Data.Number.Flint.Fmpq.Mat

Matrices over the rational numbers

data FmpqMat

Constructors

FmpqMat !(ForeignPtr CFmpqMat)

Instances

Semigroup FmpqMat

Defined in Data.Number.Flint.Fmpq.Mat.Instances

Methods

(<>) :: FmpqMat -> FmpqMat -> FmpqMat

sconcat :: NonEmpty FmpqMat -> FmpqMat

stimes :: Integral b => b -> FmpqMat -> FmpqMat

Num FmpqMat

Defined in Data.Number.Flint.Fmpq.Mat.Instances

Methods

(+) :: FmpqMat -> FmpqMat -> FmpqMat

(-) :: FmpqMat -> FmpqMat -> FmpqMat

(*) :: FmpqMat -> FmpqMat -> FmpqMat

negate :: FmpqMat -> FmpqMat

abs :: FmpqMat -> FmpqMat

signum :: FmpqMat -> FmpqMat

fromInteger :: Integer -> FmpqMat

Fractional FmpqMat

Defined in Data.Number.Flint.Fmpq.Mat.Instances

Methods

(/) :: FmpqMat -> FmpqMat -> FmpqMat

recip :: FmpqMat -> FmpqMat

fromRational :: Rational -> FmpqMat

Show FmpqMat

Defined in Data.Number.Flint.Fmpq.Mat.Instances

Methods

showsPrec :: Int -> FmpqMat -> ShowS

show :: FmpqMat -> String

showList :: [FmpqMat] -> ShowS

Eq FmpqMat

Defined in Data.Number.Flint.Fmpq.Mat.Instances

Methods

(==) :: FmpqMat -> FmpqMat -> Bool

(/=) :: FmpqMat -> FmpqMat -> Bool

data CFmpqMat

Constructors

CFmpqMat (Ptr CFmpq) CLong CLong (Ptr (Ptr CFmpq))

Instances

Storable CFmpqMat

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

Methods

sizeOf :: CFmpqMat -> Int

alignment :: CFmpqMat -> Int

peekElemOff :: Ptr CFmpqMat -> Int -> IO CFmpqMat

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

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

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

peek :: Ptr CFmpqMat -> IO CFmpqMat

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

newFmpqMat :: CLong -> CLong -> IO FmpqMat

withFmpqMat :: FmpqMat -> (Ptr CFmpqMat -> IO a) -> IO (FmpqMat, a)

withNewFmpqMat :: CLong -> CLong -> (Ptr CFmpqMat -> IO a) -> IO (FmpqMat, a)

Memory management

fmpq_mat_init :: Ptr CFmpqMat -> CLong -> CLong -> IO ()

fmpq_mat_init mat rows cols

Initialises a matrix with the given number of rows and columns for use.

fmpq_mat_init_set :: Ptr CFmpqMat -> Ptr CFmpqMat -> IO ()

fmpq_mat_init_set mat1 mat2

Initialises mat1 and sets it equal to mat2.

fmpq_mat_clear :: Ptr CFmpqMat -> IO ()

fmpq_mat_clear mat

Frees all memory associated with the matrix. The matrix must be reinitialised if it is to be used again.

fmpq_mat_swap :: Ptr CFmpqMat -> Ptr CFmpqMat -> IO ()

fmpq_mat_swap mat1 mat2

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

fmpq_mat_swap_entrywise :: Ptr CFmpqMat -> Ptr CFmpqMat -> IO ()

fmpq_mat_swap_entrywise mat1 mat2

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

Entry access

fmpq_mat_entry :: Ptr CFmpqMat -> CLong -> CLong -> IO (Ptr CFmpq)

fmpq_mat_entry mat i j

Gives a reference to the entry at row i and column j. The reference can be passed as an input or output variable to any fmpq function for direct manipulation of the matrix element. No bounds checking is performed.

fmpq_mat_entry_num :: Ptr CFmpqMat -> CLong -> CLong -> IO (Ptr CFmpz)

fmpq_mat_entry_num mat i j

Gives a reference to the numerator of the entry at row i and column j. The reference can be passed as an input or output variable to any fmpz function for direct manipulation of the matrix element. No bounds checking is performed.

fmpq_mat_entry_den :: Ptr CFmpqMat -> CLong -> CLong -> IO (Ptr CFmpz)

fmpq_mat_entry_den mat i j

Gives a reference to the denominator of the entry at row i and column j. The reference can be passed as an input or output variable to any fmpz function for direct manipulation of the matrix element. No bounds checking is performed.

fmpq_mat_set_entry :: Ptr CFmpqMat -> CLong -> CLong -> Ptr CFmpq -> IO ()

fmpq_mat_get_entry :: Ptr CFmpq -> Ptr CFmpqMat -> CLong -> CLong -> IO ()

fmpq_mat_nrows :: Ptr CFmpqMat -> IO CLong

fmpq_mat_nrows mat

Return the number of rows of the matrix mat.

fmpq_mat_ncols :: Ptr CFmpqMat -> IO CLong

fmpq_mat_ncols mat

Return the number of columns of the matrix mat.

Basic assignment

fmpq_mat_set :: Ptr CFmpqMat -> Ptr CFmpqMat -> IO ()

fmpq_mat_set dest src

Sets the entries in dest to the same values as in src, assuming the two matrices have the same dimensions.

fmpq_mat_zero :: Ptr CFmpqMat -> IO ()

fmpq_mat_zero mat

Sets mat to the zero matrix.

fmpq_mat_one :: Ptr CFmpqMat -> IO ()

fmpq_mat_one mat

Let \(m\) be the minimum of the number of rows and columns in the matrix mat. This function sets the first \(m \times m\) block to the identity matrix, and the remaining block to zero.

fmpq_mat_transpose :: Ptr CFmpqMat -> Ptr CFmpqMat -> IO ()

fmpq_mat_transpose rop op

Sets the matrix rop to the transpose of the matrix op, assuming that their dimensions are compatible.

fmpq_mat_swap_rows :: Ptr CFmpqMat -> Ptr CLong -> CLong -> CLong -> IO ()

fmpq_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.

fmpq_mat_swap_cols :: Ptr CFmpqMat -> Ptr CLong -> CLong -> CLong -> IO ()

fmpq_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.

fmpq_mat_invert_rows :: Ptr CFmpqMat -> Ptr CLong -> IO ()

fmpq_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.

fmpq_mat_invert_cols :: Ptr CFmpqMat -> Ptr CLong -> IO ()

fmpq_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.

Addition, scalar multiplication

fmpq_mat_add :: Ptr CFmpqMat -> Ptr CFmpqMat -> Ptr CFmpqMat -> IO ()

fmpq_mat_add mat mat1 mat2

Sets mat to the sum of mat1 and mat2, assuming that all three matrices have the same dimensions.

fmpq_mat_sub :: Ptr CFmpqMat -> Ptr CFmpqMat -> Ptr CFmpqMat -> IO ()

fmpq_mat_sub mat mat1 mat2

Sets mat to the difference of mat1 and mat2, assuming that all three matrices have the same dimensions.

fmpq_mat_neg :: Ptr CFmpqMat -> Ptr CFmpqMat -> IO ()

fmpq_mat_neg rop op

Sets rop to the negative of op, assuming that the two matrices have the same dimensions.

fmpq_mat_scalar_mul_fmpq :: Ptr CFmpqMat -> Ptr CFmpqMat -> Ptr CFmpq -> IO ()

fmpq_mat_scalar_mul_fmpq rop op x

Sets rop to op multiplied by the rational \(x\), assuming that the two matrices have the same dimensions.

Note that the rational x may not be aliased with any part of the entries of rop.

fmpq_mat_scalar_mul_fmpz :: Ptr CFmpqMat -> Ptr CFmpqMat -> Ptr CFmpz -> IO ()

fmpq_mat_scalar_mul_fmpz rop op x

Sets rop to op multiplied by the integer \(x\), assuming that the two matrices have the same dimensions.

Note that the integer \(x\) may not be aliased with any part of the entries of rop.

fmpq_mat_scalar_div_fmpz :: Ptr CFmpqMat -> Ptr CFmpqMat -> Ptr CFmpz -> IO ()

fmpq_mat_scalar_div_fmpz rop op x

Sets rop to op divided by the integer \(x\), assuming that the two matrices have the same dimensions and that \(x\) is non-zero.

Note that the integer \(x\) may not be aliased with any part of the entries of rop.

Input and output

fmpq_mat_get_str :: Ptr CFmpqMat -> IO CString

fmpq_mat_get_str mat

Returns a string representation.

fmpq_mat_fprint :: Ptr CFile -> Ptr CFmpqMat -> IO ()

fmpq_mat_fprint file mat

Prints the matrix mat to the stream file.

fmpq_mat_print :: Ptr CFmpqMat -> IO CInt

fmpq_mat_print mat

Prints the matrix mat to standard output.

Random matrix generation

fmpq_mat_randbits :: Ptr CFmpqMat -> Ptr CFRandState -> CFBitCnt -> IO ()

fmpq_mat_randbits mat state bits

This is equivalent to applying fmpq_randbits to all entries in the matrix.

fmpq_mat_randtest :: Ptr CFmpqMat -> Ptr CFRandState -> CFBitCnt -> IO ()

fmpq_mat_randtest mat state bits

This is equivalent to applying fmpq_randtest to all entries in the matrix.

Window

fmpq_mat_window_init :: Ptr CFmpqMat -> Ptr CFmpqMat -> CLong -> CLong -> CLong -> CLong -> IO ()

fmpq_mat_window_init window mat r1 c1 r2 c2

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.

fmpq_mat_window_clear :: Ptr CFmpqMat -> IO ()

fmpq_mat_window_clear window

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.

Concatenate

fmpq_mat_concat_vertical :: Ptr CFmpqMat -> Ptr CFmpqMat -> Ptr CFmpqMat -> IO ()

fmpq_mat_concat_vertical res mat1 mat2

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

fmpq_mat_concat_horizontal :: Ptr CFmpqMat -> Ptr CFmpqMat -> Ptr CFmpqMat -> IO ()

fmpq_mat_concat_horizontal res mat1 mat2

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

Special matrices

fmpq_mat_hilbert_matrix :: Ptr CFmpqMat -> IO ()

fmpq_mat_hilbert_matrix mat

Sets mat to a Hilbert matrix of the given size. That is, the entry at row \(i\) and column \(j\) is set to \(1/(i+j+1)\).

Basic comparison and properties

fmpq_mat_equal :: Ptr CFmpqMat -> Ptr CFmpqMat -> IO CInt

fmpq_mat_equal mat1 mat2

Returns nonzero if mat1 and mat2 have the same shape and all their entries agree, and returns zero otherwise. Assumes the entries in both mat1 and mat2 are in canonical form.

fmpq_mat_is_integral :: Ptr CFmpqMat -> IO CInt

fmpq_mat_is_integral mat

Returns nonzero if all entries in mat are integer-valued, and returns zero otherwise. Assumes that the entries in mat are in canonical form.

fmpq_mat_is_zero :: Ptr CFmpqMat -> IO CInt

fmpq_mat_is_zero mat

Returns nonzero if all entries in mat are zero, and returns zero otherwise.

fmpq_mat_is_one :: Ptr CFmpqMat -> IO CInt

fmpq_mat_is_one mat

Returns nonzero if mat ones along the diagonal and zeros elsewhere, and returns zero otherwise.

fmpq_mat_is_empty :: Ptr CFmpqMat -> IO CInt

fmpq_mat_is_empty mat

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

fmpq_mat_is_square :: Ptr CFmpqMat -> IO CInt

fmpq_mat_is_square mat

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

Integer matrix conversion

fmpq_mat_get_fmpz_mat :: Ptr CFmpzMat -> Ptr CFmpqMat -> IO CInt

fmpq_mat_get_fmpz_mat dest mat

Sets dest to mat and returns nonzero if all entries in mat are integer-valued. If not all entries in mat are integer-valued, sets dest to an undefined matrix and returns zero. Assumes that the entries in mat are in canonical form.

fmpq_mat_get_fmpz_mat_entrywise :: Ptr CFmpzMat -> Ptr CFmpzMat -> Ptr CFmpqMat -> IO ()

fmpq_mat_get_fmpz_mat_entrywise num den mat

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

fmpq_mat_get_fmpz_mat_matwise :: Ptr CFmpzMat -> Ptr CFmpz -> Ptr CFmpqMat -> IO ()

fmpq_mat_get_fmpz_mat_matwise num den mat

Converts all entries in mat to a common denominator, storing the rescaled numerators in num and the denominator in den. The denominator will be minimal if the entries in mat are in canonical form.

fmpq_mat_get_fmpz_mat_rowwise :: Ptr CFmpzMat -> Ptr CFmpz -> Ptr CFmpqMat -> IO ()

fmpq_mat_get_fmpz_mat_rowwise num den mat

Clears denominators in mat row by row. The rescaled numerators are written to num, and the denominator of row i is written to position i in den which can be a preinitialised fmpz vector. Alternatively, NULL can be passed as the den variable, in which case the denominators will not be stored.

fmpq_mat_get_fmpz_mat_rowwise_2 :: Ptr CFmpzMat -> Ptr CFmpzMat -> Ptr CFmpz -> Ptr CFmpqMat -> Ptr CFmpqMat -> IO ()

fmpq_mat_get_fmpz_mat_rowwise_2 num num2 den mat mat2

Clears denominators row by row of both mat and mat2, writing the respective numerators to num and num2. This is equivalent to concatenating mat and mat2 horizontally, calling fmpq_mat_get_fmpz_mat_rowwise, and extracting the two submatrices in the result.

fmpq_mat_get_fmpz_mat_colwise :: Ptr CFmpzMat -> Ptr CFmpz -> Ptr CFmpqMat -> IO ()

fmpq_mat_get_fmpz_mat_colwise num den mat

Clears denominators in mat column by column. The rescaled numerators are written to num, and the denominator of column i is written to position i in den which can be a preinitialised fmpz vector. Alternatively, NULL can be passed as the den variable, in which case the denominators will not be stored.

fmpq_mat_set_fmpz_mat :: Ptr CFmpqMat -> Ptr CFmpzMat -> IO ()

fmpq_mat_set_fmpz_mat dest src

Sets dest to src.

fmpq_mat_set_fmpz_mat_div_fmpz :: Ptr CFmpqMat -> Ptr CFmpzMat -> Ptr CFmpz -> IO ()

fmpq_mat_set_fmpz_mat_div_fmpz mat num den

Sets mat to the integer matrix num divided by the common denominator den.

Modular reduction and rational reconstruction

fmpq_mat_get_fmpz_mat_mod_fmpz :: Ptr CFmpzMat -> Ptr CFmpqMat -> Ptr CFmpz -> IO ()

fmpq_mat_get_fmpz_mat_mod_fmpz dest mat mod

Sets each entry in dest to the corresponding entry in mat, reduced modulo mod.

fmpq_mat_set_fmpz_mat_mod_fmpz :: Ptr CFmpqMat -> Ptr CFmpzMat -> Ptr CFmpz -> IO CInt

fmpq_mat_set_fmpz_mat_mod_fmpz X Xmod mod

Set X to the entrywise rational reconstruction integer matrix Xmod modulo mod, and returns nonzero if the reconstruction is successful. If rational reconstruction fails for any element, returns zero and sets the entries in X to undefined values.

Matrix multiplication

fmpq_mat_mul_direct :: Ptr CFmpqMat -> Ptr CFmpqMat -> Ptr CFmpqMat -> IO ()

fmpq_mat_mul_direct C A B

Sets C to the matrix product AB, computed naively using rational arithmetic. This is typically very slow and should only be used in circumstances where clearing denominators would consume too much memory.

fmpq_mat_mul_cleared :: Ptr CFmpqMat -> Ptr CFmpqMat -> Ptr CFmpqMat -> IO ()

fmpq_mat_mul_cleared C A B

Sets C to the matrix product AB, computed by clearing denominators and multiplying over the integers.

fmpq_mat_mul :: Ptr CFmpqMat -> Ptr CFmpqMat -> Ptr CFmpqMat -> IO ()

fmpq_mat_mul C A B

Sets C to the matrix product AB. This simply calls fmpq_mat_mul_cleared.

fmpq_mat_mul_fmpz_mat :: Ptr CFmpqMat -> Ptr CFmpqMat -> Ptr CFmpzMat -> IO ()

fmpq_mat_mul_fmpz_mat C A B

Sets C to the matrix product AB, with B an integer matrix. This function works efficiently by clearing denominators of A.

fmpq_mat_mul_r_fmpz_mat :: Ptr CFmpqMat -> Ptr CFmpzMat -> Ptr CFmpqMat -> IO ()

fmpq_mat_mul_r_fmpz_mat C A B

Sets C to the matrix product AB, with A an integer matrix. This function works efficiently by clearing denominators of B.

fmpq_mat_mul_fmpq_vec :: Ptr CFmpq -> Ptr CFmpqMat -> Ptr CFmpq -> CLong -> IO ()

fmpq_mat_mul_fmpq_vec 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.

fmpq_mat_fmpq_vec_mul :: Ptr CFmpq -> Ptr CFmpq -> CLong -> Ptr CFmpqMat -> IO ()

fmpq_mat_fmpq_vec_mul 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.

Kronecker product

fmpq_mat_kronecker_product :: Ptr CFmpqMat -> Ptr CFmpqMat -> Ptr CFmpqMat -> IO ()

fmpq_mat_kronecker_product C A B

Sets C to the Kronecker product of A and B.

Trace

fmpq_mat_trace :: Ptr CFmpq -> Ptr CFmpqMat -> IO ()

fmpq_mat_trace trace mat

Computes the trace of the matrix, i.e. the sum of the entries on the main diagonal. The matrix is required to be square.

Determinant

fmpq_mat_det :: Ptr CFmpq -> Ptr CFmpqMat -> IO ()

fmpq_mat_det det mat

Sets det to the determinant of mat. In the general case, the determinant is computed by clearing denominators and computing a determinant over the integers. Matrices of size 0, 1 or 2 are handled directly.

Nonsingular solving

fmpq_mat_solve_fraction_free :: Ptr CFmpqMat -> Ptr CFmpqMat -> Ptr CFmpqMat -> IO CInt

fmpq_mat_solve_fraction_free X A B

fmpq_mat_solve_dixon :: Ptr CFmpqMat -> Ptr CFmpqMat -> Ptr CFmpqMat -> IO CInt

fmpq_mat_solve_dixon X A B

fmpq_mat_solve_multi_mod :: Ptr CFmpqMat -> Ptr CFmpqMat -> Ptr CFmpqMat -> IO CInt

fmpq_mat_solve_multi_mod X A B

fmpq_mat_solve :: Ptr CFmpqMat -> Ptr CFmpqMat -> Ptr CFmpqMat -> IO CInt

fmpq_mat_solve X A B

Solves AX = B for nonsingular A. Returns nonzero if A is nonsingular or if the right hand side is empty, and zero otherwise.

All algorithms clear denominators to obtain a rescaled system over the integers. The fraction_free algorithm uses FFLU solving over the integers. The dixon and multi_mod algorithms use Dixon p-adic lifting or multimodular solving, followed by rational reconstruction with an adaptive stopping test. The dixon and multi_mod algorithms are generally the best choice for large systems.

The default method chooses an algorithm automatically.

fmpq_mat_solve_fmpz_mat_fraction_free :: Ptr CFmpqMat -> Ptr CFmpzMat -> Ptr CFmpzMat -> IO CInt

fmpq_mat_solve_fmpz_mat_fraction_free X A B

fmpq_mat_solve_fmpz_mat_dixon :: Ptr CFmpqMat -> Ptr CFmpzMat -> Ptr CFmpzMat -> IO CInt

fmpq_mat_solve_fmpz_mat_dixon X A B

fmpq_mat_solve_fmpz_mat_multi_mod :: Ptr CFmpqMat -> Ptr CFmpzMat -> Ptr CFmpzMat -> IO CInt

fmpq_mat_solve_fmpz_mat_multi_mod X A B

fmpq_mat_solve_fmpz_mat :: Ptr CFmpqMat -> Ptr CFmpzMat -> Ptr CFmpzMat -> IO CInt

fmpq_mat_solve_fmpz_mat X A B

Solves AX = B for nonsingular A, where A and B are integer matrices. Returns nonzero if A is nonsingular or if the right hand side is empty, and zero otherwise.

fmpq_mat_can_solve_multi_mod :: Ptr CFmpqMat -> Ptr CFmpqMat -> Ptr CFmpqMat -> IO CInt

fmpq_mat_can_solve_multi_mod X A B

Returns \(1\) if AX = B has a solution and if so, sets X to one such solution. The matrices can have any shape but must have the same number of rows.

fmpq_mat_can_solve_fraction_free :: Ptr CFmpqMat -> Ptr CFmpqMat -> Ptr CFmpqMat -> IO CInt

fmpq_mat_can_solve_fraction_free X A B

Returns \(1\) if AX = B has a solution and if so, sets X to one such solution. The matrices can have any shape but must have the same number of rows.

fmpq_mat_can_solve_fmpz_mat_dixon :: Ptr CFmpqMat -> Ptr CFmpzMat -> Ptr CFmpzMat -> IO CInt

fmpq_mat_can_solve_fmpz_mat_dixon X A B

Returns if \(AX = B\) has a solution and if so, sets \(X\) to one such solution. The matrices can have any shape but must have the same number of rows. The input matrices must have integer entries and cannot be an empty matrix.

fmpq_mat_can_solve_dixon :: Ptr CFmpqMat -> Ptr CFmpqMat -> Ptr CFmpqMat -> IO CInt

fmpq_mat_can_solve_dixon X A B

Returns \(1\) if \(AX = B\) has a solution and if so, sets \(X\) to one such solution. The matrices can have any shape but must have the same number of rows.

fmpq_mat_can_solve :: Ptr CFmpqMat -> Ptr CFmpqMat -> Ptr CFmpqMat -> IO CInt

fmpq_mat_can_solve X A B

Returns \(1\) if AX = B has a solution and if so, sets X to one such solution. The matrices can have any shape but must have the same number of rows.

Inverse

fmpq_mat_inv :: Ptr CFmpqMat -> Ptr CFmpqMat -> IO CInt

fmpq_mat_inv B A

Sets B to the inverse matrix of A and returns nonzero. Returns zero if A is singular. A must be a square matrix.

Echelon form

fmpq_mat_pivot :: Ptr CLong -> Ptr CFmpqMat -> CLong -> CLong -> IO CInt

fmpq_mat_pivot perm mat r c

Helper function for row reduction. Returns 1 if the entry of mat at row \(r\) and column \(c\) is nonzero. Otherwise searches for a nonzero entry in the same column among rows \(r+1, r+2, \ldots\). If a nonzero entry is found at row \(s\), swaps rows \(r\) and \(s\) and the corresponding entries in perm (unless NULL) and returns -1. If no nonzero pivot entry is found, leaves the inputs unchanged and returns 0.

fmpq_mat_rref_classical :: Ptr CFmpqMat -> Ptr CFmpqMat -> IO CLong

fmpq_mat_rref_classical B A

Sets B to the reduced row echelon form of A and returns the rank. Performs Gauss-Jordan elimination directly over the rational numbers. This algorithm is usually inefficient and is mainly intended to be used for testing purposes.

fmpq_mat_rref_fraction_free :: Ptr CFmpqMat -> Ptr CFmpqMat -> IO CLong

fmpq_mat_rref_fraction_free B A

Sets B to the reduced row echelon form of A and returns the rank. Clears denominators and performs fraction-free Gauss-Jordan elimination using fmpz_mat functions.

fmpq_mat_rref :: Ptr CFmpqMat -> Ptr CFmpqMat -> IO CLong

fmpq_mat_rref B A

Sets B to the reduced row echelon form of A and returns the rank. This function automatically chooses between the classical and fraction-free algorithms depending on the size of the matrix.

Gram-Schmidt Orthogonalisation

fmpq_mat_gso :: Ptr CFmpqMat -> Ptr CFmpqMat -> IO ()

fmpq_mat_gso B A

Takes a subset of \(\mathbb{Q}^m\) \(S = \{a_1, a_2, \ldots ,a_n\}\) (as the columns of a \(m \times n\) matrix A) and generates an orthogonal set \(S' = \{b_1, b_2, \ldots ,b_n\}\) (as the columns of the \(m \times n\) matrix B) that spans the same subspace of \(\mathbb{Q}^m\) as \(S\).

Transforms

fmpq_mat_similarity :: Ptr CFmpqMat -> CLong -> Ptr CFmpq -> IO ()

fmpq_mat_similarity A r d

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.

Characteristic polynomial

_fmpq_mat_charpoly :: Ptr CFmpz -> Ptr CFmpz -> Ptr CFmpqMat -> IO ()

_fmpq_mat_charpoly coeffs den mat

Set (coeffs, den) to the characteristic polynomial of the given \(n\times n\) matrix.

fmpq_mat_charpoly :: Ptr CFmpqPoly -> Ptr CFmpqMat -> IO ()

fmpq_mat_charpoly pol mat

Set pol to the characteristic polynomial of the given \(n\times n\) matrix. If mat is not square, an exception is raised.

Minimal polynomial

_fmpq_mat_minpoly :: Ptr CFmpz -> Ptr CFmpz -> Ptr CFmpqMat -> IO CLong

_fmpq_mat_minpoly coeffs den mat

Set (coeffs, den) to the minimal polynomial of the given \(n\times n\) matrix and return the length of the polynomial.

fmpq_mat_minpoly :: Ptr CFmpqPoly -> Ptr CFmpqMat -> IO ()

fmpq_mat_minpoly pol mat

Set pol to the minimal polynomial of the given \(n\times n\) matrix. If mat is not square, an exception is raised.