acb_mat.h – matrices over the complex numbers

An acb_mat_t represents a dense matrix over the complex numbers, implemented as an array of entries of type acb_struct.

The dimension (number of rows and columns) of a matrix is fixed at initialization, and the user must ensure that inputs and outputs to an operation have compatible dimensions. The number of rows or columns in a matrix can be zero.

Types, macros and constants

acb_mat_struct
acb_mat_t

Contains a pointer to a flat array of the entries (entries), an array of pointers to the start of each row (rows), and the number of rows (r) and columns (c).

An acb_mat_t is defined as an array of length one of type acb_mat_struct, permitting an acb_mat_t to be passed by reference.

acb_mat_entry(mat, i, j)

Macro giving a pointer to the entry at row i and column j.

acb_mat_nrows(mat)

Returns the number of rows of the matrix.

acb_mat_ncols(mat)

Returns the number of columns of the matrix.

Memory management

void acb_mat_init(acb_mat_t mat, slong r, slong c)

Initializes the matrix, setting it to the zero matrix with r rows and c columns.

void acb_mat_clear(acb_mat_t mat)

Clears the matrix, deallocating all entries.

slong acb_mat_allocated_bytes(const acb_mat_t x)

Returns the total number of bytes heap-allocated internally by this object. The count excludes the size of the structure itself. Add sizeof(acb_mat_struct) to get the size of the object as a whole.

Conversions

void acb_mat_set(acb_mat_t dest, const acb_mat_t src)
void acb_mat_set_fmpz_mat(acb_mat_t dest, const fmpz_mat_t src)
void acb_mat_set_round_fmpz_mat(acb_mat_t dest, const fmpz_mat_t src, slong prec)
void acb_mat_set_fmpq_mat(acb_mat_t dest, const fmpq_mat_t src, slong prec)
void acb_mat_set_arb_mat(acb_mat_t dest, const arb_mat_t src)
void acb_mat_set_round_arb_mat(acb_mat_t dest, const arb_mat_t src, slong prec)

Sets dest to src. The operands must have identical dimensions.

Random generation

void acb_mat_randtest(acb_mat_t mat, flint_rand_t state, slong prec, slong mag_bits)

Sets mat to a random matrix with up to prec bits of precision and with exponents of width up to mag_bits.

Input and output

void acb_mat_printd(const acb_mat_t mat, slong digits)

Prints each entry in the matrix with the specified number of decimal digits.

void acb_mat_fprintd(FILE * file, const acb_mat_t mat, slong digits)

Prints each entry in the matrix with the specified number of decimal digits to the stream file.

Comparisons

int acb_mat_equal(const acb_mat_t mat1, const acb_mat_t mat2)

Returns nonzero iff the matrices have the same dimensions and identical entries.

int acb_mat_overlaps(const acb_mat_t mat1, const acb_mat_t mat2)

Returns nonzero iff the matrices have the same dimensions and each entry in mat1 overlaps with the corresponding entry in mat2.

int acb_mat_contains(const acb_mat_t mat1, const acb_mat_t mat2)
int acb_mat_contains_fmpz_mat(const acb_mat_t mat1, const fmpz_mat_t mat2)
int acb_mat_contains_fmpq_mat(const acb_mat_t mat1, const fmpq_mat_t mat2)

Returns nonzero iff the matrices have the same dimensions and each entry in mat2 is contained in the corresponding entry in mat1.

int acb_mat_eq(const acb_mat_t mat1, const acb_mat_t mat2)

Returns nonzero iff mat1 and mat2 certainly represent the same matrix.

int acb_mat_ne(const acb_mat_t mat1, const acb_mat_t mat2)

Returns nonzero iff mat1 and mat2 certainly do not represent the same matrix.

int acb_mat_is_real(const acb_mat_t mat)

Returns nonzero iff all entries in mat have zero imaginary part.

int acb_mat_is_empty(const acb_mat_t mat)

Returns nonzero iff the number of rows or the number of columns in mat is zero.

int acb_mat_is_square(const acb_mat_t mat)

Returns nonzero iff the number of rows is equal to the number of columns in mat.

Special matrices

void acb_mat_zero(acb_mat_t mat)

Sets all entries in mat to zero.

void acb_mat_one(acb_mat_t mat)

Sets the entries on the main diagonal to ones, and all other entries to zero.

Transpose

void acb_mat_transpose(acb_mat_t dest, const acb_mat_t src)

Sets dest to the exact transpose src. The operands must have compatible dimensions. Aliasing is allowed.

Norms

void acb_mat_bound_inf_norm(mag_t b, const acb_mat_t A)

Sets b to an upper bound for the infinity norm (i.e. the largest absolute value row sum) of A.

void acb_mat_frobenius_norm(acb_t res, const acb_mat_t A, slong prec)

Sets res to the Frobenius norm (i.e. the square root of the sum of squares of entries) of A.

void acb_mat_bound_frobenius_norm(mag_t res, const acb_mat_t A)

Sets res to an upper bound for the Frobenius norm of A.

Arithmetic

void acb_mat_neg(acb_mat_t dest, const acb_mat_t src)

Sets dest to the exact negation of src. The operands must have the same dimensions.

void acb_mat_add(acb_mat_t res, const acb_mat_t mat1, const acb_mat_t mat2, slong prec)

Sets res to the sum of mat1 and mat2. The operands must have the same dimensions.

void acb_mat_sub(acb_mat_t res, const acb_mat_t mat1, const acb_mat_t mat2, slong prec)

Sets res to the difference of mat1 and mat2. The operands must have the same dimensions.

void acb_mat_mul(acb_mat_t res, const acb_mat_t mat1, const acb_mat_t mat2, slong prec)

Sets res to the matrix product of mat1 and mat2. The operands must have compatible dimensions for matrix multiplication.

void acb_mat_mul_entrywise(acb_mat_t res, const acb_mat_t mat1, const acb_mat_t mat2, slong prec)

Sets res to the entrywise product of mat1 and mat2. The operands must have the same dimensions.

void acb_mat_sqr(acb_mat_t res, const acb_mat_t mat, slong prec)

Sets res to the matrix square of mat. The operands must both be square with the same dimensions.

void acb_mat_pow_ui(acb_mat_t res, const acb_mat_t mat, ulong exp, slong prec)

Sets res to mat raised to the power exp. Requires that mat is a square matrix.

Scalar arithmetic

void acb_mat_scalar_mul_2exp_si(acb_mat_t B, const acb_mat_t A, slong c)

Sets B to A multiplied by \(2^c\).

void acb_mat_scalar_addmul_si(acb_mat_t B, const acb_mat_t A, slong c, slong prec)
void acb_mat_scalar_addmul_fmpz(acb_mat_t B, const acb_mat_t A, const fmpz_t c, slong prec)
void acb_mat_scalar_addmul_arb(acb_mat_t B, const acb_mat_t A, const arb_t c, slong prec)
void acb_mat_scalar_addmul_acb(acb_mat_t B, const acb_mat_t A, const acb_t c, slong prec)

Sets B to \(B + A \times c\).

void acb_mat_scalar_mul_si(acb_mat_t B, const acb_mat_t A, slong c, slong prec)
void acb_mat_scalar_mul_fmpz(acb_mat_t B, const acb_mat_t A, const fmpz_t c, slong prec)
void acb_mat_scalar_mul_arb(acb_mat_t B, const acb_mat_t A, const arb_t c, slong prec)
void acb_mat_scalar_mul_acb(acb_mat_t B, const acb_mat_t A, const acb_t c, slong prec)

Sets B to \(A \times c\).

void acb_mat_scalar_div_si(acb_mat_t B, const acb_mat_t A, slong c, slong prec)
void acb_mat_scalar_div_fmpz(acb_mat_t B, const acb_mat_t A, const fmpz_t c, slong prec)
void acb_mat_scalar_div_arb(acb_mat_t B, const acb_mat_t A, const arb_t c, slong prec)
void acb_mat_scalar_div_acb(acb_mat_t B, const acb_mat_t A, const acb_t c, slong prec)

Sets B to \(A / c\).

Gaussian elimination and solving

int acb_mat_lu(slong * perm, acb_mat_t LU, const acb_mat_t A, slong prec)

Given an \(n \times n\) matrix \(A\), computes an LU decomposition \(PLU = A\) using Gaussian elimination with partial pivoting. The input and output matrices can be the same, performing the decomposition in-place.

Entry \(i\) in the permutation vector perm is set to the row index in the input matrix corresponding to row \(i\) in the output matrix.

The algorithm succeeds and returns nonzero if it can find \(n\) invertible (i.e. not containing zero) pivot entries. This guarantees that the matrix is invertible.

The algorithm fails and returns zero, leaving the entries in \(P\) and \(LU\) undefined, if it cannot find \(n\) invertible pivot elements. In this case, either the matrix is singular, the input matrix was computed to insufficient precision, or the LU decomposition was attempted at insufficient precision.

void acb_mat_solve_lu_precomp(acb_mat_t X, const slong * perm, const acb_mat_t LU, const acb_mat_t B, slong prec)

Solves \(AX = B\) given the precomputed nonsingular LU decomposition \(A = PLU\). The matrices \(X\) and \(B\) are allowed to be aliased with each other, but \(X\) is not allowed to be aliased with \(LU\).

int acb_mat_solve(acb_mat_t X, const acb_mat_t A, const acb_mat_t B, slong prec)

Solves \(AX = B\) where \(A\) is a nonsingular \(n \times n\) matrix and \(X\) and \(B\) are \(n \times m\) matrices, using LU decomposition.

If \(m > 0\) and \(A\) cannot be inverted numerically (indicating either that \(A\) is singular or that the precision is insufficient), the values in the output matrix are left undefined and zero is returned. A nonzero return value guarantees that \(A\) is invertible and that the exact solution matrix is contained in the output.

int acb_mat_inv(acb_mat_t X, const acb_mat_t A, slong prec)

Sets \(X = A^{-1}\) where \(A\) is a square matrix, computed by solving the system \(AX = I\).

If \(A\) cannot be inverted numerically (indicating either that \(A\) is singular or that the precision is insufficient), the values in the output matrix are left undefined and zero is returned. A nonzero return value guarantees that the matrix is invertible and that the exact inverse is contained in the output.

void acb_mat_det(acb_t det, const acb_mat_t A, slong prec)

Computes the determinant of the matrix, using Gaussian elimination with partial pivoting. If at some point an invertible pivot element cannot be found, the elimination is stopped and the magnitude of the determinant of the remaining submatrix is bounded using Hadamard’s inequality.

Characteristic polynomial

void _acb_mat_charpoly(acb_ptr cp, const acb_mat_t mat, slong prec)
void acb_mat_charpoly(acb_poly_t cp, const acb_mat_t mat, slong prec)

Sets cp to the characteristic polynomial of mat which must be a square matrix. If the matrix has n rows, the underscore method requires space for \(n + 1\) output coefficients. Employs a division-free algorithm using \(O(n^4)\) operations.

Special functions

void acb_mat_exp_taylor_sum(acb_mat_t S, const acb_mat_t A, slong N, slong prec)

Sets S to the truncated exponential Taylor series \(S = \sum_{k=0}^{N-1} A^k / k!\). See arb_mat_exp_taylor_sum() for implementation notes.

void acb_mat_exp(acb_mat_t B, const acb_mat_t A, slong prec)

Sets B to the exponential of the matrix A, defined by the Taylor series

\[\exp(A) = \sum_{k=0}^{\infty} \frac{A^k}{k!}.\]

The function is evaluated as \(\exp(A/2^r)^{2^r}\), where \(r\) is chosen to give rapid convergence of the Taylor series. Error bounds are computed as for arb_mat_exp().

void acb_mat_trace(acb_t trace, const acb_mat_t mat, slong prec)

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