fmpz_poly_mat.h – matrices of polynomials over the integers¶

The fmpz_poly_mat_t data type represents matrices whose entries are integer polynomials.

The fmpz_poly_mat_t type is defined as an array of fmpz_poly_mat_struct’s of length one. This permits passing parameters of type fmpz_poly_mat_t by reference.

An integer polynomial matrix internally consists of a single array of fmpz_poly_struct’s, representing a dense matrix in row-major order. This array is only directly indexed during memory allocation and deallocation. A separate array holds pointers to the start of each row, and is used for all indexing. This allows the rows of a matrix to be permuted quickly by swapping pointers.

Matrices having zero rows or columns are allowed.

The shape of a matrix is fixed upon initialisation. The user is assumed to provide input and output variables whose dimensions are compatible with the given operation.

Simple example¶

The following example constructs the matrix \(\begin{pmatrix} 2x+1 & x \\ 1-x & -1 \end{pmatrix}\) and computes its determinant.

#include "fmpz_poly.h"
#include "fmpz_poly_mat.h"
int main()
{
    fmpz_poly_mat_t A;
    fmpz_poly_t P;

    fmpz_poly_mat_init(A, 2, 2);
    fmpz_poly_init(P);

    fmpz_poly_set_str(fmpz_poly_mat_entry(A, 0, 0), "2  1 2");
    fmpz_poly_set_str(fmpz_poly_mat_entry(A, 0, 1), "2  0 1");
    fmpz_poly_set_str(fmpz_poly_mat_entry(A, 1, 0), "2  1 -1");
    fmpz_poly_set_str(fmpz_poly_mat_entry(A, 1, 1), "1  -1");

    fmpz_poly_mat_det(P, A);
    fmpz_poly_print_pretty(P, "x");

    fmpz_poly_clear(P);
    fmpz_poly_mat_clear(A);
}

The output is:

x^2-3*x-1

Types, macros and constants¶

type fmpz_poly_mat_struct¶

type fmpz_poly_mat_t¶

Memory management¶

void fmpz_poly_mat_init(fmpz_poly_mat_t mat, slong rows, slong cols)¶: Initialises a matrix with the given number of rows and columns for use.

void fmpz_poly_mat_init_set(fmpz_poly_mat_t mat, const fmpz_poly_mat_t src)¶: Initialises a matrix mat of the same dimensions as src, and sets it to a copy of src.

void fmpz_poly_mat_clear(fmpz_poly_mat_t mat)¶: Frees all memory associated with the matrix. The matrix must be reinitialised if it is to be used again.

Basic properties¶

slong fmpz_poly_mat_nrows(const fmpz_poly_mat_t mat)¶: Returns the number of rows in mat.

slong fmpz_poly_mat_ncols(const fmpz_poly_mat_t mat)¶: Returns the number of columns in mat.

Basic assignment and manipulation¶

fmpz_poly_struct *fmpz_poly_mat_entry(const fmpz_poly_mat_t mat, slong i, slong j)¶: Gives a reference to the entry at row i and column j. The reference can be passed as an input or output variable to any fmpz_poly function for direct manipulation of the matrix element. No bounds checking is performed.

void fmpz_poly_mat_set(fmpz_poly_mat_t mat1, const fmpz_poly_mat_t mat2)¶: Sets mat1 to a copy of mat2.

void fmpz_poly_mat_swap(fmpz_poly_mat_t mat1, fmpz_poly_mat_t mat2)¶: Swaps mat1 and mat2 efficiently.

void fmpz_poly_mat_swap_entrywise(fmpz_poly_mat_t mat1, fmpz_poly_mat_t mat2)¶: Swaps two matrices by swapping the individual entries rather than swapping the contents of the structs.

Input and output¶

void fmpz_poly_mat_print(const fmpz_poly_mat_t mat, const char *x)¶: Prints the matrix mat to standard output, using the variable x.

Random matrix generation¶

void fmpz_poly_mat_randtest(fmpz_poly_mat_t mat, flint_rand_t state, slong len, flint_bitcnt_t bits)¶: This is equivalent to applying fmpz_poly_randtest to all entries in the matrix.

void fmpz_poly_mat_randtest_unsigned(fmpz_poly_mat_t mat, flint_rand_t state, slong len, flint_bitcnt_t bits)¶: This is equivalent to applying fmpz_poly_randtest_unsigned to all entries in the matrix.

void fmpz_poly_mat_randtest_sparse(fmpz_poly_mat_t A, flint_rand_t state, slong len, flint_bitcnt_t bits, float density)¶

Creates a random matrix with the amount of nonzero entries given approximately by the density variable, which should be a fraction between 0 (most sparse) and 1 (most dense).

The nonzero entries will have random lengths between 1 and len.

Special matrices¶

void fmpz_poly_mat_zero(fmpz_poly_mat_t mat)¶: Sets mat to the zero matrix.

void fmpz_poly_mat_one(fmpz_poly_mat_t mat)¶: Sets mat to the unit or identity matrix of given shape, having the element 1 on the main diagonal and zeros elsewhere. If mat is nonsquare, it is set to the truncation of a unit matrix.

Basic comparison and properties¶

int fmpz_poly_mat_equal(const fmpz_poly_mat_t mat1, const fmpz_poly_mat_t mat2)¶: Returns nonzero if mat1 and mat2 have the same shape and all their entries agree, and returns zero otherwise.

int fmpz_poly_mat_is_zero(const fmpz_poly_mat_t mat)¶: Returns nonzero if all entries in mat are zero, and returns zero otherwise.

int fmpz_poly_mat_is_one(const fmpz_poly_mat_t mat)¶: Returns nonzero if all entries of mat on the main diagonal are the constant polynomial 1 and all remaining entries are zero, and returns zero otherwise. The matrix need not be square.

int fmpz_poly_mat_is_empty(const fmpz_poly_mat_t mat)¶: Returns a non-zero value if the number of rows or the number of columns in mat is zero, and otherwise returns zero.

int fmpz_poly_mat_is_square(const fmpz_poly_mat_t mat)¶: Returns a non-zero value if the number of rows is equal to the number of columns in mat, and otherwise returns zero.

Norms¶

slong fmpz_poly_mat_max_bits(const fmpz_poly_mat_t A)¶: Returns the maximum number of bits among the coefficients of the entries in A, or the negative of that value if any coefficient is negative.

slong fmpz_poly_mat_max_length(const fmpz_poly_mat_t A)¶: Returns the maximum polynomial length among all the entries in A.

Transpose¶

void fmpz_poly_mat_transpose(fmpz_poly_mat_t B, const fmpz_poly_mat_t A)¶: Sets \(B\) to \(A^t\).

Evaluation¶

void fmpz_poly_mat_evaluate_fmpz(fmpz_mat_t B, const fmpz_poly_mat_t A, const fmpz_t x)¶: Sets the fmpz_mat_t B to A evaluated entrywise at the point x.

Arithmetic¶

void fmpz_poly_mat_scalar_mul_fmpz_poly(fmpz_poly_mat_t B, const fmpz_poly_mat_t A, const fmpz_poly_t c)¶: Sets B to A multiplied entrywise by the polynomial c.

void fmpz_poly_mat_scalar_mul_fmpz(fmpz_poly_mat_t B, const fmpz_poly_mat_t A, const fmpz_t c)¶: Sets B to A multiplied entrywise by the integer c.

void fmpz_poly_mat_add(fmpz_poly_mat_t C, const fmpz_poly_mat_t A, const fmpz_poly_mat_t B)¶: Sets C to the sum of A and B. All matrices must have the same shape. Aliasing is allowed.

void fmpz_poly_mat_sub(fmpz_poly_mat_t C, const fmpz_poly_mat_t A, const fmpz_poly_mat_t B)¶: Sets C to the sum of A and B. All matrices must have the same shape. Aliasing is allowed.

void fmpz_poly_mat_neg(fmpz_poly_mat_t B, const fmpz_poly_mat_t A)¶: Sets B to the negation of A. The matrices must have the same shape. Aliasing is allowed.

void fmpz_poly_mat_mul(fmpz_poly_mat_t C, const fmpz_poly_mat_t A, const fmpz_poly_mat_t B)¶: Sets C to the matrix product of A and B. The matrices must have compatible dimensions for matrix multiplication. Aliasing is allowed. This function automatically chooses between classical and KS multiplication.

void fmpz_poly_mat_mul_classical(fmpz_poly_mat_t C, const fmpz_poly_mat_t A, const fmpz_poly_mat_t B)¶: Sets C to the matrix product of A and B, computed using the classical algorithm. The matrices must have compatible dimensions for matrix multiplication. Aliasing is allowed.

void fmpz_poly_mat_mul_KS(fmpz_poly_mat_t C, const fmpz_poly_mat_t A, const fmpz_poly_mat_t B)¶: Sets C to the matrix product of A and B, computed using Kronecker segmentation. The matrices must have compatible dimensions for matrix multiplication. Aliasing is allowed.

void fmpz_poly_mat_mullow(fmpz_poly_mat_t C, const fmpz_poly_mat_t A, const fmpz_poly_mat_t B, slong len)¶: Sets C to the matrix product of A and B, truncating each entry in the result to length len. Uses classical matrix multiplication. The matrices must have compatible dimensions for matrix multiplication. Aliasing is allowed.

void fmpz_poly_mat_sqr(fmpz_poly_mat_t B, const fmpz_poly_mat_t A)¶: Sets B to the square of A, which must be a square matrix. Aliasing is allowed. This function automatically chooses between classical and KS squaring.

void fmpz_poly_mat_sqr_classical(fmpz_poly_mat_t B, const fmpz_poly_mat_t A)¶: Sets B to the square of A, which must be a square matrix. Aliasing is allowed. This function uses direct formulas for very small matrices, and otherwise classical matrix multiplication.

void fmpz_poly_mat_sqr_KS(fmpz_poly_mat_t B, const fmpz_poly_mat_t A)¶: Sets B to the square of A, which must be a square matrix. Aliasing is allowed. This function uses Kronecker segmentation.

void fmpz_poly_mat_sqrlow(fmpz_poly_mat_t B, const fmpz_poly_mat_t A, slong len)¶: Sets B to the square of A, which must be a square matrix, truncating all entries to length len. Aliasing is allowed. This function uses direct formulas for very small matrices, and otherwise classical matrix multiplication.

void fmpz_poly_mat_pow(fmpz_poly_mat_t B, const fmpz_poly_mat_t A, ulong exp)¶: Sets B to A raised to the power exp, where A is a square matrix. Uses exponentiation by squaring. Aliasing is allowed.

void fmpz_poly_mat_pow_trunc(fmpz_poly_mat_t B, const fmpz_poly_mat_t A, ulong exp, slong len)¶: Sets B to A raised to the power exp, truncating all entries to length len, where A is a square matrix. Uses exponentiation by squaring. Aliasing is allowed.

void fmpz_poly_mat_prod(fmpz_poly_mat_t res, fmpz_poly_mat_t *const factors, slong n)¶: Sets res to the product of the n matrices given in the vector factors, all of which must be square and of the same size. Uses binary splitting.

Row reduction¶

slong fmpz_poly_mat_find_pivot_any(const fmpz_poly_mat_t mat, slong start_row, slong end_row, slong c)¶

Attempts to find a pivot entry for row reduction. Returns a row index \(r\) between start_row (inclusive) and stop_row (exclusive) such that column \(c\) in mat has a nonzero entry on row \(r\), or returns -1 if no such entry exists.

This implementation simply chooses the first nonzero entry it encounters. This is likely to be a nearly optimal choice if all entries in the matrix have roughly the same size, but can lead to unnecessary coefficient growth if the entries vary in size.

slong fmpz_poly_mat_find_pivot_partial(const fmpz_poly_mat_t mat, slong start_row, slong end_row, slong c)¶

Attempts to find a pivot entry for row reduction. Returns a row index \(r\) between start_row (inclusive) and stop_row (exclusive) such that column \(c\) in mat has a nonzero entry on row \(r\), or returns -1 if no such entry exists.

This implementation searches all the rows in the column and chooses the nonzero entry of smallest degree. If there are several entries with the same minimal degree, it chooses the entry with the smallest coefficient bit bound. This heuristic typically reduces coefficient growth when the matrix entries vary in size.

slong fmpz_poly_mat_fflu(fmpz_poly_mat_t B, fmpz_poly_t den, slong *perm, const fmpz_poly_mat_t A, int rank_check)¶

Uses fraction-free Gaussian elimination to set (B, den) to a fraction-free LU decomposition of A and returns the rank of A. Aliasing of A and B is allowed.

Pivot elements are chosen with fmpz_poly_mat_find_pivot_partial. If perm is non-NULL, the permutation of rows in the matrix will also be applied to perm.

If rank_check is set, the function aborts and returns 0 if the matrix is detected not to have full rank without completing the elimination.

The denominator den is set to \(\pm \operatorname{det}(A)\), where the sign is decided by the parity of the permutation. Note that the determinant is not generally the minimal denominator.

slong fmpz_poly_mat_rref(fmpz_poly_mat_t B, fmpz_poly_t den, const fmpz_poly_mat_t A)¶

Sets (B, den) to the reduced row echelon form of A and returns the rank of A. Aliasing of A and B is allowed.

The denominator den is set to \(\pm \operatorname{det}(A)\). Note that the determinant is not generally the minimal denominator.

Trace¶

void fmpz_poly_mat_trace(fmpz_poly_t trace, const fmpz_poly_mat_t mat)¶: Computes the trace of the matrix, i.e. the sum of the entries on the main diagonal. The matrix is required to be square.

Determinant and rank¶

void fmpz_poly_mat_det(fmpz_poly_t det, const fmpz_poly_mat_t A)¶: Sets det to the determinant of the square matrix A. Uses a direct formula, fraction-free LU decomposition, or interpolation, depending on the size of the matrix.

void fmpz_poly_mat_det_fflu(fmpz_poly_t det, const fmpz_poly_mat_t A)¶: Sets det to the determinant of the square matrix A. The determinant is computed by performing a fraction-free LU decomposition on a copy of A.

void fmpz_poly_mat_det_interpolate(fmpz_poly_t det, const fmpz_poly_mat_t A)¶: Sets det to the determinant of the square matrix A. The determinant is computed by determining a bound \(n\) for its length, evaluating the matrix at \(n\) distinct points, computing the determinant of each integer matrix, and forming the interpolating polynomial.

slong fmpz_poly_mat_rank(const fmpz_poly_mat_t A)¶: Returns the rank of A. Performs fraction-free LU decomposition on a copy of A.

Inverse¶

int fmpz_poly_mat_inv(fmpz_poly_mat_t Ainv, fmpz_poly_t den, const fmpz_poly_mat_t A)¶

Sets (Ainv, den) to the inverse matrix of A. Returns 1 if A is nonsingular and 0 if A is singular. Aliasing of Ainv and A is allowed.

More precisely, det will be set to the determinant of A and Ainv will be set to the adjugate matrix of A. Note that the determinant is not necessarily the minimal denominator.

Uses fraction-free LU decomposition, followed by solving for the identity matrix.

Nullspace¶

slong fmpz_poly_mat_nullspace(fmpz_poly_mat_t res, const fmpz_poly_mat_t mat)¶

Computes the right rational nullspace of the matrix mat and returns the nullity.

More precisely, assume that mat has rank \(r\) and nullity \(n\). Then this function sets the first \(n\) columns of res to linearly independent vectors spanning the nullspace of mat. As a result, we always have rank(res) \(= n\), and mat \(\times\) res is the zero matrix.

The computed basis vectors will not generally be in a reduced form. In general, the polynomials in each column vector in the result will have a nontrivial common GCD.

Solving¶

int fmpz_poly_mat_solve(fmpz_poly_mat_t X, fmpz_poly_t den, const fmpz_poly_mat_t A, const fmpz_poly_mat_t B)¶

Solves the equation \(AX = B\) for nonsingular \(A\). More precisely, computes (X, den) such that \(AX = B \times \operatorname{den}\). Returns 1 if \(A\) is nonsingular and 0 if \(A\) is singular. The computed denominator will not generally be minimal.

Uses fraction-free LU decomposition followed by fraction-free forward and back substitution.

int fmpz_poly_mat_solve_fflu(fmpz_poly_mat_t X, fmpz_poly_t den, const fmpz_poly_mat_t A, const fmpz_poly_mat_t B)¶