fq_mat.h

Initialises mat as a rows by cols zero matrix.

Initialises mat as a copy of src, with the same dimensions.

Frees the memory used by mat.

Initialises window as the submatrix of mat formed by the rows $r_1 \le i < r_2$ and the columns $c_1 \le j < c_2$. The entries are shared with mat, not copied, so changes to one are visible in the other. mat must not be cleared while the window is in use.

Clears a window. The entries, which belong to the underlying matrix, are not freed.

Returns a pointer to the entry in row $i$ and column $j$, indexed from zero. No bounds checking is performed.

Sets the entry in row $i$ and column $j$ to $x$.

Return the number of rows, respectively columns, of mat.

Copies mat2 into mat1, which must have the same dimensions.

Swaps the two matrices by swapping their structures; the dimensions may differ. Not for windows.

Swaps the entries of two matrices of the same dimensions one by one; this also works for windows.

Set A to the zero matrix, respectively to the matrix with ones on the main diagonal and zeros elsewhere (also for non-square A).

Swap rows, respectively columns, $r$ and $s$ of mat. If perm is not NULL, its entries $r$ and $s$ are swapped as well, so that it records the permutation.

Reverse the order of the rows, respectively columns, of mat, by swapping row $i$ and row $r - 1 - i$ for $0 \le i < \lfloor r/2 \rfloor$ (and likewise for columns). If perm is not NULL, the same permutation is applied to it.

Set mat1 to the image of the matrix mat2 over $\mathbb{F}_p$ (or over $\mathbb{Z}/n\mathbb{Z}$, whose entries are mapped to $\mathbb{F}_p$ by reduction) in $\mathbb{F}_q$. The dimensions must agree.

Set res to the block matrix $\begin{pmatrix} A \\ B \end{pmatrix}$, respectively $(A \ B)$, for $A$ = mat1 and $B$ = mat2. For vertical concatenation of an $m \times n$ and a $k \times n$ matrix, res must be $(m + k) \times n$; for horizontal concatenation of an $m \times n$ and an $m \times k$ matrix, $m \times (n + k)$.

Returns whether the matrices have the same dimensions and entries.

Return whether all entries are zero, respectively whether the diagonal entries are one and the others zero. An empty matrix is both zero and one.

Return whether mat has zero rows or zero columns, respectively whether it has as many rows as columns.

Sets the entries of mat to random elements for testing (see fq_randtest).

Sets mat to a random row and column permutation of the diagonal matrix whose first $n$ diagonal entries are diag and whose other entries are zero; $n$ must not exceed the number of rows or columns. Returns the parity of the permutation (0 for even, 1 for odd).

Sets mat to a random sparse matrix of the given rank, with exactly rank nonzero entries, which are random nonzero elements. fq_mat_randops turns it into a dense matrix of the same rank.

Applies count random elementary row or column operations (additions or subtractions of a row or column to another) to mat. This preserves the rank and, for square matrices, the determinant.

Set mat to a random lower, respectively upper, triangular matrix, with ones on the diagonal if unit is nonzero and random nonzero diagonal entries otherwise.

Print mat to file, respectively stdout: the number of rows and columns separated by a space, two spaces, and the entries in row-major order in the format of fq_fprint, separated by spaces. Return a positive value on success and a non-positive value on failure.

Print mat to file, respectively stdout, as a bracketed list of bracketed rows, with the entries in the format of fq_fprint_pretty. Return a positive value on success.

Sets $B$ to $A^T$. $B$ must have the transposed dimensions of $A$; aliasing is allowed for square matrices.

Set $C$ to $A + B$, respectively $A - B$. All three matrices must have the same dimensions.

Sets $B$ to $-A$.

Set $B$ to $cA$, where $c$ is an element of $\mathbb{F}_q$ or an integer mapped to $\mathbb{F}_p$.

Sets $C = AB$, where $A$ is $m \times k$, $B$ is $k \times n$ and $C$ is $m \times n$. Aliasing is allowed. The function uses fq_mat_mul_KS if FQ_MAT_MUL_KS_CUTOFF returns 1 for $m$ and $n$, and fq_mat_mul_classical otherwise.

Returns 1 if fq_mat_mul should use Kronecker substitution for an $r \times k$ times $k \times c$ product, which is the case if $5 \min(r, c) > 8d + 29$. (FQ_NMOD_MAT_MUL_KS_CUTOFF and FQ_ZECH_MAT_MUL_KS_CUTOFF are the corresponding tuned functions of the other representations.)

Sets $C = AB$ by computing each entry as a dot product with _fq_vec_dot, which reduces modulo the defining polynomial only once per entry. Aliasing is allowed.

Sets $C = AB$ by Kronecker substitution. $C$ must not be aliased with $A$ or $B$.

Sets $D = C - AB$. $D$ may be aliased with $C$, $A$ or $B$.

Set the vector $c$ to the matrix-vector product $Ab$. The vector (b, blen) is truncated or extended by zeros to the number of columns of $A$, and $c$ receives as many entries as $A$ has rows. The _ptr version takes arrays of pointers to the entries. $c$ must not alias $b$.

Set the vector $c$ to the vector-matrix product $aB$. The vector (a, alen) is truncated or extended by zeros to the number of rows of $B$, and $c$ receives as many entries as $B$ has columns. $c$ must not alias $a$.

Computes a generalised LU decomposition $PA = LU$ of $A$ in place, and returns the rank of $A$. P must have room for as many entries as $A$ has rows; it receives the row permutation.

Sets $B$ to the reduced row echelon form of $A$ and returns the rank of $A$. The form is computed from an LU decomposition. $B$ may be aliased with $A$. If $A$ is zero, $B$ is set to zero and 0 is returned.

Reduces row $m$ of $A$ against the earlier rows, which must already be in reduced form (pivot entries 1, not necessarily in order), and returns the column of the new pivot, or $-1$ if row $m$ becomes zero. P[i] is the row with its pivot in column $i$, or $-1$; it is updated when a new pivot is found, and the new pivot row is scaled to make the pivot 1. L[i] is the number of possibly nonzero columns of row $i$, which allows faster reduction of matrices that are zero on the right; the entries of L must be nondecreasing, and can all be set to the number of columns. The arithmetic is done without intermediate reductions modulo the defining polynomial.

Returns the rank of $A$, computed by LU decomposition.

If the square matrix $A$ is invertible, sets $B = A^{-1}$ and returns 1; otherwise returns 0 and the entries of $B$ are undefined. $A$ is not modified (although not declared const), and $B$ may be aliased with $A$.

Set $X = L^{-1} B$, respectively $X = U^{-1} B$, for a square lower, respectively upper, triangular matrix with nonzero diagonal entries. If unit is nonzero, the diagonal is assumed to consist of ones and is not read. Only the triangular part of the matrix is read. Aliasing of $X$ and $B$ is allowed. Classical substitution is used if $B$ has fewer than 64 rows or columns, and block recursion otherwise.

Solves $AX = C$ for a square matrix $A$. Returns 1 and sets $X$ if $A$ is nonsingular; otherwise returns 0 and $X$ is undefined.

Solves $AX = B$ for a matrix $A$ of any shape: if a solution exists, sets $X$ to one and returns 1, otherwise returns 0 and $X$ is undefined. For an $m \times n$ matrix $A$ and an $m \times k$ matrix $B$, $X$ must be $n \times k$.

Computes a basis of the right nullspace $\{v : Av = 0\}$ of $A$ and returns its dimension $k$ (the nullity). $X$ must have as many rows and at least as many columns as $A$ has columns; its first $k$ columns receive the basis and the remaining columns are set to zero.

Replaces the square matrix $A$ by $P^{-1} A P$, where $P$ is the identity matrix with the entries in row $r$ and every column $j \notin \{r - 1, r\}$ replaced by $d$. Such transforms preserve the characteristic and minimal polynomials.