fmpq_mat.h

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

Initialises mat1 as a copy of mat2.

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.

Returns a pointer to the numerator, respectively denominator, of the entry in row i and column j.

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.

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. Wrappers of gr_mat_concat_horizontal and gr_mat_concat_vertical.

Prints mat to stdout: a header <r x c matrix over Q>, then each row in square brackets with entries separated by commas, then an empty line.

Sets the entries to random fractions with numerator and denominator of bits bits (fmpq_randbits), respectively of at most bits bits with a bias towards special values (fmpq_randtest).

Sets mat to the Hilbert matrix, with entry (i, j) equal to 1/(i + j + 1). mat need not be square.

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

Returns 1 if mat is zero, respectively has ones on the main diagonal and zeros elsewhere (\mathit{mat} need not be square), and 0 otherwise. Empty matrices are both zero and one.

Returns 1 if all entries have denominator 1, and 0 otherwise.

Copies src to dest, which must have the same dimensions.

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

Sets rop to the transpose of op, which may be rop if it is square. Wrapper of gr_mat_transpose.

Swaps rows, respectively columns, r and s of mat, and entries r and s of perm if it is not NULL.

Reverses the order of the rows, respectively columns, of mat, and of the entries of perm if it is not NULL.

Sets mat to mat1 + mat2, respectively mat1 - mat2, and rop to -op. Aliasing is allowed.

Sets rop to op multiplied by x. Aliasing is allowed. The si and ui versions wrap gr_mat_mul_si and gr_mat_mul_ui.

Sets rop to op divided by x \ne 0. Aliasing is allowed.

Sets dest to mat and returns 1 if all entries are integers; otherwise returns 0, and dest may have been partially written.

Sets dest to the integer matrix src.

Sets num and den to the matrices of numerators and denominators of the entries of mat.

Sets den to the least common denominator d of all entries and num to the integer matrix d \cdot \mathit{mat}. For an empty matrix, d = 1.

Clears denominators row by row, respectively column by column: den[i] is set to the least common denominator of row (column) i, and row (column) i of num to the corresponding integer vector. den may be NULL if the denominators are not wanted.

Clears denominators row by row in the two matrices mat and mat2 with the same number of rows jointly, as if they were concatenated horizontally: den[i] is the least common denominator of row i of both, and num, num2 receive the scaled rows. This turns a system AX = B over \mathbb{Q} into an equivalent integer system. den may be NULL.

Sets X to the matrix Xmod divided by the nonzero integer div, reducing every entry to canonical form. This converts a pair (X, den) returned by the fmpz_mat solvers to a rational matrix.

Sets each entry of dest to the corresponding entry of mat modulo mod, in [0, \mathit{mod}) (see fmpq_mod_fmpz). All denominators must be invertible modulo mod; otherwise the corresponding entries are undefined.

Sets X to the rational matrix congruent to Xmod modulo mod whose entries are obtained by rational reconstruction with balanced bounds (_fmpq_reconstruct_fmpz, Rational reconstruction), using a running common denominator: each entry is multiplied by the product of the denominators found so far before it is reconstructed. Returns 1 on success and 0 if some entry cannot be reconstructed, in which case X is partially written. The result is correct if every numerator and every partial common denominator is less than \sqrt{\mathit{mod}/2}.

Sets C = AB. fmpq_mat_mul calls fmpq_mat_mul_cleared, which clears denominators of the rows of A and of the columns of B, multiplies the integer matrices with fmpz_mat_mul and divides entry (i, j) by the product of the i-th row and j-th column denominators. fmpq_mat_mul_direct computes the dot products in rational arithmetic and throws if C is A or B. fmpq_mat_mul and fmpq_mat_mul_cleared allow aliasing.

Sets C = AB where B, respectively A, is an integer matrix, by clearing the denominators of the rows of A, respectively the columns of B, and one fmpz_mat_mul.

Sets the column vector c of length A->r to Ab, where b is truncated or zero-extended to the number of columns of A. The ptr versions take arrays of pointers. The fmpq_vec versions clear the denominator of b first. c must not overlap b.

Sets the row vector c of length B->c to aB, where a is truncated or zero-extended to the number of rows of B. The ptr versions take arrays of pointers. c must not overlap a.

Sets C to the Kronecker product A \otimes B; for A of size m \times n and B of size k \times l, C must be mk \times nl. C must not be aliased with A or B.

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

Sets det to the determinant of the square matrix mat. For dimensions up to 2 an explicit formula is used; otherwise the denominators of the rows are cleared, giving \operatorname{diag}(d_i) \cdot \mathit{mat}, and the determinant of that integer matrix (fmpz_mat_det) is divided by \prod_i d_i.

Sets res to the permanent of the square matrix A, computed like the determinant with fmpz_mat_permanent after clearing denominators row by row, and returns 1.

For a square A, sets X to the solution of AX = B and returns 1 if A is nonsingular, and returns 0 if A is singular. The functions without algorithm suffix choose; the others use fmpz_mat_solve (which uses fraction-free elimination up to dimension 15), fmpz_mat_solve_dixon_den, respectively fmpz_mat_solve_multi_mod_den.

For an m \times n matrix A of any rank and an m \times k matrix B, returns 1 and sets the n \times k matrix X to a solution of AX = B if one exists, and returns 0 if the system is inconsistent. The answer is proven. The solution has zeros in the rows corresponding to the non-pivot columns of A.

Sets B to the inverse of the square matrix A and returns 1 if A is nonsingular; returns 0 if A is singular. B may be A. For dimension at least 3, the denominators of the rows of A are cleared, \operatorname{diag}(d) A = A', and A' X = \operatorname{diag}(d) is solved with fmpq_mat_solve_fmpz_mat.

Helper for Gaussian elimination. Returns 1 if the entry (r, c) is nonzero. Otherwise, if some entry (j, c) with j > r is nonzero, swaps the first such row with row r (and the entries r, j of perm if it is not NULL) and returns -1; returns 0 if all these entries are zero.

Sets B to the reduced row echelon form of A and returns the rank. B may be A. fmpq_mat_rref_classical uses Gauss–Jordan elimination with rational arithmetic; fmpq_mat_rref_fraction_free clears the denominators of the rows and calls fmpz_mat_rref. fmpq_mat_rref uses the classical algorithm if A has at most 2 rows or columns and the fraction-free one otherwise.

Sets B to the Gram–Schmidt orthogonalisation of the columns of A: column i of B is column i of A minus its projections onto the previous columns of B, without normalisation. Columns that become zero are skipped in later projections. B may be A. Throws if the dimensions differ.

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 (see fmpz_mat_similarity). Wrapper of gr_mat_apply_row_similarity; throws if A is not square or r is out of range.

Sets pol to the characteristic polynomial of the square matrix mat; the underscore version writes the n + 1 coefficients of the numerator to coeffs and the denominator to den, not necessarily in canonical form. Throws if mat is not square.

Sets pol to the minimal polynomial of the square matrix mat; the underscore version writes the coefficients of the numerator to coeffs (space for n + 1 of them) and the denominator to den, and returns the length. The minimal polynomial is computed like the characteristic polynomial, from that of the integer matrix qA with _fmpz_mat_minpoly. Throws if mat is not square.