Data.Number.Flint.Fq.Mat

An FqMat represents an matrix over a finite field. This module implements operations on matrices over a finite field.

Matrices over finite fields

data FqMat

Constructors

FqMat !(ForeignPtr CFqMat)

data CFqMat

Constructors

CFqMat (Ptr CFq) CLong CLong (Ptr (Ptr CFq))

Instances

Storable CFqMat

Defined in Data.Number.Flint.Fq.Mat.FFI

Methods

sizeOf :: CFqMat -> Int

alignment :: CFqMat -> Int

peekElemOff :: Ptr CFqMat -> Int -> IO CFqMat

pokeElemOff :: Ptr CFqMat -> Int -> CFqMat -> IO ()

peekByteOff :: Ptr b -> Int -> IO CFqMat

pokeByteOff :: Ptr b -> Int -> CFqMat -> IO ()

peek :: Ptr CFqMat -> IO CFqMat

poke :: Ptr CFqMat -> CFqMat -> IO ()

newFqMat :: CLong -> CLong -> FqCtx -> IO FqMat

withFqMat :: FqMat -> (Ptr CFqMat -> IO a) -> IO (FqMat, a)

withNewFqMat :: CLong -> CLong -> FqCtx -> (Ptr CFqMat -> IO a) -> IO (FqMat, a)

Memory management

fq_mat_init :: Ptr CFqMat -> CLong -> CLong -> Ptr CFqCtx -> IO ()

fq_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_mat_init_set :: Ptr CFqMat -> Ptr CFqMat -> Ptr CFqCtx -> IO ()

fq_mat_init_set mat src ctx

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

fq_mat_clear :: Ptr CFqMat -> Ptr CFqCtx -> IO ()

fq_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_mat_t object.

fq_mat_set :: Ptr CFqMat -> Ptr CFqMat -> Ptr CFqCtx -> IO ()

fq_mat_set mat src ctx

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

Basic properties and manipulation

fq_mat_entry :: Ptr CFqMat -> CLong -> CLong -> IO (Ptr CFq)

fq_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_mat_entry_set :: Ptr CFqMat -> CLong -> CLong -> Ptr CFq -> Ptr CFqCtx -> IO ()

fq_mat_entry_set mat i j x ctx

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

fq_mat_nrows :: Ptr CFqMat -> Ptr CFqCtx -> IO CLong

fq_mat_nrows mat ctx

Returns the number of rows in mat.

fq_mat_ncols :: Ptr CFqMat -> Ptr CFqCtx -> IO CLong

fq_mat_ncols mat ctx

Returns the number of columns in mat.

fq_mat_swap :: Ptr CFqMat -> Ptr CFqMat -> Ptr CFqCtx -> IO ()

fq_mat_swap mat1 mat2 ctx

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

fq_mat_swap_entrywise :: Ptr CFqMat -> Ptr CFqMat -> IO ()

fq_mat_swap_entrywise mat1 mat2

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

fq_mat_zero :: Ptr CFqMat -> Ptr CFqCtx -> IO ()

fq_mat_zero mat ctx

Sets all entries of mat to 0.

fq_mat_one :: Ptr CFqMat -> Ptr CFqCtx -> IO ()

fq_mat_one mat ctx

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

fq_mat_swap_rows :: Ptr CFqMat -> Ptr CLong -> CLong -> CLong -> IO ()

fq_mat_swap_rows mat perm r s

Swaps rows r and s of mat. If perm is non-NULL, the permutation of the rows will also be applied to perm.

fq_mat_swap_cols :: Ptr CFqMat -> Ptr CLong -> CLong -> CLong -> IO ()

fq_mat_swap_cols mat perm r s

Swaps columns r and s of mat. If perm is non-NULL, the permutation of the columns will also be applied to perm.

fq_mat_invert_rows :: Ptr CFqMat -> Ptr CLong -> IO ()

fq_mat_invert_rows mat perm

Swaps rows i and r - i of mat for 0 <= i < r/2, where r is the number of rows of mat. If perm is non-NULL, the permutation of the rows will also be applied to perm.

fq_mat_invert_cols :: Ptr CFqMat -> Ptr CLong -> IO ()

fq_mat_invert_cols mat perm

Swaps columns i and c - i of mat for 0 <= i < c/2, where c is the number of columns of mat. If perm is non-NULL, the permutation of the columns will also be applied to perm.

Conversions

fq_mat_set_nmod_mat :: Ptr CFqMat -> Ptr CNModMat -> Ptr CFqCtx -> IO ()

fq_mat_set_nmod_mat mat1 mat2 ctx

Sets the matrix mat1 to the matrix mat2.

fq_mat_set_fmpz_mod_mat :: Ptr CFqMat -> Ptr CFmpzModMat -> Ptr CFqCtx -> IO ()

fq_mat_set_fmpz_mod_mat mat1 mat2 ctx

Sets the matrix mat1 to the matrix mat2.

Concatenate

fq_mat_concat_vertical :: Ptr CFqMat -> Ptr CFqMat -> Ptr CFqMat -> Ptr CFqCtx -> IO ()

fq_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_mat_concat_horizontal :: Ptr CFqMat -> Ptr CFqMat -> Ptr CFqMat -> Ptr CFqCtx -> IO ()

fq_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 : \(m \times (n + k)\).

Printing

fq_mat_print_pretty :: Ptr CFqMat -> Ptr CFqCtx -> IO ()

fq_mat_print_pretty mat ctx

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

fq_mat_fprint_pretty :: Ptr CFile -> Ptr CFqMat -> Ptr CFqCtx -> IO CInt

fq_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_mat_print :: Ptr CFqMat -> Ptr CFqCtx -> IO ()

fq_mat_print mat ctx

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

fq_mat_fprint :: Ptr CFile -> Ptr CFqMat -> Ptr CFqCtx -> IO CInt

fq_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.

Window

fq_mat_window_init :: Ptr CFqMat -> Ptr CFqMat -> CLong -> CLong -> CLong -> CLong -> Ptr CFqCtx -> IO ()

fq_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_mat_window_clear :: Ptr CFqMat -> Ptr CFqCtx -> IO ()

fq_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.

Random matrix generation

fq_mat_randtest :: Ptr CFqMat -> Ptr CFRandState -> Ptr CFqCtx -> IO ()

fq_mat_randtest mat state ctx

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

fq_mat_randpermdiag :: Ptr CFqMat -> Ptr CFRandState -> Ptr (Ptr CFq) -> CLong -> Ptr CFqCtx -> IO CInt

fq_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_mat_randrank :: Ptr CFqMat -> Ptr CFRandState -> CLong -> Ptr CFqCtx -> IO ()

fq_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_mat_randops.

fq_mat_randops :: Ptr CFqMat -> CLong -> Ptr CFRandState -> Ptr CFqCtx -> IO ()

fq_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_mat_randtril :: Ptr CFqMat -> Ptr CFRandState -> CInt -> Ptr CFqCtx -> IO ()

fq_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_mat_randtriu :: Ptr CFqMat -> Ptr CFRandState -> CInt -> Ptr CFqCtx -> IO ()

fq_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.

Comparison

fq_mat_equal :: Ptr CFqMat -> Ptr CFqMat -> Ptr CFqCtx -> IO CInt

fq_mat_equal mat1 mat2 ctx

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

fq_mat_is_zero :: Ptr CFqMat -> Ptr CFqCtx -> IO CInt

fq_mat_is_zero mat ctx

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

fq_mat_is_one :: Ptr CFqMat -> Ptr CFqCtx -> IO CInt

fq_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_mat_is_empty :: Ptr CFqMat -> Ptr CFqCtx -> IO CInt

fq_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_mat_is_square :: Ptr CFqMat -> Ptr CFqCtx -> IO CInt

fq_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.

Addition and subtraction

fq_mat_add :: Ptr CFqMat -> Ptr CFqMat -> Ptr CFqMat -> Ptr CFqCtx -> IO ()

fq_mat_add C A B ctx

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

fq_mat_sub :: Ptr CFqMat -> Ptr CFqMat -> Ptr CFqMat -> Ptr CFqCtx -> IO ()

fq_mat_sub C A B ctx

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

fq_mat_neg :: Ptr CFqMat -> Ptr CFqMat -> Ptr CFqCtx -> IO ()

fq_mat_neg A B ctx

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

Matrix multiplication

fq_mat_mul :: Ptr CFqMat -> Ptr CFqMat -> Ptr CFqMat -> Ptr CFqCtx -> IO ()

fq_mat_mul C A B ctx

Sets \(C = AB\). Dimensions must be compatible for matrix multiplication. Aliasing is allowed. This function automatically chooses between classical and KS multiplication.

fq_mat_mul_classical :: Ptr CFqMat -> Ptr CFqMat -> Ptr CFqMat -> Ptr CFqCtx -> IO ()

fq_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_mat_mul_KS :: Ptr CFqMat -> Ptr CFqMat -> Ptr CFqMat -> Ptr CFqCtx -> IO ()

fq_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_mat_submul :: Ptr CFqMat -> Ptr CFqMat -> Ptr CFqMat -> Ptr CFqMat -> Ptr CFqCtx -> IO ()

fq_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_mat_mul_vec :: Ptr (Ptr CFq) -> Ptr CFqMat -> Ptr (Ptr CFq) -> CLong -> IO ()

fq_mat_mul_vec c A b blen

fq_mat_mul_vec_ptr :: Ptr (Ptr (Ptr CFq)) -> Ptr CFqMat -> Ptr (Ptr (Ptr CFq)) -> CLong -> IO ()

fq_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_mat_vec_mul :: Ptr (Ptr CFq) -> Ptr (Ptr CFq) -> CLong -> Ptr CFqMat -> IO ()

fq_mat_vec_mul c a alen B

fq_mat_vec_mul_ptr :: Ptr (Ptr (Ptr CFq)) -> Ptr (Ptr (Ptr CFq)) -> CLong -> Ptr CFqMat -> IO ()

fq_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.

Inverse

fq_mat_inv :: Ptr CFqMat -> Ptr CFqMat -> Ptr CFqCtx -> IO CInt

fq_mat_inv B A ctx

Sets \(B = A^{-1}\) and returns \(1\) if \(A\) is invertible. If \(A\) is singular, returns \(0\) and sets the elements of \(B\) to undefined values.

\(A\) and \(B\) must be square matrices with the same dimensions.

LU decomposition

fq_mat_lu :: Ptr CLong -> Ptr CFqMat -> CInt -> Ptr CFqCtx -> IO CLong

fq_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_mat_lu_recursive.

fq_mat_lu_classical :: Ptr CLong -> Ptr CFqMat -> CInt -> Ptr CFqCtx -> IO CLong

fq_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_mat_lu. Uses Gaussian elimination.

fq_mat_lu_recursive :: Ptr CLong -> Ptr CFqMat -> CInt -> Ptr CFqCtx -> IO CLong

fq_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_mat_lu. Uses recursive block decomposition, switching to classical Gaussian elimination for sufficiently small blocks.

Reduced row echelon form

fq_mat_rref :: Ptr CFqMat -> Ptr CFqCtx -> IO CLong

fq_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_mat_reduce_row :: Ptr CFqMat -> Ptr CLong -> Ptr CLong -> CLong -> Ptr CFqCtx -> IO CLong

fq_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.

Triangular solving

fq_mat_solve_tril :: Ptr CFqMat -> Ptr CFqMat -> Ptr CFqMat -> CInt -> Ptr CFqCtx -> IO ()

fq_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_mat_solve_tril_classical :: Ptr CFqMat -> Ptr CFqMat -> Ptr CFqMat -> CInt -> Ptr CFqCtx -> IO ()

fq_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_mat_solve_tril_recursive :: Ptr CFqMat -> Ptr CFqMat -> Ptr CFqMat -> CInt -> Ptr CFqCtx -> IO ()

fq_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_mat_solve_triu :: Ptr CFqMat -> Ptr CFqMat -> Ptr CFqMat -> CInt -> Ptr CFqCtx -> IO ()

fq_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_mat_solve_triu_classical :: Ptr CFqMat -> Ptr CFqMat -> Ptr CFqMat -> CInt -> Ptr CFqCtx -> IO ()

fq_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_mat_solve_triu_recursive :: Ptr CFqMat -> Ptr CFqMat -> Ptr CFqMat -> CInt -> Ptr CFqCtx -> IO ()

fq_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.

Solving

fq_mat_solve :: Ptr CFqMat -> Ptr CFqMat -> Ptr CFqMat -> Ptr CFqCtx -> IO CInt

fq_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_mat_can_solve :: Ptr CFqMat -> Ptr CFqMat -> Ptr CFqMat -> Ptr CFqCtx -> IO CInt

fq_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.

Transforms

fq_mat_similarity :: Ptr CFqMat -> CLong -> Ptr CFq -> Ptr CFqCtx -> IO ()

fq_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.

Characteristic polynomial

fq_mat_charpoly_danilevsky :: Ptr CFqPoly -> Ptr CFqMat -> Ptr CFqCtx -> IO ()

fq_mat_charpoly_danilevsky p M ctx

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

fq_mat_charpoly :: Ptr CFqPoly -> Ptr CFqMat -> Ptr CFqCtx -> IO ()

fq_mat_charpoly p M ctx

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

Minimal polynomial

fq_mat_minpoly :: Ptr CFqPoly -> Ptr CFqMat -> Ptr CFqCtx -> IO ()

fq_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.