arb_mat.h – matrices over the real numbers

An arb_mat_t represents a dense matrix over the real numbers, implemented as an array of entries of type arb_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

arb_mat_struct
arb_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 arb_mat_t is defined as an array of length one of type arb_mat_struct, permitting an arb_mat_t to be passed by reference.

arb_mat_entry(mat, i, j)

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

arb_mat_nrows(mat)

Returns the number of rows of the matrix.

arb_mat_ncols(mat)

Returns the number of columns of the matrix.

Memory management

void arb_mat_init(arb_mat_t mat, slong r, slong c)

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

void arb_mat_clear(arb_mat_t mat)

Clears the matrix, deallocating all entries.

Conversions

void arb_mat_set(arb_mat_t dest, const arb_mat_t src)
void arb_mat_set_fmpz_mat(arb_mat_t dest, const fmpz_mat_t src)
void arb_mat_set_round_fmpz_mat(arb_mat_t dest, const fmpz_mat_t src, slong prec)
void arb_mat_set_fmpq_mat(arb_mat_t dest, const fmpq_mat_t src, slong prec)

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

Random generation

void arb_mat_randtest(arb_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 arb_mat_printd(const arb_mat_t mat, slong digits)

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

Comparisons

int arb_mat_equal(const arb_mat_t mat1, const arb_mat_t mat2)

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

int arb_mat_overlaps(const arb_mat_t mat1, const arb_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 arb_mat_contains(const arb_mat_t mat1, const arb_mat_t mat2)
int arb_mat_contains_fmpz_mat(const arb_mat_t mat1, const fmpz_mat_t mat2)
int arb_mat_contains_fmpq_mat(const arb_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 arb_mat_eq(const arb_mat_t mat1, const arb_mat_t mat2)

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

int arb_mat_ne(const arb_mat_t mat1, const arb_mat_t mat2)

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

Special matrices

void arb_mat_zero(arb_mat_t mat)

Sets all entries in mat to zero.

void arb_mat_one(arb_mat_t mat)

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

Transpose

void arb_mat_transpose(arb_mat_t dest, const arb_mat_t src)

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

Norms

void arb_mat_bound_inf_norm(mag_t b, const arb_mat_t A)

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

Arithmetic

void arb_mat_neg(arb_mat_t dest, const arb_mat_t src)

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

void arb_mat_add(arb_mat_t res, const arb_mat_t mat1, const arb_mat_t mat2, slong prec)

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

void arb_mat_sub(arb_mat_t res, const arb_mat_t mat1, const arb_mat_t mat2, slong prec)

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

void arb_mat_mul_classical(arb_mat_t C, const arb_mat_t A, const arb_mat_t B, slong prec)
void arb_mat_mul_threaded(arb_mat_t C, const arb_mat_t A, const arb_mat_t B, slong prec)
void arb_mat_mul(arb_mat_t res, const arb_mat_t mat1, const arb_mat_t mat2, slong prec)

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

The threaded version splits the computation over the number of threads returned by flint_get_num_threads(). The default version automatically calls the threaded version if the matrices are sufficiently large and more than one thread can be used.

void arb_mat_sqr_classical(arb_mat_t B, const arb_mat_t A, slong prec)
void arb_mat_sqr(arb_mat_t res, const arb_mat_t mat, slong prec)

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

void arb_mat_pow_ui(arb_mat_t res, const arb_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 arb_mat_scalar_mul_2exp_si(arb_mat_t B, const arb_mat_t A, slong c)

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

void arb_mat_scalar_addmul_si(arb_mat_t B, const arb_mat_t A, slong c, slong prec)
void arb_mat_scalar_addmul_fmpz(arb_mat_t B, const arb_mat_t A, const fmpz_t c, slong prec)
void arb_mat_scalar_addmul_arb(arb_mat_t B, const arb_mat_t A, const arb_t c, slong prec)

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

void arb_mat_scalar_mul_si(arb_mat_t B, const arb_mat_t A, slong c, slong prec)
void arb_mat_scalar_mul_fmpz(arb_mat_t B, const arb_mat_t A, const fmpz_t c, slong prec)
void arb_mat_scalar_mul_arb(arb_mat_t B, const arb_mat_t A, const arb_t c, slong prec)

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

void arb_mat_scalar_div_si(arb_mat_t B, const arb_mat_t A, slong c, slong prec)
void arb_mat_scalar_div_fmpz(arb_mat_t B, const arb_mat_t A, const fmpz_t c, slong prec)
void arb_mat_scalar_div_arb(arb_mat_t B, const arb_mat_t A, const arb_t c, slong prec)

Sets B to \(A / c\).

Gaussian elimination and solving

int arb_mat_lu(slong * perm, arb_mat_t LU, const arb_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 arb_mat_solve_lu_precomp(arb_mat_t X, const slong * perm, const arb_mat_t LU, const arb_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 arb_mat_solve(arb_mat_t X, const arb_mat_t A, const arb_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 arb_mat_inv(arb_mat_t X, const arb_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 arb_mat_det(arb_t det, const arb_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 _arb_mat_charpoly(arb_ptr cp, const arb_mat_t mat, slong prec)
void arb_mat_charpoly(arb_poly_t cp, const arb_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 arb_mat_exp(arb_mat_t B, const arb_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. The series is evaluated using rectangular splitting.

The elementwise error when truncating the Taylor series after N terms is bounded by the error in the infinity norm, for which we have

\[\left\|\exp(2^{-r}A) - \sum_{k=0}^{N-1} \frac{\left(2^{-r} A\right)^k}{k!} \right\|_{\infty} = \left\|\sum_{k=N}^{\infty} \frac{\left(2^{-r} A\right)^k}{k!}\right\|_{\infty} \le \sum_{k=N}^{\infty} \frac{(2^{-r} \|A\|_{\infty})^k}{k!}.\]

We bound the sum on the right using mag_exp_tail().

void arb_mat_trace(arb_t trace, const arb_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.