fq_zech_mat_init mat rows cols ctx

Initialises mat to a rows-by-cols matrix with coefficients in \(\mathbf{F}_{q}\) given by ctx. All elements are set to zero.

fq_zech_mat_init_set mat src ctx

Initialises mat and sets its dimensions and elements to those of src.

fq_zech_mat_clear mat ctx

Clears the matrix and releases any memory it used. The matrix cannot be used again until it is initialised. This function must be called exactly once when finished using an fq_zech_mat_t object.

fq_zech_mat_set mat src ctx

Sets mat to a copy of src. It is assumed that mat and src have identical dimensions.

fq_zech_mat_entry mat i j

Directly accesses the entry in mat in row \(i\) and column \(j\), indexed from zero. No bounds checking is performed.

fq_zech_mat_entry_set mat i j x ctx

Sets the entry in mat in row \(i\) and column \(j\) to x.

fq_zech_mat_nrows mat ctx

Returns the number of rows in mat.

fq_zech_mat_ncols mat ctx

Returns the number of columns in mat.

fq_zech_mat_swap mat1 mat2 ctx

Swaps two matrices. The dimensions of mat1 and mat2 are allowed to be different.

fq_zech_mat_swap_entrywise mat1 mat2

Swaps two matrices by swapping the individual entries rather than swapping the contents of the structs.

fq_zech_mat_zero mat ctx

Sets all entries of mat to 0.

fq_zech_mat_one mat ctx

Sets all diagonal entries of mat to 1 and all other entries to 0.

fq_zech_mat_set_nmod_mat mat1 mat2 ctx

Sets the matrix mat1 to the matrix mat2.

fq_zech_mat_set_fmpz_mod_mat mat1 mat2 ctx

Sets the matrix mat1 to the matrix mat2.

fq_zech_mat_concat_vertical res mat1 mat2 ctx

Sets res to vertical concatenation of (mat1, mat2) in that order. Matrix dimensions : mat1 : \(m \times n\), mat2 : \(k \times n\), res : \((m + k) \times n\).

fq_zech_mat_concat_horizontal res mat1 mat2 ctx

Sets res to horizontal concatenation of (mat1, mat2) in that order. Matrix dimensions : mat1 : \(m \times n\), mat2 : \(m \times k\), res : \(m \times (n + k)\).

fq_zech_mat_print_pretty mat ctx

Pretty-prints mat to stdout. A header is printed followed by the rows enclosed in brackets.

fq_zech_mat_fprint_pretty file mat ctx

Pretty-prints mat to file. A header is printed followed by the rows enclosed in brackets.

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

fq_zech_mat_print mat ctx

Prints mat to stdout. A header is printed followed by the rows enclosed in brackets.

fq_zech_mat_fprint file mat ctx

Prints mat to file. A header is printed followed by the rows enclosed in brackets.

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

fq_zech_mat_window_init window mat r1 c1 r2 c2 ctx

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.

fq_zech_mat_window_clear window ctx

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.

fq_zech_mat_randtest mat state ctx

Sets the elements of mat to random elements of \(\mathbf{F}_{q}\), given by ctx.

fq_zech_mat_randpermdiag mat state diag n ctx

Sets mat to a random permutation of the diagonal matrix with \(n\) leading entries given by the vector diag. It is assumed that the main diagonal of mat has room for at least \(n\) entries.

Returns \(0\) or \(1\), depending on whether the permutation is even or odd respectively.

fq_zech_mat_randrank mat state rank ctx

Sets mat to a random sparse matrix with the given rank, having exactly as many non-zero elements as the rank, with the non-zero elements being uniformly random elements of \(\mathbf{F}_{q}\).

The matrix can be transformed into a dense matrix with unchanged rank by subsequently calling fq_zech_mat_randops.

fq_zech_mat_randops mat count state ctx

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, determinant) unchanged.

fq_zech_mat_randtril mat state unit ctx

Sets mat to a random lower triangular matrix. If unit is 1, it will have ones on the main diagonal, otherwise it will have random nonzero entries on the main diagonal.

fq_zech_mat_randtriu mat state unit ctx

Sets mat to a random upper triangular matrix. If unit is 1, it will have ones on the main diagonal, otherwise it will have random nonzero entries on the main diagonal.

fq_zech_mat_equal mat1 mat2 ctx

Returns nonzero if mat1 and mat2 have the same dimensions and elements, and zero otherwise.

fq_zech_mat_is_zero mat ctx

Returns a non-zero value if all entries mat are zero, and otherwise returns zero.

fq_zech_mat_is_one mat ctx

Returns a non-zero value if all entries mat are zero except the diagonal entries which must be one, otherwise returns zero.

fq_zech_mat_is_empty mat ctx

Returns a non-zero value if the number of rows or the number of columns in mat is zero, and otherwise returns zero.

fq_zech_mat_is_square mat ctx

Returns a non-zero value if the number of rows is equal to the number of columns in mat, and otherwise returns zero.

fq_zech_mat_add C A B ctx

Computes \(C = A + B\). Dimensions must be identical.

fq_zech_mat_sub C A B ctx

Computes \(C = A - B\). Dimensions must be identical.

fq_zech_mat_neg A B ctx

Sets \(B = -A\). Dimensions must be identical.

fq_zech_mat_mul C A B ctx

Sets \(C = AB\). Dimensions must be compatible for matrix multiplication. \(C\) is not allowed to be aliased with \(A\) or \(B\). This function automatically chooses between classical and KS multiplication.

fq_zech_mat_mul_classical C A B ctx

Sets \(C = AB\). Dimensions must be compatible for matrix multiplication. \(C\) is not allowed to be aliased with \(A\) or \(B\). Uses classical matrix multiplication.

fq_zech_mat_mul_KS C A B ctx

Sets \(C = AB\). Dimensions must be compatible for matrix multiplication. \(C\) is not allowed to be aliased with \(A\) or \(B\). Uses Kronecker substitution to perform the multiplication over the integers.

fq_zech_mat_submul D C A B ctx

Sets \(D = C + AB\). \(C\) and \(D\) may be aliased with each other but not with \(A\) or \(B\).

fq_zech_mat_mul_vec c A b blen

fq_zech_mat_mul_vec_ptr c A b blen

Compute a matrix-vector product of A and (b, blen) and store the result in c. The vector (b, blen) is either truncated or zero-extended to the number of columns of A. The number entries written to c is always equal to the number of rows of A.

fq_zech_mat_vec_mul c a alen B

fq_zech_mat_vec_mul_ptr c a alen B

Compute a vector-matrix product of (a, alen) and B and and store the result in c. The vector (a, alen) is either truncated or zero-extended to the number of rows of B. The number entries written to c is always equal to the number of columns of B.

fq_zech_mat_lu P A rank_check ctx

Computes a generalised LU decomposition \(LU = PA\) of a given matrix \(A\), returning the rank of \(A\).

If \(A\) is a nonsingular square matrix, it will be overwritten with a unit diagonal lower triangular matrix \(L\) and an upper triangular matrix \(U\) (the diagonal of \(L\) will not be stored explicitly).

If \(A\) is an arbitrary matrix of rank \(r\), \(U\) will be in row echelon form having \(r\) nonzero rows, and \(L\) will be lower triangular but truncated to \(r\) columns, having implicit ones on the \(r\) first entries of the main diagonal. All other entries will be zero.

If a nonzero value for rank_check is passed, the function will abandon the output matrix in an undefined state and return 0 if \(A\) is detected to be rank-deficient.

This function calls fq_zech_mat_lu_recursive.

fq_zech_mat_lu_classical P A rank_check ctx

Computes a generalised LU decomposition \(LU = PA\) of a given matrix \(A\), returning the rank of \(A\). The behavior of this function is identical to that of fq_zech_mat_lu. Uses Gaussian elimination.

fq_zech_mat_lu_recursive P A rank_check ctx

Computes a generalised LU decomposition \(LU = PA\) of a given matrix \(A\), returning the rank of \(A\). The behavior of this function is identical to that of fq_zech_mat_lu. Uses recursive block decomposition, switching to classical Gaussian elimination for sufficiently small blocks.

fq_zech_mat_rref A ctx

Puts \(A\) in reduced row echelon form and returns the rank of \(A\).

The rref is computed by first obtaining an unreduced row echelon form via LU decomposition and then solving an additional triangular system.

fq_zech_mat_reduce_row A P L n ctx

Reduce row n of the matrix \(A\), assuming the prior rows are in Gauss form. However those rows may not be in order. The entry \(i\) of the array \(P\) is the row of \(A\) which has a pivot in the \(i\)-th column. If no such row exists, the entry of \(P\) will be \(-1\). The function returns the column in which the \(n\)-th row has a pivot after reduction. This will always be chosen to be the first available column for a pivot from the left. This information is also updated in \(P\). Entry \(i\) of the array \(L\) contains the number of possibly nonzero columns of \(A\) row \(i\). This speeds up reduction in the case that \(A\) is chambered on the right. Otherwise the entries of \(L\) can all be set to the number of columns of \(A\). We require the entries of \(L\) to be monotonic increasing.

fq_zech_mat_solve_tril X L B unit ctx

Sets \(X = L^{-1} B\) where \(L\) is a full rank lower triangular square matrix. If unit = 1, \(L\) is assumed to have ones on its main diagonal, and the main diagonal will not be read. \(X\) and \(B\) are allowed to be the same matrix, but no other aliasing is allowed. Automatically chooses between the classical and recursive algorithms.

fq_zech_mat_solve_tril_classical X L B unit ctx

Sets \(X = L^{-1} B\) where \(L\) is a full rank lower triangular square matrix. If unit = 1, \(L\) is assumed to have ones on its main diagonal, and the main diagonal will not be read. \(X\) and \(B\) are allowed to be the same matrix, but no other aliasing is allowed. Uses forward substitution.

fq_zech_mat_solve_tril_recursive X L B unit ctx

Sets \(X = L^{-1} B\) where \(L\) is a full rank lower triangular square matrix. If unit = 1, \(L\) is assumed to have ones on its main diagonal, and the main diagonal will not be read. \(X\) and \(B\) are allowed to be the same matrix, but no other aliasing is allowed.

Uses the block inversion formula

\[\begin{aligned} ` \begin{pmatrix} A & 0 \\ C & D \end{pmatrix}^{-1} \begin{pmatrix} X \\ Y \end{pmatrix} = \begin{pmatrix} A^{-1} X \\ D^{-1} ( Y - C A^{-1} X ) \end{pmatrix} \end{aligned}\]

to reduce the problem to matrix multiplication and triangular solving of smaller systems.

fq_zech_mat_solve_triu X U B unit ctx

Sets \(X = U^{-1} B\) where \(U\) is a full rank upper triangular square matrix. If unit = 1, \(U\) is assumed to have ones on its main diagonal, and the main diagonal will not be read. \(X\) and \(B\) are allowed to be the same matrix, but no other aliasing is allowed. Automatically chooses between the classical and recursive algorithms.

fq_zech_mat_solve_triu_classical X U B unit ctx

Sets \(X = U^{-1} B\) where \(U\) is a full rank upper triangular square matrix. If unit = 1, \(U\) is assumed to have ones on its main diagonal, and the main diagonal will not be read. \(X\) and \(B\) are allowed to be the same matrix, but no other aliasing is allowed. Uses forward substitution.

fq_zech_mat_solve_triu_recursive X U B unit ctx

Sets \(X = U^{-1} B\) where \(U\) is a full rank upper triangular square matrix. If unit = 1, \(U\) is assumed to have ones on its main diagonal, and the main diagonal will not be read. \(X\) and \(B\) are allowed to be the same matrix, but no other aliasing is allowed.

Uses the block inversion formula

\[\begin{aligned} ` \begin{pmatrix} A & B \\ 0 & D \end{pmatrix}^{-1} \begin{pmatrix} X \\ Y \end{pmatrix} = \begin{pmatrix} A^{-1} (X - B D^{-1} Y) \\ D^{-1} Y \end{pmatrix} \end{aligned}\]

to reduce the problem to matrix multiplication and triangular solving of smaller systems.

fq_zech_mat_solve X A B ctx

Solves the matrix-matrix equation \(AX = B\).

Returns \(1\) if \(A\) has full rank; otherwise returns \(0\) and sets the elements of \(X\) to undefined values.

The matrix \(A\) must be square.

fq_zech_mat_can_solve X A B ctx

Solves the matrix-matrix equation \(AX = B\) over \(Fq\).

Returns \(1\) if a solution exists; otherwise returns \(0\) and sets the elements of \(X\) to zero. If more than one solution exists, one of the valid solutions is given.

There are no restrictions on the shape of \(A\) and it may be singular.

fq_zech_mat_similarity M r d ctx

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.

The value \(d\) is required to be reduced modulo the modulus of the entries in the matrix.

fq_zech_mat_charpoly_danilevsky p M ctx

Compute the characteristic polynomial \(p\) of the matrix \(M\). The matrix is assumed to be square.

fq_zech_mat_charpoly p M

Compute the characteristic polynomial \(p\) of the matrix \(M\). The matrix is required to be square, otherwise an exception is raised.

fq_zech_mat_minpoly p M ctx

Compute the minimal polynomial \(p\) of the matrix \(M\). The matrix is required to be square, otherwise an exception is raised.