fmpz_mat.h – matrices over the integers¶

Description.

Types, macros and constants¶

fmpz_mat_struct¶

fmpz_mat_t¶: Description.

Memory management¶

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

void fmpz_mat_clear(fmpz_mat_t mat)¶: Clears the given matrix.

Basic assignment and manipulation¶

void fmpz_mat_set(fmpz_mat_t mat1, const fmpz_mat_t mat2)¶: Sets mat1 to a copy of mat2. The dimensions of mat1 and mat2 must be the same.

void fmpz_mat_init_set(fmpz_mat_t mat, const fmpz_mat_t src)¶: Initialises the matrix mat to the same size as src and sets it to a copy of src.

void fmpz_mat_swap(fmpz_mat_t mat1, fmpz_mat_t mat2)¶: Swaps two matrices. The dimensions of mat1 and mat2 are allowed to be different.

fmpz * fmpz_mat_entry(fmpz_mat_t mat, slong i, slong j)¶

Returns a reference to the entry of mat at row $i$ and column $j$. This reference can be passed as an input or output variable to any function in the fmpz module for direct manipulation.

Both $i$ and $j$ must not exceed the dimensions of the matrix.

This function is implemented as a macro.

void fmpz_mat_zero(fmpz_mat_t mat)¶: Sets all entries of mat to 0.

void fmpz_mat_one(fmpz_mat_t mat)¶: Sets mat to the unit matrix, having ones on the main diagonal and zeroes elsewhere. If mat is nonsquare, it is set to the truncation of a unit matrix.

Window¶

void fmpz_mat_window_init(fmpz_mat_t window, const fmpz_mat_t mat, slong r1, slong c1, slong r2, slong c2)¶: Initializes the matrix window to be an r2 - r1 by c2 - c1 submatrix of mat whose (0,0) entry is the (r1, c1) entry of mat. The memory for the elements of window is shared with mat.

void fmpz_mat_window_clear(fmpz_mat_t window)¶: Clears the matrix window and releases any memory that it uses. Note that the memory to the underlying matrix that window points to is not freed.

Random matrix generation¶

void fmpz_mat_randbits(fmpz_mat_t mat, flint_rand_t state, mp_bitcnt_t bits)¶: Sets the entries of mat to random signed integers whose absolute values have the given number of binary bits.

void fmpz_mat_randtest(fmpz_mat_t mat, flint_rand_t state, mp_bitcnt_t bits)¶: Sets the entries of mat to random signed integers whose absolute values have a random number of bits up to the given number of bits inclusive.

void fmpz_mat_randintrel(fmpz_mat_t mat, flint_rand_t state, mp_bitcnt_t bits)¶

Sets mat to be a random emph{integer relations} matrix, with signed entries up to the given number of bits.

The number of columns of mat must be equal to one more than the number of rows. The format of the matrix is a set of random integers in the left hand column and an identity matrix in the remaining square submatrix.

void fmpz_mat_randsimdioph(fmpz_mat_t mat, flint_rand_t state, mp_bitcnt_t bits, mp_bitcnt_t bits2)¶

Sets mat to a random emph{simultaneous diophantine} matrix.

The matrix must be square. The top left entry is set to 2^bits2. The remainder of that row is then set to signed random integers of the given number of binary bits. The remainder of the first column is zero. Running down the rest of the diagonal are the values 2^bits with all remaining entries zero.

void fmpz_mat_randntrulike(fmpz_mat_t mat, flint_rand_t state, mp_bitcnt_t bits, ulong q)¶

Sets a square matrix mat of even dimension to a random emph{NTRU like} matrix.

The matrix is broken into four square submatrices. The top left submatrix is set to the identity. The bottom left submatrix is set to the zero matrix. The bottom right submatrix is set to $q$ times the identity matrix. Finally the top right submatrix has the following format. A random vector $h$ of length $r/2$ is created, with random signed entries of the given number of bits. Then entry $(i, j)$ of the submatrix is set to $h[i + j \bmod{r/2}]$.

void fmpz_mat_randntrulike2(fmpz_mat_t mat, flint_rand_t state, mp_bitcnt_t bits, ulong q)¶

Sets a square matrix mat of even dimension to a random emph{NTRU like} matrix.

The matrix is broken into four square submatrices. The top left submatrix is set to $q$ times the identity matrix. The top right submatrix is set to the zero matrix. The bottom right submatrix is set to the identity matrix. Finally the bottom left submatrix has the following format. A random vector $h$ of length $r/2$ is created, with random signed entries of the given number of bits. Then entry $(i, j)$ of the submatrix is set to $h[i + j \bmod{r/2}]$.

void fmpz_mat_randajtai(fmpz_mat_t mat, flint_rand_t state, double alpha)¶: Sets a square matrix mat to a random emph{ajtai} matrix. The diagonal entries $(i, i)$ are set to a random entry in the range $[1, 2^{b-1}]$ inclusive where $b = \floor{(2 r - i)^\alpha}$ for some double parameter~`alpha`. The entries below the diagonal in column~`i` are set to a random entry in the range $(-2^b + 1, 2^b - 1)$ whilst the entries to the right of the diagonal in row~`i` are set to zero.

int fmpz_mat_randpermdiag(fmpz_mat_t mat, flint_rand_t state, const fmpz * diag, slong n)¶

Sets mat to a random permutation of the rows and columns of a given diagonal matrix. The diagonal matrix is specified in the form of an array of the $n$ initial entries on the main diagonal.

The return value is $0$ or $1$ depending on whether the permutation is even or odd.

void fmpz_mat_randrank(fmpz_mat_t mat, flint_rand_t state, slong rank, mp_bitcnt_t bits)¶

Sets mat to a random sparse matrix with the given rank, having exactly as many non-zero elements as the rank, with the nonzero elements being random integers of the given bit size.

The matrix can be transformed into a dense matrix with unchanged rank by subsequently calling fmpz_mat_randops().

void fmpz_mat_randdet(fmpz_mat_t mat, flint_rand_t state, const fmpz_t det)¶

Sets mat to a random sparse matrix with minimal number of nonzero entries such that its determinant has the given value.

Note that the matrix will be zero if det is zero. In order to generate a non-zero singular matrix, the function fmpz_mat_randrank() can be used.

The matrix can be transformed into a dense matrix with unchanged determinant by subsequently calling fmpz_mat_randops().

void fmpz_mat_randops(fmpz_mat_t mat, flint_rand_t state, slong count)¶: Randomises mat by performing elementary row or column operations. More precisely, at most count random additions or subtractions of distinct rows and columns will be performed. This leaves the rank (and for square matrices, the determinant) unchanged.

Input and output¶

int fmpz_mat_fprint(FILE * file, const fmpz_mat_t mat)¶

Prints the given matrix to the stream file. The format is the number of rows, a space, the number of columns, two spaces, then a space separated list of coefficients, one row after the other.

In case of success, returns a positive value; otherwise, returns a non-positive value.

int fmpz_mat_fprint_pretty(FILE * file, const fmpz_mat_t mat)¶

Prints the given matrix to the stream file. The format is an opening square bracket then on each line a row of the matrix, followed by a closing square bracket. Each row is written as an opening square bracket followed by a space separated list of coefficients followed by a closing square bracket.

In case of success, returns a positive value; otherwise, returns a non-positive value.

int fmpz_mat_print(const fmpz_mat_t mat)¶: Prints the given matrix to the stream stdout. For further details, see fmpz_mat_fprint().

int fmpz_mat_print_pretty(const fmpz_mat_t mat)¶: Prints the given matrix to stdout. For further details, see fmpz_mat_fprint_pretty().

int fmpz_mat_fread(FILE* file, fmpz_mat_t mat)¶

Reads a matrix from the stream file, storing the result in mat. The expected format is the number of rows, a space, the number of columns, two spaces, then a space separated list of coefficients, one row after the other.

In case of success, returns a positive number. In case of failure, returns a non-positive value.

int fmpz_mat_read(fmpz_mat_t mat)¶

Reads a matrix from stdin, storing the result in mat.

In case of success, returns a positive number. In case of failure, returns a non-positive value.

Comparison¶

int fmpz_mat_equal(const fmpz_mat_t mat1, const fmpz_mat_t mat2)¶: Returns a non-zero value if mat1 and mat2 have the same dimensions and entries, and zero otherwise.

int fmpz_mat_is_zero(const fmpz_mat_t mat)¶: Returns a non-zero value if all entries mat are zero, and otherwise returns zero.

int fmpz_mat_is_one(const fmpz_mat_t mat)¶: Returns a non-zero value if mat is the unit matrix or the truncation of a unit matrix, and otherwise returns zero.

int fmpz_mat_is_empty(const fmpz_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_mat_is_square(const fmpz_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.

int fmpz_mat_is_zero_row(const fmpz_mat_t mat, slong i)¶: Returns a non-zero value if row $i$ of mat is zero.

int fmpz_mat_col_equal(fmpz_mat_t M, slong m, slong n)¶: Returns $1$ if columns $m$ and $n$ of the matrix $M$ are equal, otherwise returns $0$.

int fmpz_mat_row_equal(fmpz_mat_t M, slong m, slong n)¶: Returns $1$ if rows $m$ and $n$ of the matrix $M$ are equal, otherwise returns $0$.

Transpose¶

void fmpz_mat_transpose(fmpz_mat_t B, const fmpz_mat_t A)¶: Sets $B$ to $A^T$, the transpose of $A$. Dimensions must be compatible. $A$ and $B$ are allowed to be the same object if $A$ is a square matrix.

Concatenate¶

void fmpz_mat_concat_vertical(fmpz_mat_t res, const fmpz_mat_t mat1, const fmpz_mat_t mat2) Sets code{res} to vertical concatenation of (code{mat1}, code{mat2}) in that order. Matrix dimensions : code{mat1} : $m times n$, code{mat2} : $k times n$, code{res} : $(m + k) times n$.

void fmpz_mat_concat_horizontal(fmpz_mat_t res, const fmpz_mat_t mat1, const fmpz_mat_t mat2) Sets code{res} to horizontal concatenation of (code{mat1}, code{mat2}) in that order. Matrix dimensions : code{mat1} : $m times n$, code{mat2} : $m times k$, code{res} : $m times (n + k)$.

Modular reduction and reconstruction¶

void fmpz_mat_get_nmod_mat(nmod_mat_t Amod, const fmpz_mat_t A)¶: Sets the entries of Amod to the entries of A reduced by the modulus of Amod.

void fmpz_mat_set_nmod_mat(fmpz_mat_t A, const nmod_mat_t Amod)¶: Sets the entries of Amod to the residues in Amod, normalised to the interval $-m/2 <= r < m/2$ where $m$ is the modulus.

void fmpz_mat_set_nmod_mat_unsigned(fmpz_mat_t A, const nmod_mat_t Amod)¶: Sets the entries of Amod to the residues in Amod, normalised to the interval $0 <= r < m$ where $m$ is the modulus.

void fmpz_mat_CRT_ui(fmpz_mat_t res, const fmpz_mat_t mat1, const fmpz_t m1, const nmod_mat_t mat2, int sign)¶: Given mat1 with entries modulo m and mat2 with modulus $n$, sets res to the CRT reconstruction modulo $mn$ with entries satisfying $-mn/2 <= c < mn/2$ (if sign = 1) or $0 <= c < mn$ (if sign = 0).

void fmpz_mat_multi_mod_ui_precomp(nmod_mat_t * residues, slong nres, const fmpz_mat_t mat, fmpz_comb_t comb, fmpz_comb_temp_t temp)¶: Sets each of the nres matrices in residues to mat reduced modulo the modulus of the respective matrix, given precomputed comb and comb_temp structures.

void fmpz_mat_multi_mod_ui(nmod_mat_t * residues, slong nres, const fmpz_mat_t mat)¶

Sets each of the nres matrices in residues to mat reduced modulo the modulus of the respective matrix.

This function is provided for convenience purposes. For reducing or reconstructing multiple integer matrices over the same set of moduli, it is faster to use\ fmpz_mat_multi_mod_precomp.

void fmpz_mat_multi_CRT_ui_precomp(fmpz_mat_t mat, nmod_mat_t * const residues, slong nres, fmpz_comb_t comb, fmpz_comb_temp_t temp, int sign)¶: Reconstructs mat from its images modulo the nres matrices in residues, given precomputed comb and comb_temp structures.

void fmpz_mat_multi_CRT_ui(fmpz_mat_t mat, nmod_mat_t * const residues, slong nres, int sign)¶

Reconstructs mat from its images modulo the nres matrices in residues.

This function is provided for convenience purposes. For reducing or reconstructing multiple integer matrices over the same set of moduli, it is faster to use\ fmpz_mat_multi_CRT_ui_precomp.

Addition and subtraction¶

void fmpz_mat_add(fmpz_mat_t C, const fmpz_mat_t A, const fmpz_mat_t B)¶: Sets C to the elementwise sum $A + B$. All inputs must be of the same size. Aliasing is allowed.

void fmpz_mat_sub(fmpz_mat_t C, const fmpz_mat_t A, const fmpz_mat_t B)¶: Sets C to the elementwise difference $A - B$. All inputs must be of the same size. Aliasing is allowed.

void fmpz_mat_neg(fmpz_mat_t B, const fmpz_mat_t A)¶: Sets B to the elementwise negation of A. Both inputs must be of the same size. Aliasing is allowed.

Matrix-scalar arithmetic¶

void fmpz_mat_scalar_mul_si(fmpz_mat_t B, const fmpz_mat_t A, slong c)¶

void fmpz_mat_scalar_mul_ui(fmpz_mat_t B, const fmpz_mat_t A, ulong c)¶

void fmpz_mat_scalar_mul_fmpz(fmpz_mat_t B, const fmpz_mat_t A, const fmpz_t c)¶: Set A = B*c where B is an fmpz_mat_t and c is a scalar respectively of type slong, ulong, or fmpz_t. The dimensions of A and B must be compatible.

void fmpz_mat_scalar_addmul_si(fmpz_mat_t B, const fmpz_mat_t A, slong c)¶

void fmpz_mat_scalar_addmul_ui(fmpz_mat_t B, const fmpz_mat_t A, ulong c)¶

void fmpz_mat_scalar_addmul_fmpz(fmpz_mat_t B, const fmpz_mat_t A, const fmpz_t c)¶: Set A = A + B*c where B is an fmpz_mat_t and c is a scalar respectively of type slong, ulong, or fmpz_t. The dimensions of A and B must be compatible.

void fmpz_mat_scalar_submul_si(fmpz_mat_t B, const fmpz_mat_t A, slong c)¶

void fmpz_mat_scalar_submul_ui(fmpz_mat_t B, const fmpz_mat_t A, ulong c)¶

void fmpz_mat_scalar_submul_fmpz(fmpz_mat_t B, const fmpz_mat_t A, const fmpz_t c)¶: Set A = A - B*c where B is an fmpz_mat_t and c is a scalar respectively of type slong, ulong, or fmpz_t. The dimensions of A and B must be compatible.

void fmpz_mat_scalar_addmul_nmod_mat_ui(fmpz_mat_t B, const nmod_mat_t A, ulong c) void fmpz_mat_scalar_addmul_nmod_mat_fmpz(fmpz_mat_t B, const nmod_mat_t A, const fmpz_t c)¶: Set A = A + B*c where B is an nmod_mat_t and c is a scalar respectively of type ulong or fmpz_t. The dimensions of A and B must be compatible.

void fmpz_mat_scalar_divexact_si(fmpz_mat_t B, const fmpz_mat_t A, slong c)¶

void fmpz_mat_scalar_divexact_ui(fmpz_mat_t B, const fmpz_mat_t A, ulong c)¶

void fmpz_mat_scalar_divexact_fmpz(fmpz_mat_t B, const fmpz_mat_t A, const fmpz_t c)¶: Set A = B / c, where B is an fmpz_mat_t and c is a scalar respectively of type slong, ulong, or fmpz_t, which is assumed to divide all elements of B exactly.

void fmpz_mat_scalar_mul_2exp(fmpz_mat_t B, const fmpz_mat_t A, ulong exp)¶: Set the matrix B to the matrix A, of the same dimensions, multiplied by $2^{exp}$.

void fmpz_mat_scalar_tdiv_q_2exp(fmpz_mat_t B, const fmpz_mat_t A, ulong exp)¶: Set the matrix B to the matrix A, of the same dimensions, divided by $2^{exp}$, rounding down towards zero.

void fmpz_mat_scalar_smod(fmpz_mat_t B, const fmpz_mat_t A, const fmpz_t P)¶: Set the matrix B to the matrix A, of the same dimensions, with each entry reduced modulo $P$ in the symmetric moduli system. We require $P > 0$.

Matrix multiplication¶

void fmpz_mat_mul(fmpz_mat_t C, const fmpz_mat_t A, const fmpz_mat_t B)¶

Sets C to the matrix product $C = A B$. The matrices must have compatible dimensions for matrix multiplication. Aliasing is allowed.

This function automatically switches between classical and multimodular multiplication, based on a heuristic comparison of the dimensions and entry sizes.

void fmpz_mat_mul_classical(fmpz_mat_t C, const fmpz_mat_t A, const fmpz_mat_t B)¶

Sets C to the matrix product $C = A B$ computed using classical matrix algorithm.

The matrices must have compatible dimensions for matrix multiplication. No aliasing is allowed.

void fmpz_mat_mul_strassen(fmpz_mat_t C, const fmpz_mat_t A, const fmpz_mat_t B)¶: Sets $C = AB$. Dimensions must be compatible for matrix multiplication. $C$ is not allowed to be aliased with $A$ or $B$. Uses Strassen multiplication (the Strassen-Winograd variant).

void _fmpz_mat_mul_multi_mod(fmpz_mat_t C, const fmpz_mat_t A, const fmpz_mat_t B, mp_bitcnt_t bits)¶

void fmpz_mat_mul_multi_mod(fmpz_mat_t C, const fmpz_mat_t A, const fmpz_mat_t B)¶

Sets C to the matrix product $C = AB$ computed using a multimodular algorithm. $C$ is computed modulo several small prime numbers and reconstructed using the Chinese Remainder Theorem. This generally becomes more efficient than classical multiplication for large matrices.

The bits parameter is a bound for the bit size of largest element of $C$, or twice the absolute value of the largest element if any elements of $C$ are negative. The function fmpz_mat_mul_multi_mod calculates a rigorous bound automatically. If the default bound is too pessimistic, _fmpz_mat_mul_multi_mod can be used with a custom bound.

The matrices must have compatible dimensions for matrix multiplication. No aliasing is allowed.

void fmpz_mat_sqr(fmpz_mat_t B, const fmpz_mat_t A)¶: Sets B to the square of the matrix A, which must be a square matrix. Aliasing is allowed. The function calls fmpz_mat_mul for dimensions less than 12 and calls fmpz_mat_sqr_bodrato for cases in which the latter is faster.

void fmpz_mat_sqr_bodrato(fmpz_mat_t B, const fmpz_mat_t A)¶: Sets B to the square of the matrix A, which must be a square matrix. Aliasing is allowed. The bodrato algorithm is described in cite{Bodrato2010}. It is highly efficient for squaring matrices which satisfy both the following conditions : (a) large elements (b) dimensions less than 150.

void fmpz_mat_pow(fmpz_mat_t B, const fmpz_mat_t A, ulong e)¶: Sets B to the matrix A raised to the power e, where A must be a square matrix. Aliasing is allowed.

Inverse¶

int fmpz_mat_inv(fmpz_mat_t Ainv, fmpz_t den, const fmpz_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.

The denominator is not guaranteed to be minimal, but is guaranteed to be a divisor of the determinant of A.

This function uses a direct formula for matrices of size two or less, and otherwise solves for the identity matrix using fraction-free LU decomposition.

Kronecker product¶

void fmpz_mat_kronecker_product(fmpz_mat_t C, const fmpz_mat_t A, const fmpz_mat_t B)¶: Sets C to the Kronecker product of A and B.

Content¶

void fmpz_mat_content(fmpz_t mat_gcd, const fmpz_mat_t A)¶: Sets mat_gcd as the gcd of all the elements of the matrix A. Returns 0 if the matrix is empty.

Trace¶

void fmpz_mat_trace(fmpz_t trace, const fmpz_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¶

void fmpz_mat_det(fmpz_t det, const fmpz_mat_t A)¶

Sets det to the determinant of the square matrix $A$. The matrix of dimension $0 \times 0$ is defined to have determinant 1.

This function automatically chooses between fmpz_mat_det_cofactor,\ fmpz_mat_det_bareiss, fmpz_mat_det_modular and\ fmpz_mat_det_modular_accelerated (with proved = 1), depending on the size of the matrix and its entries.

void fmpz_mat_det_cofactor(fmpz_t det, const fmpz_mat_t A)¶: Sets det to the determinant of the square matrix $A$ computed using direct cofactor expansion. This function only supports matrices up to size $4 \times 4$.

void fmpz_mat_det_bareiss(fmpz_t det, const fmpz_mat_t A)¶: Sets det to the determinant of the square matrix $A$ computed using the Bareiss algorithm. A copy of the input matrix is row reduced using fraction-free Gaussian elimination, and the determinant is read off from the last element on the main diagonal.

void fmpz_mat_det_modular(fmpz_t det, const fmpz_mat_t A, int proved)¶

Sets det to the determinant of the square matrix $A$ (if proved = 1), or a probabilistic value for the determinant (proved = 0), computed using a multimodular algorithm.

The determinant is computed modulo several small primes and reconstructed using the Chinese Remainder Theorem. With proved = 1, sufficiently many primes are chosen to satisfy the bound computed by fmpz_mat_det_bound. With proved = 0, the determinant is considered determined if it remains unchanged modulo several consecutive primes (currently if their product exceeds $2^{100}$).

void fmpz_mat_det_modular_accelerated(fmpz_t det, const fmpz_mat_t A, int proved)¶

Sets det to the determinant of the square matrix $A$ (if proved = 1), or a probabilistic value for the determinant (proved = 0), computed using a multimodular algorithm.

This function uses the same basic algorithm as fmpz_mat_det_modular, but instead of computing $\det(A)$ directly, it generates a divisor $d$ of $\det(A)$ and then computes $x = \det(A) / d$ modulo several small primes not dividing $d$. This typically accelerates the computation by requiring fewer primes for large matrices, since $d$ with high probability will be nearly as large as the determinant. This trick is described in citep{AbbottBronsteinMulders1999}.

void fmpz_mat_det_modular_given_divisor(fmpz_t det, const fmpz_mat_t A, const fmpz_t d, int proved)¶: Given a positive divisor $d$ of $\det(A)$, sets det to the determinant of the square matrix $A$ (if proved = 1), or a probabilistic value for the determinant (proved = 0), computed using a multimodular algorithm.

void fmpz_mat_det_bound(fmpz_t bound, const fmpz_mat_t A)¶: Sets bound to a nonnegative integer $B$ such that $|\det(A)| \le B$. Assumes $A$ to be a square matrix. The bound is computed from the Hadamard inequality $|\det(A)| \le \prod \|a_i\|_2$ where the product is taken over the rows $a_i$ of $A$.

void fmpz_mat_det_divisor(fmpz_t d, const fmpz_mat_t A)¶

Sets $d$ to some positive divisor of the determinant of the given square matrix $A$, if the determinant is nonzero. If $|\det(A)| = 0$, $d$ will always be set to zero.

A divisor is obtained by solving $Ax = b$ for an arbitrarily chosen right-hand side $b$ using Dixon’s algorithm and computing the least common multiple of the denominators in $x$. This yields a divisor $d$ such that $|\det(A)| / d$ is tiny with very high probability.

Transforms¶

void fmpz_mat_similarity(fmpz_mat_t A, slong r, fmpz_t d)¶

Applies a similarity transform to the $n\times n$ matrix $M$ in-place.

If $P$ is the $n\times n$ identity matrix the zero entries of whose row $r$ ($0$-indexed) have been replaced by $d$, this transform is equivalent to $M = P^{-1}MP$.

Similarity transforms preserve the determinant, characteristic polynomial and minimal polynomial.

Characteristic polynomial¶

void _fmpz_mat_charpoly_berkowitz(fmpz * cp, const fmpz_mat_t mat)¶: Sets (cp, n+1) to the characteristic polynomial of an $n \times n$ square matrix.

void fmpz_mat_charpoly_berkowitz(fmpz_poly_t cp, const fmpz_mat_t mat)¶: Computes the characteristic polynomial of length $n + 1$ of an $n \times n$ square matrix. Uses an $O(n^4)$ algorithm based on the method of Berkowitz.

void _fmpz_mat_charpoly_modular(fmpz * cp, const fmpz_mat_t mat)¶: Sets (cp, n+1) to the characteristic polynomial of an $n \times n$ square matrix.

void fmpz_mat_charpoly_modular(fmpz_poly_t cp, const fmpz_mat_t mat)¶: Computes the characteristic polynomial of length $n + 1$ of an $n \times n$ square matrix. Uses a modular method based on an $O(n^3)$ method over $\mathbb{Z}/n\mathbb{Z}$.

void _fmpz_mat_charpoly(fmpz * cp, const fmpz_mat_t mat)¶: Sets (cp, n+1) to the characteristic polynomial of an $n \times n$ square matrix.

void fmpz_mat_charpoly(fmpz_poly_t cp, const fmpz_mat_t mat)¶: Computes the characteristic polynomial of length $n + 1$ of an $n \times n$ square matrix.

Minimal polynomial¶

slong _fmpz_mat_minpoly_modular(fmpz * cp, const fmpz_mat_t mat)¶: Sets (cp, n+1) to the modular polynomial of an $n \times n$ square matrix and returns its length.

void fmpz_mat_minpoly_modular(fmpz_poly_t cp, const fmpz_mat_t mat)¶: Computes the minimal polynomial of an $n \times n$ square matrix. Uses a modular method based on an average time $O~(n^3)$, worst case $O(n^4)$ method over $\mathbb{Z}/n\mathbb{Z}$.

slong _fmpz_mat_minpoly(fmpz * cp, const fmpz_mat_t mat)¶: Sets cp to the minimal polynomial of an $n \times n$ square matrix and returns its length.

void fmpz_mat_minpoly(fmpz_poly_t cp, const fmpz_mat_t mat)¶: Computes the minimal polynomial of an $n \times n$ square matrix.

Rank¶

slong fmpz_mat_rank(const fmpz_mat_t A)¶: Returns the rank, that is, the number of linearly independent columns (equivalently, rows), of $A$. The rank is computed by row reducing a copy of $A$.

Column partitioning¶

int fmpz_mat_col_partition(slong * part, fmpz_mat_t M, int short_circuit)¶: Returns the number $p$ of distinct columns of $M$ (or $0$ if the flag short_circuit is set and this number is greater than the number of rows of $M$). The entries of array part are set to values in $[0, p)$ such that two entries of part are equal iff the corresponding columns of $M$ are equal. This function is used in van Hoeij polynomial factoring.

Nonsingular solving¶

The following functions allow solving matrix-matrix equations $AX = B$ where the system matrix $A$ is square and has full rank. The solving is implicitly done over the field of rational numbers: except where otherwise noted, an integer matrix $\hat X$ and a separate denominator $d$ (den) are computed such that $A(\hat X/d) = b$, equivalently such that $A\hat X = bd$ holds over the integers. No guarantee is made that the numerators and denominator are reduced to lowest terms, but the denominator is always guaranteed to be a divisor of the determinant of $A$. If $A$ is singular, den will be set to zero and the elements of the solution vector or matrix will have undefined values. No aliasing is allowed between arguments.

int fmpz_mat_solve(fmpz_mat_t X, fmpz_t den, const fmpz_mat_t A, const fmpz_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.

This function uses Cramer’s rule for small systems and fraction-free LU decomposition followed by fraction-free forward and back substitution for larger systems.

Note that for very large systems, it is faster to compute a modular solution using fmpz_mat_solve_dixon.

int fmpz_mat_solve_fflu(fmpz_mat_t X, fmpz_t den, const fmpz_mat_t A, const fmpz_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.

void fmpz_mat_solve_fflu_precomp(fmpz_mat_t X, const slong * perm, const fmpz_mat_t FFLU, const fmpz_mat_t B)¶: Performs fraction-free forward and back substitution given a precomputed fraction-free LU decomposition and corresponding permutation.

int fmpz_mat_solve_cramer(fmpz_mat_t X, fmpz_t den, const fmpz_mat_t A, const fmpz_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.

Uses Cramer’s rule. Only systems of size up to $3 \times 3$ are allowed.

void fmpz_mat_solve_bound(fmpz_t N, fmpz_t D, const fmpz_mat_t A, const fmpz_mat_t B)¶: Assuming that $A$ is nonsingular, computes integers $N$ and $D$ such that the reduced numerators and denominators $n/d$ in $A^{-1} B$ satisfy the bounds $0 \le |n| \le N$ and $0 \le d \le D$.

int fmpz_mat_solve_dixon(fmpz_mat_t X, fmpz_t M, const fmpz_mat_t A, const fmpz_mat_t B)¶

Solves $AX = B$ given a nonsingular square matrix $A$ and a matrix $B$ of compatible dimensions, using a modular algorithm. In particular, Dixon’s p-adic lifting algorithm is used (currently a non-adaptive version). This is generally the preferred method for large dimensions.

More precisely, this function computes an integer $M$ and an integer matrix $X$ such that $AX = B \bmod M$ and such that all the reduced numerators and denominators of the elements $x = p/q$ in the full solution satisfy $2|p|q < M$. As such, the explicit rational solution matrix can be recovered uniquely by passing the output of this function to fmpq_mat_set_fmpz_mat_mod.

A nonzero value is returned if $A$ is nonsingular. If $A$ is singular, zero is returned and the values of the output variables will be undefined.

Aliasing between input and output matrices is allowed.

Row reduction¶

slong fmpz_mat_find_pivot_any(const fmpz_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 fmpz_mat_fflu(fmpz_mat_t B, fmpz_t den, slong * perm, const fmpz_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_mat_find_pivot_any. 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 $\pm \operatorname{det}(S)$ where $S$ is an appropriate submatrix of $A$ ($S = A$ if $A$ is square) and the sign is decided by the parity of the permutation. Note that the determinant is not generally the minimal denominator.

The fraction-free LU decomposition is defined in citep{NakTurWil1997}.

slong fmpz_mat_rref(fmpz_mat_t B, fmpz_t den, const fmpz_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 algorithm used chooses between fmpz_mat_rref_fflu and fmpz_mat_rref_mul based on the dimensions of the input matrix.

slong fmpz_mat_rref_fflu(fmpz_mat_t B, fmpz_t den, const fmpz_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 algorithm proceeds by first computing a row echelon form using fmpz_mat_fflu. Letting the upper part of this matrix be $(U | V) P$ where $U$ is full rank upper triangular and $P$ is a permutation matrix, we obtain the rref by setting $V$ to $U^{-1} V$ using back substitution. Scaling each completed row in the back substitution to the denominator den, we avoid introducing new fractions. This strategy is equivalent to the fraction-free Gauss-Jordan elimination in citep{NakTurWil1997}, but faster since only the part $V$ corresponding to the null space has to be updated.

The denominator den is set to $\pm \operatorname{det}(S)$ where $S$ is an appropriate submatrix of $A$ ($S = A$ if $A$ is square). Note that the determinant is not generally the minimal denominator.

slong fmpz_mat_rref_mul(fmpz_mat_t B, fmpz_t den, const fmpz_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 algorithm works by computing the reduced row echelon form of A modulo a prime $p$ using nmod_mat_rref. The pivot columns and rows of this matrix will then define a non-singular submatrix of A, nonsingular solving and matrix multiplication can then be used to determine the reduced row echelon form of the whole of A. This procedure is described in cite{Stein2007}.

int fmpz_mat_is_in_rref_with_rank(const fmpz_mat_t A, const fmpz_t den, slong rank)¶: Checks that the matrix $A/den$ is in reduced row echelon form of rank rank, returns 1 if so and 0 otherwise.

Modular gaussian elimination¶

slong fmpz_mat_rref_mod(slong * perm, fmpz_mat_t A, const fmpz_t p)¶

Uses fraction-free Gauss-Jordan elimination to set A to its reduced row echelon form and returns the rank of A. All computations are done modulo p.

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

Strong echelon form and Howell form¶

void fmpz_mat_strong_echelon_form_mod(fmpz_mat_t A, const fmpz_t mod)¶

Transforms $A$ such that $A$ modulo mod is the strong echelon form of the input matrix modulo mod. The Howell form and the strong echelon form are equal up to permutation of the rows, see cite{FieHof2014} for a definition of the strong echelon form and the algorithm used here.

$A$ must have at least as many rows as columns.

slong fmpz_mat_howell_form_mod(nmod_mat_t A, const fmpz_t mod)¶

Transforms $A$ such that $A$ modulo mod is the Howell form of the input matrix modulo mod. For a definition of the Howell form see cite{StoMul1998}. The Howell form is computed by first putting $A$ into strong echelon form and then ordering the rows.

$A$ must have at least as many rows as columns.

Nullspace¶

slong fmpz_mat_nullspace(fmpz_mat_t B, const fmpz_mat_t A)¶

Computes a basis for the right rational nullspace of $A$ and returns the dimension of the nullspace (or nullity). $B$ is set to a matrix with linearly independent columns and maximal rank such that $AB = 0$ (i.e. $Ab = 0$ for each column $b$ in $B$), and the rank of $B$ is returned.

In general, the entries in $B$ will not be minimal: in particular, the pivot entries in $B$ will generally differ from unity. $B$ must be allocated with sufficient space to represent the result (at most $n \times n$ where $n$ is the number of column of $A$).

Echelon form¶

slong fmpz_mat_rref_fraction_free(slong * perm, fmpz_mat_t B, fmpz_t den, const fmpz_mat_t A)¶

Computes an integer matrix B and an integer den such that B / den is the unique row reduced echelon form (RREF) of A and returns the rank, i.e. the number of nonzero rows in B.

Aliasing of B and A is allowed, with an in-place computation being more efficient. The size of B must be the same as that of A.

The permutation order will be written to perm unless this argument is NULL. That is, row i of the output matrix will correspond to row perm[i] of the input matrix.

The denominator will always be a divisor of the determinant of (some submatrix of) $A$, but is not guaranteed to be minimal or canonical in any other sense.

Hermite normal form¶

void fmpz_mat_hnf(fmpz_mat_t H, const fmpz_mat_t A)¶

Computes an integer matrix H such that H is the unique (row) Hermite normal form of A. The algorithm used is selected from the implementations in FLINT to be the one most likely to be optimal, based on the characteristics of the input matrix.