fmpz_poly_mat.h

Initialises mat as a rows by cols matrix of zero polynomials. Throws if $\mathit{rows} \cdot \mathit{cols}$ overflows an slong.

Initialises mat as a copy of src.

Frees the memory used by mat. Must not be called on a window.

Swaps two matrices in constant time by swapping their structs (not for windows), respectively swaps the entries of two matrices of the same dimensions one by one.

Returns a pointer to the entry in row $i$ and column $j$, which can be used with the functions of fmpz_poly.h.

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

Returns 1 if mat has no rows or no columns, respectively as many rows as columns, and 0 otherwise.

Returns $\pm b$, where $b$ is the largest bit size of the absolute value of a coefficient of an entry of $A$, negative if some coefficient is negative.

Returns the largest length of an entry of $A$ (0 if all entries are zero).

Initialises window as the submatrix of mat formed by rows $r_1$ to $r_2 - 1$ and columns $c_1$ to $c_2 - 1$, sharing its entries with mat. The bounds are checked only by FLINT_ASSERT.

Releases a window. This does nothing at present.

Sets res to mat1 placed to the left of, respectively above, mat2, whose dimensions must be compatible; res must have the dimensions of the result.

Prints mat to stdout: a header <r x c matrix over Z[x]> (with the variable name x), then each row in square brackets with the entries printed by fmpz_poly_print_pretty and separated by commas, then an empty line.

Sets each entry to a random polynomial of length up to len with signed, respectively nonnegative, coefficients of up to bits bits.

Sets each entry, with probability density, to a random nonzero polynomial of length up to len with coefficients of up to bits bits, and to zero otherwise.

Returns 1 if the matrices have the same dimensions and entries, and 0 otherwise.

Returns 1 if mat is zero, respectively has the constant 1 on the main diagonal and zeros elsewhere (it need not be square), and 0 otherwise.

Sets mat to the zero matrix, respectively to the matrix with the constant 1 on the main diagonal and zeros elsewhere.

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

Sets $B$ to the transpose of $A$; $B$ may be $A$ if it is square.

Truncates every entry of $A$ to length len (reduces it modulo $x^{\mathit{len}}$), in place.

Sets the integer matrix $B$ to $A$ evaluated at $x$.

Sets $C = A + B$, $C = A - B$, respectively $B = -A$. Aliasing is allowed.

Sets $B = cA$ for a polynomial, respectively integer, $c$. Aliasing is allowed.

Sets trace to the sum of the diagonal entries of the square matrix mat. Throws if mat is not square.

Sets $C = AB$. $C$ may be $A$ or $B$. fmpz_poly_mat_mul_classical computes each entry as a sum of polynomial products; fmpz_poly_mat_mul_KS packs the entries into integers by Kronecker substitution and multiplies the integer matrices with fmpz_mat_mul. fmpz_poly_mat_mul uses the classical algorithm if a dimension is less than 8 and Kronecker substitution otherwise. See Multiplication.

Sets $C$ to $AB$ with every entry truncated to length len, computing only the needed coefficients of each polynomial product. $C$ may be $A$ or $B$.

Sets $B = A^2$ for a square matrix $A$; $B$ may be $A$. The classical version uses five polynomial products for $2 \times 2$ matrices; fmpz_poly_mat_sqr uses it for dimension less than 8 and Kronecker substitution otherwise.

Sets $B$ to $A^2$ with every entry truncated to length len.

Sets $B = A^{\mathit{exp}}$ for a square matrix $A$ by binary powering. $A^0$ is the identity. $B$ may be $A$ but must not be a window.

Sets $B$ to $A^{\mathit{exp}}$ with every entry truncated to length len, truncating after every multiplication. $B$ may be $A$ but must not be a window.

Sets res to the product of the $n$ matrices factors[0], ..., factors[n-1], all of the same square dimension, computed by a balanced product tree. The empty product is the identity.

Returns the index of the first row $r$ with start_row $\le r <$ end_row whose entry in column $c$ is nonzero, or $-1$ if there is none.

Returns the index of a row $r$ with start_row $\le r <$ end_row whose entry in column $c$ is nonzero, choosing one of least length and, among those, with coefficients of least bit size; returns $-1$ if all entries are zero. Small pivots keep the entries of fraction-free elimination small.

Computes a fraction-free LU decomposition of $A$ in $B$ and returns the rank $r$ of $A$ over $\mathbb{Q}(x)$. den is set to the last pivot, a nonzero $r \times r$ minor of $A$ (1 if $r = 0$). Row exchanges are applied to perm if it is not NULL. Pivots are chosen with fmpz_poly_mat_find_pivot_partial. If rank_check is set, the function returns 0 with den = 0 as soon as a column without pivot is found. $B$ may be $A$.

Sets ($B$, den) to the reduced row echelon form of $A$ over $\mathbb{Q}(x)$ and returns the rank: $B/\mathit{den}$ is in reduced row echelon form, with den on the pivot positions. Uses fraction-free elimination followed by fraction-free back substitution. $B$ may be $A$.

Sets det to the determinant of the square matrix $A$. fmpz_poly_mat_det_fflu uses fraction-free elimination; fmpz_poly_mat_det_interpolate evaluates $A$ at $L = n(\ell - 1) + 1$ consecutive integers around 0, where $\ell$ is the largest length of an entry, computes the integer determinants with fmpz_mat_det, and interpolates. fmpz_poly_mat_det uses explicit formulas for $n \le 2$, fraction-free elimination for $n < 15$ and interpolation otherwise.

Returns the rank of $A$ over $\mathbb{Q}(x)$, computed by fraction-free elimination.

For a square $A$ of dimension $n$ and an $n \times k$ matrix $B$, sets ($X$, den) with $AX = \mathit{den} \cdot B$ and $\mathit{den} = \det A$, and returns 1 if $A$ is nonsingular; returns 0 with den = 0 if $A$ is singular. Both use fraction-free elimination (fmpz_poly_mat_solve calls fmpz_poly_mat_solve_fflu). $X$ may be $A$ but not $B$.

Solves $AX = \mathit{den} \cdot B$ given the output FFLU, den, perm of fmpz_poly_mat_fflu for a nonsingular $A$, with fraction-free forward and back substitution. The sign of the permutation is not taken into account. perm must not be NULL, and $X$ must not be $B$; both raise an exception.

Sets (Ainv, den) to the inverse of the square matrix $A$, so that $A \cdot \mathit{Ainv} = \mathit{den} \cdot I$ with $\mathit{den} = \det A$, and returns 1 if $A$ is nonsingular; returns 0 if it is singular. Ainv may be $A$.

Sets the first $n - r$ columns of the $n \times n$ matrix res to a basis of the right nullspace of the $m \times n$ matrix mat over $\mathbb{Q}(x)$, with polynomial entries, and returns $n - r$; the other columns are set to zero. The basis is read off from fmpz_poly_mat_rref.