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.