.. _arb-mat:

**arb_mat.h** -- matrices over the real numbers
===============================================================================

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

.. note::

    Methods prefixed with *arb_mat_approx* treat all input entries
    as floating-point numbers (ignoring the radii of the balls) and
    compute floating-point output (balls with zero radius) representing
    approximate solutions *without error bounds*.
    All other methods compute rigorous error bounds.
    The *approx* methods are typically useful for computing initial
    values or preconditioners for rigorous solvers.
    Some users may also find *approx* methods useful
    for doing ordinary numerical linear algebra in applications where
    error bounds are not needed.

Types, macros and constants
-------------------------------------------------------------------------------

.. type:: arb_mat_struct

.. type:: 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.

.. macro:: arb_mat_entry(mat, i, j)

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

.. macro:: arb_mat_nrows(mat)

    Returns the number of rows of the matrix.

.. macro:: arb_mat_ncols(mat)

    Returns the number of columns of the matrix.


Memory management
-------------------------------------------------------------------------------

.. function:: 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.

.. function:: void arb_mat_clear(arb_mat_t mat)

    Clears the matrix, deallocating all entries.

.. function:: slong arb_mat_allocated_bytes(const arb_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(arb_mat_struct)`` to get the size of the object as a whole.

.. function:: void arb_mat_window_init(arb_mat_t window, const arb_mat_t mat, slong r1, slong c1, slong r2, slong c2)

    Initializes *window* to a window matrix into the submatrix of *mat*
    starting at the corner at row *r1* and column *c1* (inclusive) and ending
    at row *r2* and column *c2* (exclusive).

.. function:: void arb_mat_window_clear(arb_mat_t window)

    Frees the window matrix.

Conversions
-------------------------------------------------------------------------------

.. function:: void arb_mat_set(arb_mat_t dest, const arb_mat_t src)

.. function:: void arb_mat_set_fmpz_mat(arb_mat_t dest, const fmpz_mat_t src)

.. function:: void arb_mat_set_round_fmpz_mat(arb_mat_t dest, const fmpz_mat_t src, slong prec)

.. function:: 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
-------------------------------------------------------------------------------

.. function:: 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*.

.. function:: void arb_mat_randtest_cho(arb_mat_t mat, flint_rand_t state, slong prec, slong mag_bits)

    Sets *mat* to a random lower-triangular matrix with precise entries
    and positive diagonal entries. Requires that *mat* is square.

.. function:: void arb_mat_randtest_spd(arb_mat_t mat, flint_rand_t state, slong prec, slong mag_bits)

    Sets *mat* to a random symmetric positive definite matrix, obtained as a
    product `L L^T` where *L* is a random Cholesky matrix. Requires that
    *mat* is square.

Input and output
-------------------------------------------------------------------------------

.. function:: void arb_mat_printd(const arb_mat_t mat, slong digits)

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

.. function:: void arb_mat_fprintd(FILE * file, const arb_mat_t mat, slong digits)

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

Comparisons
-------------------------------------------------------------------------------

Predicate methods return 1 if the property certainly holds and 0 otherwise.

.. function:: int arb_mat_equal(const arb_mat_t mat1, const arb_mat_t mat2)

    Returns whether the matrices have the same dimensions
    and identical intervals as entries.

.. function:: int arb_mat_overlaps(const arb_mat_t mat1, const arb_mat_t mat2)

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

.. function:: int arb_mat_contains(const arb_mat_t mat1, const arb_mat_t mat2)

.. function:: int arb_mat_contains_fmpz_mat(const arb_mat_t mat1, const fmpz_mat_t mat2)

.. function:: int arb_mat_contains_fmpq_mat(const arb_mat_t mat1, const fmpq_mat_t mat2)

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

.. function:: int arb_mat_eq(const arb_mat_t mat1, const arb_mat_t mat2)

    Returns whether *mat1* and *mat2* certainly represent the same matrix.

.. function:: int arb_mat_ne(const arb_mat_t mat1, const arb_mat_t mat2)

    Returns whether *mat1* and *mat2* certainly do not represent the same matrix.

.. function:: int arb_mat_is_empty(const arb_mat_t mat)

    Returns whether the number of rows or the number of columns in *mat* is zero.

.. function:: int arb_mat_is_square(const arb_mat_t mat)

    Returns whether the number of rows is equal to the number of columns in *mat*.

.. function:: int arb_mat_is_exact(const arb_mat_t mat)

    Returns whether all entries in *mat* have zero radius.

.. function:: int arb_mat_is_zero(const arb_mat_t mat)

    Returns whether all entries in *mat* are exactly zero.

.. function:: int arb_mat_is_finite(const arb_mat_t mat)

    Returns whether all entries in *mat* are finite.

.. function:: int arb_mat_is_triu(const arb_mat_t mat)

    Returns whether *mat* is upper triangular; that is, all entries
    below the main diagonal are exactly zero.

.. function:: int arb_mat_is_tril(const arb_mat_t mat)

    Returns whether *mat* is lower triangular; that is, all entries
    above the main diagonal are exactly zero.

.. function:: int arb_mat_is_diag(const arb_mat_t mat)

    Returns whether *mat* is a diagonal matrix; that is, all entries
    off the main diagonal are exactly zero.

Special matrices
-------------------------------------------------------------------------------

.. function:: void arb_mat_zero(arb_mat_t mat)

    Sets all entries in mat to zero.

.. function:: void arb_mat_one(arb_mat_t mat)

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

.. function:: void arb_mat_ones(arb_mat_t mat)

    Sets all entries in the matrix to ones.

.. function:: void arb_mat_indeterminate(arb_mat_t mat)

    Sets all entries in the matrix to indeterminate (NaN).

.. function:: void arb_mat_hilbert(arb_mat_t mat, slong prec)

    Sets *mat* to the Hilbert matrix, which has entries `A_{j,k} = 1/(j+k+1)`.

.. function:: void arb_mat_pascal(arb_mat_t mat, int triangular, slong prec)

    Sets *mat* to a Pascal matrix, whose entries are binomial coefficients.
    If *triangular* is 0, constructs a full symmetric matrix
    with the rows of Pascal's triangle as successive antidiagonals.
    If *triangular* is 1, constructs the upper triangular matrix with
    the rows of Pascal's triangle as columns, and if *triangular* is -1,
    constructs the lower triangular matrix with the rows of Pascal's
    triangle as rows.

    The entries are computed using recurrence relations.
    When the dimensions get large, some precision loss is possible; in that
    case, the user may wish to create the matrix at slightly higher precision
    and then round it to the final precision.

.. function:: void arb_mat_stirling(arb_mat_t mat, int kind, slong prec)

    Sets *mat* to a Stirling matrix, whose entries are Stirling numbers.
    If *kind* is 0, the entries are set to the unsigned Stirling numbers
    of the first kind. If *kind* is 1, the entries are set to the signed
    Stirling numbers of the first kind. If *kind* is 2, the entries are
    set to the Stirling numbers of the second kind.

    The entries are computed using recurrence relations.
    When the dimensions get large, some precision loss is possible; in that
    case, the user may wish to create the matrix at slightly higher precision
    and then round it to the final precision.

.. function:: void arb_mat_dct(arb_mat_t mat, int type, slong prec)

    Sets *mat* to the DCT (discrete cosine transform) matrix of order *n*
    where *n* is the smallest dimension of *mat* (if *mat* is not square,
    the matrix is extended periodically along the larger dimension).
    There are many different conventions for defining DCT matrices; here,
    we use the normalized "DCT-II" transform matrix

    .. math::

        A_{j,k} = \sqrt{\frac{2}{n}} \cos\left(\frac{\pi j}{n} \left(k+\frac{1}{2}\right)\right)

    which satisfies `A^{-1} = A^T`.
    The *type* parameter is currently ignored and should be set to 0.
    In the future, it might be used to select a different convention.

Transpose
-------------------------------------------------------------------------------

.. function:: 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 for square matrices.

Norms
-------------------------------------------------------------------------------

.. function:: 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*.

.. function:: void arb_mat_frobenius_norm(arb_t res, const arb_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*.

.. function:: void arb_mat_bound_frobenius_norm(mag_t res, const arb_mat_t A)

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

Arithmetic
-------------------------------------------------------------------------------

.. function:: 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.

.. function:: 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.

.. function:: 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.

.. function:: void arb_mat_mul_classical(arb_mat_t C, const arb_mat_t A, const arb_mat_t B, slong prec)

.. function:: void arb_mat_mul_threaded(arb_mat_t C, const arb_mat_t A, const arb_mat_t B, slong prec)

.. function:: void arb_mat_mul_block(arb_mat_t C, const arb_mat_t A, const arb_mat_t B, slong prec)

.. function:: 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 *classical* version performs matrix multiplication in the trivial way.

    The *block* version decomposes the input matrices into one or several
    blocks of uniformly scaled matrices and multiplies 
    large blocks via *fmpz_mat_mul*. It also invokes
    :func:`_arb_mat_addmul_rad_mag_fast` for the radius matrix multiplications.

    The *threaded* version performs classical multiplication but splits the
    computation over the number of threads returned by *flint_get_num_threads()*.

    The default version chooses an algorithm automatically.

.. function:: void arb_mat_mul_entrywise(arb_mat_t C, const arb_mat_t A, const arb_mat_t B, slong prec)

    Sets *C* to the entrywise product of *A* and *B*.
    The operands must have the same dimensions.

.. function:: void arb_mat_sqr_classical(arb_mat_t B, const arb_mat_t A, slong prec)

.. function:: 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.

.. function:: 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.

.. function:: void _arb_mat_addmul_rad_mag_fast(arb_mat_t C, mag_srcptr A, mag_srcptr B, slong ar, slong ac, slong bc)

    Helper function for matrix multiplication.
    Adds to the radii of *C* the matrix product of the matrices represented
    by *A* and *B*, where *A* is a linear array of coefficients in row-major
    order and *B* is a linear array of coefficients in column-major order. 
    This function assumes that all exponents are small and is unsafe
    for general use.

.. function:: void arb_mat_approx_mul(arb_mat_t res, const arb_mat_t mat1, const arb_mat_t mat2, slong prec)

    Approximate matrix multiplication. The input radii are ignored and
    the output matrix is set to an approximate floating-point result.
    The radii in the output matrix will *not* necessarily be zeroed.

Scalar arithmetic
-------------------------------------------------------------------------------

.. function:: 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`.

.. function:: void arb_mat_scalar_addmul_si(arb_mat_t B, const arb_mat_t A, slong c, slong prec)

.. function:: void arb_mat_scalar_addmul_fmpz(arb_mat_t B, const arb_mat_t A, const fmpz_t c, slong prec)

.. function:: 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`.

.. function:: void arb_mat_scalar_mul_si(arb_mat_t B, const arb_mat_t A, slong c, slong prec)

.. function:: void arb_mat_scalar_mul_fmpz(arb_mat_t B, const arb_mat_t A, const fmpz_t c, slong prec)

.. function:: 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`.

.. function:: void arb_mat_scalar_div_si(arb_mat_t B, const arb_mat_t A, slong c, slong prec)

.. function:: void arb_mat_scalar_div_fmpz(arb_mat_t B, const arb_mat_t A, const fmpz_t c, slong prec)

.. function:: 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`.

Vector arithmetic
-------------------------------------------------------------------------------

.. function:: void _arb_mat_vector_mul_row(arb_ptr res, arb_srcptr v, const arb_mat_t A, slong prec)

.. function:: void _arb_mat_vector_mul_col(arb_ptr res, const arb_mat_t A, arb_srcptr v, slong prec)

.. function:: void arb_mat_vector_mul_row(arb_ptr res, arb_srcptr v, const arb_mat_t A, slong prec)

.. function:: void arb_mat_vector_mul_col(arb_ptr res, const arb_mat_t A, arb_srcptr v, slong prec)

    Sets *res* to the product `vA`, (resp. `Av`), where *res* and *v* are seen
    as row (resp. column) vectors. The lengths of the vectors must match the
    dimensions of *A*.

    The underscore methods do not allow aliasing between *res* and *v*.

Gaussian elimination and solving
-------------------------------------------------------------------------------

.. function:: int arb_mat_lu_classical(slong * perm, arb_mat_t LU, const arb_mat_t A, slong prec)

.. function:: int arb_mat_lu_recursive(slong * perm, arb_mat_t LU, const arb_mat_t A, slong prec)

.. function:: 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.

    The *classical* version uses Gaussian elimination directly while
    the *recursive* version performs the computation in a block recursive
    way to benefit from fast matrix multiplication. The default version
    chooses an algorithm automatically.

.. function:: void arb_mat_solve_tril_classical(arb_mat_t X, const arb_mat_t L, const arb_mat_t B, int unit, slong prec)

.. function:: void arb_mat_solve_tril_recursive(arb_mat_t X, const arb_mat_t L, const arb_mat_t B, int unit, slong prec)

.. function:: void arb_mat_solve_tril(arb_mat_t X, const arb_mat_t L, const arb_mat_t B, int unit, slong prec)

.. function:: void arb_mat_solve_triu_classical(arb_mat_t X, const arb_mat_t U, const arb_mat_t B, int unit, slong prec)

.. function:: void arb_mat_solve_triu_recursive(arb_mat_t X, const arb_mat_t U, const arb_mat_t B, int unit, slong prec)

.. function:: void arb_mat_solve_triu(arb_mat_t X, const arb_mat_t U, const arb_mat_t B, int unit, slong prec)

    Solves the lower triangular system `LX = B` or the upper triangular system
    `UX = B`, respectively. If *unit* is set, the main diagonal of *L* or *U*
    is taken to consist of all ones, and in that case the actual entries on
    the diagonal are not read at all and can contain other data.

    The *classical* versions perform the computations iteratively while the
    *recursive* versions perform the computations in a block recursive
    way to benefit from fast matrix multiplication. The default versions
    choose an algorithm automatically.

.. function:: 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`.

.. function:: int arb_mat_solve(arb_mat_t X, const arb_mat_t A, const arb_mat_t B, slong prec)

.. function:: int arb_mat_solve_lu(arb_mat_t X, const arb_mat_t A, const arb_mat_t B, slong prec)

.. function:: int arb_mat_solve_precond(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.

    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.

    Three algorithms are provided:

    * The *lu* version performs LU decomposition directly in ball arithmetic.
      This is fast, but the bounds typically blow up exponentially with *n*,
      even if the system is well-conditioned. This algorithm is usually
      the best choice at very high precision.
    * The *precond* version computes an approximate inverse to precondition
      the system [HS1967]_. This is usually several times slower than direct LU
      decomposition, but the bounds do not blow up with *n* if the system is
      well-conditioned. This algorithm is usually
      the best choice for large systems at low to moderate precision.
    * The default version selects between *lu* and *precomp* automatically.

    The automatic choice should be reasonable most of the time, but users
    may benefit from trying either *lu* or *precond* in specific applications.
    For example, the *lu* solver often performs better for ill-conditioned
    systems where use of very high precision is unavoidable.

.. function:: int arb_mat_solve_preapprox(arb_mat_t X, const arb_mat_t A, const arb_mat_t B, const arb_mat_t R, const arb_mat_t T, slong prec)

    Solves `AX = B` where `A` is a nonsingular `n \times n` matrix
    and `X` and `B` are `n \times m` matrices, given an approximation
    `R` of the matrix inverse of `A`, and given the approximation `T`
    of the solution `X`.

    If `m > 0` and `A` cannot be inverted numerically (indicating either that
    `A` is singular or that the precision is insufficient, or that `R` is
    not a close enough approximation of the inverse of `A`), 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.

.. function:: 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.

.. function:: void arb_mat_det_lu(arb_t det, const arb_mat_t A, slong prec)

.. function:: void arb_mat_det_precond(arb_t det, const arb_mat_t A, slong prec)

.. function:: void arb_mat_det(arb_t det, const arb_mat_t A, slong prec)

    Sets *det* to the determinant of the matrix *A*.

    The *lu* version uses 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.

    The *precond* version computes an approximate LU factorization of *A*
    and multiplies by the inverse *L* and *U* martices as preconditioners
    to obtain a matrix close to the identity matrix [Rum2010]_. An enclosure
    for this determinant is computed using Gershgorin circles. This is about
    four times slower than direct Gaussian elimination, but much more
    numerically stable.

    The default version automatically selects between the *lu* and *precond*
    versions and additionally handles small or triangular matrices
    by direct formulas.

.. function:: void arb_mat_approx_solve_triu(arb_mat_t X, const arb_mat_t U, const arb_mat_t B, int unit, slong prec)

.. function:: void arb_mat_approx_solve_tril(arb_mat_t X, const arb_mat_t L, const arb_mat_t B, int unit, slong prec)

.. function:: int arb_mat_approx_lu(slong * P, arb_mat_t LU, const arb_mat_t A, slong prec)

.. function:: void arb_mat_approx_solve_lu_precomp(arb_mat_t X, const slong * perm, const arb_mat_t A, const arb_mat_t B, slong prec)

.. function:: int arb_mat_approx_solve(arb_mat_t X, const arb_mat_t A, const arb_mat_t B, slong prec)

.. function:: int arb_mat_approx_inv(arb_mat_t X, const arb_mat_t A, slong prec)

    These methods perform approximate solving *without any error control*.
    The radii in the input matrices are ignored, the computations are done
    numerically with floating-point arithmetic (using ordinary
    Gaussian elimination and triangular solving, accelerated through
    the use of block recursive strategies for large matrices), and the
    output matrices are set to the approximate floating-point results with
    zeroed error bounds.

    Approximate solutions are useful for computing preconditioning matrices
    for certified solutions. Some users may also find these methods useful
    for doing ordinary numerical linear algebra in applications where
    error bounds are not needed.

Cholesky decomposition and solving
-------------------------------------------------------------------------------

.. function:: int _arb_mat_cholesky_banachiewicz(arb_mat_t A, slong prec)

.. function:: int arb_mat_cho(arb_mat_t L, const arb_mat_t A, slong prec)

    Computes the Cholesky decomposition of *A*, returning nonzero iff
    the symmetric matrix defined by the lower triangular part of *A*
    is certainly positive definite.

    If a nonzero value is returned, then *L* is set to the lower triangular
    matrix such that `A = L * L^T`.

    If zero is returned, then either the matrix is not symmetric positive
    definite, the input matrix was computed to insufficient precision,
    or the decomposition was attempted at insufficient precision.

    The underscore method computes *L* from *A* in-place, leaving the
    strict upper triangular region undefined.

.. function:: void arb_mat_solve_cho_precomp(arb_mat_t X, const arb_mat_t L, const arb_mat_t B, slong prec)

    Solves `AX = B` given the precomputed Cholesky decomposition `A = L L^T`.
    The matrices *X* and *B* are allowed to be aliased with each other,
    but *X* is not allowed to be aliased with *L*.

.. function:: int arb_mat_spd_solve(arb_mat_t X, const arb_mat_t A, const arb_mat_t B, slong prec)

    Solves `AX = B` where *A* is a symmetric positive definite matrix
    and *X* and *B* are `n \times m` matrices, using Cholesky decomposition.

    If `m > 0` and *A* cannot be factored using Cholesky decomposition
    (indicating either that *A* is not symmetric positive definite 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 symmetric matrix defined through the lower
    triangular part of *A* is invertible and that the exact solution matrix
    is contained in the output.

.. function:: void arb_mat_inv_cho_precomp(arb_mat_t X, const arb_mat_t L, slong prec)

    Sets `X = A^{-1}` where `A` is a symmetric positive definite matrix
    whose Cholesky decomposition *L* has been computed with
    :func:`arb_mat_cho`.
    The inverse is calculated using the method of [Kri2013]_ which is more
    efficient than solving `AX = I` with :func:`arb_mat_solve_cho_precomp`.

.. function:: int arb_mat_spd_inv(arb_mat_t X, const arb_mat_t A, slong prec)

    Sets `X = A^{-1}` where *A* is a symmetric positive definite matrix.
    It is calculated using the method of [Kri2013]_ which computes fewer
    intermediate results than solving `AX = I` with :func:`arb_mat_spd_solve`.

    If *A* cannot be factored using Cholesky decomposition
    (indicating either that *A* is not symmetric positive definite 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 symmetric matrix defined through the lower
    triangular part of *A* is invertible and that the exact inverse
    is contained in the output.

.. function:: int _arb_mat_ldl_inplace(arb_mat_t A, slong prec)

.. function:: int _arb_mat_ldl_golub_and_van_loan(arb_mat_t A, slong prec)

.. function:: int arb_mat_ldl(arb_mat_t res, const arb_mat_t A, slong prec)

    Computes the `LDL^T` decomposition of *A*, returning nonzero iff
    the symmetric matrix defined by the lower triangular part of *A*
    is certainly positive definite.

    If a nonzero value is returned, then *res* is set to a lower triangular
    matrix that encodes the `L * D * L^T` decomposition of *A*.
    In particular, `L` is a lower triangular matrix with ones on its diagonal
    and whose strictly lower triangular region is the same as that of *res*.
    `D` is a diagonal matrix with the same diagonal as that of *res*.

    If zero is returned, then either the matrix is not symmetric positive
    definite, the input matrix was computed to insufficient precision,
    or the decomposition was attempted at insufficient precision.

    The underscore methods compute *res* from *A* in-place, leaving the
    strict upper triangular region undefined.
    The default method uses algorithm 4.1.2 from [GVL1996]_.

.. function:: void arb_mat_solve_ldl_precomp(arb_mat_t X, const arb_mat_t L, const arb_mat_t B, slong prec)

    Solves `AX = B` given the precomputed `A = LDL^T` decomposition
    encoded by *L*.  The matrices *X* and *B* are allowed to be aliased
    with each other, but *X* is not allowed to be aliased with *L*.

.. function:: void arb_mat_inv_ldl_precomp(arb_mat_t X, const arb_mat_t L, slong prec)

    Sets `X = A^{-1}` where `A` is a symmetric positive definite matrix
    whose `LDL^T` decomposition encoded by *L* has been computed with
    :func:`arb_mat_ldl`.
    The inverse is calculated using the method of [Kri2013]_ which is more
    efficient than solving `AX = I` with :func:`arb_mat_solve_ldl_precomp`.

Characteristic polynomial and companion matrix
-------------------------------------------------------------------------------

.. function:: void _arb_mat_charpoly(arb_ptr poly, const arb_mat_t mat, slong prec)

.. function:: void arb_mat_charpoly(arb_poly_t poly, const arb_mat_t mat, slong prec)

    Sets *poly* 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.

.. function:: void _arb_mat_companion(arb_mat_t mat, arb_srcptr poly, slong prec)

.. function:: void arb_mat_companion(arb_mat_t mat, const arb_poly_t poly, slong prec)

    Sets the *n* by *n* matrix *mat* to the companion matrix of the polynomial
    *poly* which must have degree *n*.
    The underscore method reads `n + 1` input coefficients.

Special functions
-------------------------------------------------------------------------------

.. function:: void arb_mat_exp_taylor_sum(arb_mat_t S, const arb_mat_t A, slong N, slong prec)

    Sets *S* to the truncated exponential Taylor series `S = \sum_{k=0}^{N-1} A^k / k!`.
    Uses rectangular splitting to compute the sum using `O(\sqrt{N})`
    matrix multiplications. The recurrence relation for factorials
    is used to get scalars that are small integers instead of full
    factorials. As in [Joh2014b]_, all divisions are postponed to
    the end by computing partial factorials of length `O(\sqrt{N})`.
    The scalars could be reduced by doing more divisions, but this
    appears to be slower in most cases.

.. function:: 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

    .. math::

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

    .. math::
        \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 :func:`mag_exp_tail`.
    Truncation error is not added to entries whose values are determined
    by the sparsity structure of `A`.

.. function:: 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.

.. function:: void _arb_mat_diag_prod(arb_t res, const arb_mat_t mat, slong a, slong b, slong prec)

.. function:: void arb_mat_diag_prod(arb_t res, const arb_mat_t mat, slong prec)

    Sets *res* to the product of the entries on the main diagonal of *mat*.
    The underscore method computes the product of the entries between
    index *a* inclusive and *b* exclusive (the indices must be in range).

Sparsity structure
-------------------------------------------------------------------------------

.. function:: slong arb_mat_count_is_zero(const arb_mat_t mat)

    Returns the number of entries of *mat* that are certainly zero
    according to :func:`arb_is_zero`.

.. function:: slong arb_mat_count_not_is_zero(const arb_mat_t mat)

    Returns the number of entries of *mat* that are not certainly zero.

Component and error operations
-------------------------------------------------------------------------------

.. function:: void arb_mat_get_mid(arb_mat_t B, const arb_mat_t A)

    Sets the entries of *B* to the exact midpoints of the entries of *A*.

.. function:: void arb_mat_add_error_mag(arb_mat_t mat, const mag_t err)

    Adds *err* in-place to the radii of the entries of *mat*.

Eigenvalues and eigenvectors
-------------------------------------------------------------------------------

To compute eigenvalues and eigenvectors, one can convert to an
:type:`acb_mat_t` and use the functions in :ref:`acb_mat.h: Eigenvalues and eigenvectors<acb-mat-eigenvalues>`.
In the future dedicated methods for real matrices will be added here.

LLL reduction
-------------------------------------------------------------------------------

.. function:: int arb_mat_spd_get_fmpz_mat(fmpz_mat_t B, const arb_mat_t A, slong prec)

    Attempts to set *B* to a symmetric and positive definite matrix obtained by
    rounding the midpoints of entries of `2^{\mathit{prec}}\cdot A` to
    integers. Returns 1 on success. Returns 0 and leaves *B* undefined if *A*
    is not symmetric or the result of rounding is not a positive definite
    matrix. The warnings of :func:`arf_get_fmpz` apply.

.. function:: void arb_mat_spd_lll_reduce(fmpz_mat_t U, const arb_mat_t A, double delta, double eta, slong prec)

    Given a symmetric positive definite matrix *A*, sets *U* to an invertible
    matrix such that `U^T A U` is close to being LLL-reduced for the given
    parameters `\delta,\eta`. If :func:`arb_mat_spd_get_fmpz_mat` succeeds at
    the chosen precision, we call :func:`fmpz_lll`, and otherwise set *U* to
    the identity matrix. The warnings of :func:`arf_get_fmpz` apply.

.. function:: int arb_mat_spd_is_lll_reduced(const arb_mat_t A, double delta, double eta, slong prec)

    Given a symmetric positive definite matrix *A*, returns nonzero iff *A* is
    certainly LLL-reduced for the given LLL parameters `\eta, \delta`, meaning
    that it satisfies the inequalities `|\mu_{j,k}|\leq \eta` and `\delta
    \lVert b_{k-1}^*\rVert^2 \leq \lVert b_k^*\rVert^2 + \mu_{k,k-1}^2 \lVert
    b_{k-1}^*\rVert^2` (with the usual notation).
