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.

🤖 Generated with Claude Code

[rocky]

This name seems specialized, but the concept isn't - it is a matrix in nested list form.

BTW, whenever you see List or Tuple, that should now be list and tuple since we no longer support Python 3.9 where it had to be List, and Tuple.

[tombouctou]

Done in d141405. The type is now NestedListMatrix = list[list[Any]] in mathics.core.convert.matrix, and List/Tuple are now list/tuple.

[rocky]

Given that this is really a matrix in list-of-lists form, this conversion function, which is also generic, should be moved under mathics.core.convert somewhere.

[tombouctou]

Moved to mathics.core.convert.matrix as to_mathics_matrix(rows, elements_conversion_fn=from_python). It is built on to_mathics_list, and the entry conversion is a parameter, so it is not tied to integers.