nmod_poly_mat.h – matrices of univariate polynomials over integers mod n (word-size n)¶

The nmod_poly_mat_t data type represents matrices whose entries are polynomials having coefficients in \(\mathbb{Z}/n\mathbb{Z}\). We generally assume that \(n\) is a prime number.

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

A matrix internally consists of a single array of nmod_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.

Types, macros and constants¶

type nmod_poly_mat_struct¶

type nmod_poly_mat_t¶

Memory management¶

void nmod_poly_mat_init(nmod_poly_mat_t mat, slong rows, slong cols, mp_limb_t n)¶: Initialises a matrix with the given number of rows and columns for use. The modulus is set to \(n\).

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

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

Truncate, shift¶

void nmod_poly_mat_set_trunc(nmod_poly_mat_t res, const nmod_poly_mat_t pmat, long len)¶: Set res to the truncation of pmat to length len. Entries of res are normalized.

void nmod_poly_mat_truncate(nmod_poly_mat_t pmat, long len)¶: Truncates pmat to the given length len, and normalize its entries. If len is greater than the maximum length of the entries of pmat, then nothing happens.

void nmod_poly_mat_shift_left(nmod_poly_mat_t res, const nmod_poly_mat_t pmat, slong k)¶: Sets res to pmat shifted left by k coefficients, that is, multiplied by \(x^k\).

void nmod_poly_mat_shift_right(nmod_poly_mat_t res, const nmod_poly_mat_t pmat, slong k)¶: Sets res to pmat shifted right by k coefficients, that is, divide by \(x^k\) and throw away the remainder. If k is greater than or equal to the length of pmat, the result is the zero polynomial matrix.

Basic properties¶

slong nmod_poly_mat_nrows(const nmod_poly_mat_t mat)¶: Returns the number of rows in mat.

slong nmod_poly_mat_ncols(const nmod_poly_mat_t mat)¶: Returns the number of columns in mat.

mp_limb_t nmod_poly_mat_modulus(const nmod_poly_mat_t mat)¶: Returns the modulus of mat.

Basic assignment and manipulation¶

nmod_poly_struct *nmod_poly_mat_entry(const nmod_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 nmod_poly function for direct manipulation of the matrix element. No bounds checking is performed.

void nmod_poly_mat_set(nmod_poly_mat_t mat1, const nmod_poly_mat_t mat2)¶: Sets mat1 to a copy of mat2.

void nmod_poly_mat_set_nmod_mat(nmod_poly_mat_t pmat, const nmod_mat_t cmat)¶: Sets the already-initialized polynomial matrix pmat to a constant matrix with the same entries as cmat. Both input matrices must have the same dimensions and modulus.

void nmod_poly_mat_swap(nmod_poly_mat_t mat1, nmod_poly_mat_t mat2)¶: Swaps mat1 and mat2 efficiently.

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

Input and output¶

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

Random matrix generation¶

void nmod_poly_mat_randtest(nmod_poly_mat_t mat, flint_rand_t state, slong len)¶: This is equivalent to applying nmod_poly_randtest to all entries in the matrix.

void nmod_poly_mat_randtest_sparse(nmod_poly_mat_t A, flint_rand_t state, slong len, 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 nmod_poly_mat_zero(nmod_poly_mat_t mat)¶: Sets mat to the zero matrix.

void nmod_poly_mat_one(nmod_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 nmod_poly_mat_equal(const nmod_poly_mat_t mat1, const nmod_poly_mat_t mat2)¶: Returns nonzero if mat1 and mat2 have the same shape and all their entries agree, and returns zero otherwise.

int nmod_poly_mat_equal_nmod_mat(const nmod_poly_mat_t pmat, const nmod_mat_t cmat)¶: Returns nonzero if pmat is a constant matrix with the same dimensions and entries as cmat; returns zero otherwise.

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

int nmod_poly_mat_is_one(const nmod_poly_mat_t mat)¶: Returns nonzero if all entry 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 nmod_poly_mat_is_empty(const nmod_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 nmod_poly_mat_is_square(const nmod_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.

void nmod_poly_mat_get_coeff_mat(nmod_mat_t coeff, const nmod_poly_mat_t pmat, slong deg)¶: Sets coeff to be the coefficient of pmat of degree deg, where pmat is seen as a polynomial with matrix coefficients and coefficients are numbered from zero. coeff must be already initialized with the right dimensions and modulus. For entries of pmat of degree less than deg, the corresponding entry of coeff is zero.

void nmod_poly_mat_set_coeff_mat(nmod_poly_mat_t pmat, const nmod_mat_t coeff, slong deg)¶: Sets the coefficient of pmat of degree deg to coeff, where pmat is seen as a polynomial with matrix coefficients and coefficients are numbered from zero. For each entry of pmat, if deg is larger than its degree, this entry is first resized to the appropriate length, with intervening coefficients being set to zero.

Norms¶

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

slong nmod_poly_mat_degree(const nmod_poly_mat_t pmat)¶: Returns the degree of the polynomial matrix pmat. The zero matrix is deemed to have degree \(-1\).

Evaluation¶

void nmod_poly_mat_evaluate_nmod(nmod_mat_t B, const nmod_poly_mat_t A, mp_limb_t x)¶: Sets the nmod_mat_t B to A evaluated entrywise at the point x.

Arithmetic¶

void nmod_poly_mat_scalar_mul_nmod_poly(nmod_poly_mat_t B, const nmod_poly_mat_t A, const nmod_poly_t c)¶: Sets B to A multiplied entrywise by the polynomial c.

void nmod_poly_mat_scalar_mul_nmod(nmod_poly_mat_t B, const nmod_poly_mat_t A, mp_limb_t c)¶: Sets B to A multiplied entrywise by the coefficient c, which is assumed to be reduced modulo the modulus.

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

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

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

void nmod_poly_mat_mul(nmod_poly_mat_t C, const nmod_poly_mat_t A, const nmod_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, KS and evaluation-interpolation multiplication.

void nmod_poly_mat_mul_classical(nmod_poly_mat_t C, const nmod_poly_mat_t A, const nmod_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 nmod_poly_mat_mul_KS(nmod_poly_mat_t C, const nmod_poly_mat_t A, const nmod_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 nmod_poly_mat_mul_interpolate(nmod_poly_mat_t C, const nmod_poly_mat_t A, const nmod_poly_mat_t B)¶: Sets C to the matrix product of A and B, computed through evaluation and interpolation. The matrices must have compatible dimensions for matrix multiplication. For interpolation to be well-defined, we require that the modulus is a prime at least as large as \(m + n - 1\) where \(m\) and \(n\) are the maximum lengths of polynomials in the input matrices. Aliasing is allowed.

void nmod_poly_mat_sqr(nmod_poly_mat_t B, const nmod_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 nmod_poly_mat_sqr_classical(nmod_poly_mat_t B, const nmod_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 nmod_poly_mat_sqr_KS(nmod_poly_mat_t B, const nmod_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 nmod_poly_mat_sqr_interpolate(nmod_poly_mat_t B, const nmod_poly_mat_t A)¶: Sets B to the square of A, which must be a square matrix, computed through evaluation and interpolation. For interpolation to be well-defined, we require that the modulus is a prime at least as large as \(2n - 1\) where \(n\) is the maximum length of polynomials in the input matrix. Aliasing is allowed.

void nmod_poly_mat_pow(nmod_poly_mat_t B, const nmod_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.

Row reduction¶

slong nmod_poly_mat_find_pivot_any(const nmod_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 from 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 nmod_poly_mat_find_pivot_partial(const nmod_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. This heuristic typically reduces coefficient growth when the matrix entries vary in size.

slong nmod_poly_mat_fflu(nmod_poly_mat_t B, nmod_poly_t den, slong *perm, const nmod_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 nmod_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 nmod_poly_mat_rref(nmod_poly_mat_t B, nmod_poly_t den, const nmod_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 nmod_poly_mat_trace(nmod_poly_t trace, const nmod_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 nmod_poly_mat_det(nmod_poly_t det, const nmod_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 nmod_poly_mat_det_fflu(nmod_poly_t det, const nmod_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 nmod_poly_mat_det_interpolate(nmod_poly_t det, const nmod_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 coefficient matrix, and forming the interpolating polynomial.

If the coefficient ring does not contain \(n\) distinct points (that is, if working over \(\mathbf{Z}/p\mathbf{Z}\) where \(p < n\)), this function automatically falls back to nmod_poly_mat_det_fflu.

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

Inverse¶

int nmod_poly_mat_inv(nmod_poly_mat_t Ainv, nmod_poly_t den, const nmod_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 nmod_poly_mat_nullspace(nmod_poly_mat_t res, const nmod_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 nmod_poly_mat_solve(nmod_poly_mat_t X, nmod_poly_t den, const nmod_poly_mat_t A, const nmod_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 nmod_poly_mat_solve_fflu(nmod_poly_mat_t X, nmod_poly_t den, const nmod_poly_mat_t A, const nmod_poly_mat_t B)¶