m . ns == 0, m . x == b, m . Inverse[m] == 1 mod p) on 60 random matrices. pytest and the doctests give the same results as on master; the only failures are environmental ones that also fail on master. mypy is clean.

Motivation and a larger check: Strickland's LambdaAlgebra.m (https://strickland1.org/toda/) computes the homology of the mod-2 lambda algebra with all five functions and Modulus -> 2, and his LambdaData.m stores the results computed in WMA. With this PR (and the Transpose and MapAt fixes in #2007 and #2008) Mathics3 reproduces exactly, in the same order, his homology, boundary and non-boundary bases and his canonical forms: 218 of the 296 tridegrees in LambdaData.m (4225 canonical forms), with no difference. The remaining 78 are too slow to finish under Mathics3 in reasonable time; the largest would take days. For comparison, under Wolfram Engine 15 the same script computes all 296 in under a minute and matches LambdaData.m exactly.

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