.. _gr-mat:

**gr_mat.h** -- dense matrices over generic rings
===============================================================================

A :type:`gr_mat_t` represents a matrix implemented as a dense
array of entries in a generic ring *R*.

* In this module, the context object ``ctx`` always represents the
  coefficient ring *R* unless otherwise stated.
  Creating a context object representing a matrix
  space only becomes necessary when one
  wants to manipulate matrices using generic ring methods
  like ``gr_add`` instead of the designated matrix
  methods like ``gr_mat_add``.
* Matrix functions generally assume that input as well
  as output operands have compatible shapes.
  Some functions return ``GR_DOMAIN`` for matrices with the
  wrong shape, but this is not always consistent.
* Some operations (like rank, LU factorization) generally only make
  sense when the base ring is an integral domain.
  Typically the algorithms designed for integral domains also work
  over non-integral domains as long as all inversions of nonzero
  elements succeed. If an inversion fails, the algorithm will return
  the ``GR_DOMAIN`` or ``GR_UNABLE`` flag.
  This might not yet be entirely consistent.

Type compatibility
-------------------------------------------------------------------------------

The ``gr_mat`` type has the same data layout as most
FLINT, Arb and Calcium matrix types.
Methods in this module can therefore be mixed freely with
methods in the corresponding FLINT, Arb and Calcium modules
when the underlying coefficient type is the same.

It is not directly compatible with the ``nmod_mat`` type,
which stores modulus data as part of the matrix object.

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

.. type:: gr_mat_struct

.. type:: gr_mat_t

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

    A ``gr_mat_t`` is defined as an array of length one of type
    ``gr_mat_struct``, permitting a ``gr_mat_t`` to
    be passed by reference.

.. macro:: GR_MAT_ENTRY(mat, i, j, sz)

    Macro to access the entry at row *i* and column *j* of the
    matrix *mat* whose entries have size *sz* bytes.

.. function:: gr_ptr gr_mat_entry_ptr(gr_mat_t mat, slong i, slong j, gr_ctx_t ctx)

    Function returning a pointer to the entry at row *i* and column
    *j* of the matrix *mat*. The indices must be in bounds.

.. macro:: gr_mat_nrows(mat, ctx)

    Macro accessing the number of rows of *mat*.

.. macro:: gr_mat_ncols(mat, ctx)

    Macro accessing the number of columns of *mat*.

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

.. function:: void gr_mat_init(gr_mat_t mat, slong rows, slong cols, gr_ctx_t ctx)

    Initializes *mat* to a matrix with the given number of rows and
    columns.

.. function:: int gr_mat_init_set(gr_mat_t res, const gr_mat_t mat, gr_ctx_t ctx)

    Initializes *res* to a copy of the matrix *mat*.

.. function:: void gr_mat_clear(gr_mat_t mat, gr_ctx_t ctx)

    Clears the matrix.

.. function:: void gr_mat_swap(gr_mat_t mat1, gr_mat_t mat2, gr_ctx_t ctx)

    Swaps *mat1* and *mat12* efficiently.

.. function:: int gr_mat_swap_entrywise(gr_mat_t mat1, const gr_mat_t mat2, gr_ctx_t ctx)

    Performs a deep swap of *mat1* and *mat2*, swapping the individual
    entries rather than the top-level structures.

Window matrices
-------------------------------------------------------------------------------

.. function:: void gr_mat_window_init(gr_mat_t window, const gr_mat_t mat, slong r1, slong c1, slong r2, slong c2, gr_ctx_t ctx)

    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).
    The indices must be within bounds.

.. function:: void gr_mat_window_clear(gr_mat_t window, gr_ctx_t ctx)

    Frees the window matrix.

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

.. function:: int gr_mat_write(gr_stream_t out, const gr_mat_t mat, gr_ctx_t ctx)

    Write *mat* to the stream *out*.

.. function:: int gr_mat_print(const gr_mat_t mat, gr_ctx_t ctx)

    Prints *mat* to standard output.

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

.. function:: truth_t gr_mat_equal(const gr_mat_t mat1, const gr_mat_t mat2, gr_ctx_t ctx)

    Returns whether *mat1* and *mat2* are equal.

Assignment and special values
-------------------------------------------------------------------------------

.. function:: truth_t gr_mat_is_zero(const gr_mat_t mat, gr_ctx_t ctx)
              truth_t gr_mat_is_one(const gr_mat_t mat, gr_ctx_t ctx)
              truth_t gr_mat_is_neg_one(const gr_mat_t mat, gr_ctx_t ctx)

    Returns whether *mat* respectively is the zero matrix or
    the scalar matrix with 1 or -1 on the main diagonal.

.. function:: truth_t gr_mat_is_scalar(const gr_mat_t mat, gr_ctx_t ctx)

    Returns whether *mat* is a scalar matrix, being a diagonal matrix
    with identical elements on the main diagonal.

.. function:: int gr_mat_zero(gr_mat_t res, gr_ctx_t ctx)

    Sets *res* to the zero matrix.

.. function:: int gr_mat_one(gr_mat_t res, gr_ctx_t ctx)

    Sets *res* to the scalar matrix with 1 on the main diagonal
    and zero elsewhere.

.. function:: int gr_mat_set(gr_mat_t res, const gr_mat_t mat, gr_ctx_t ctx)
              int gr_mat_set_fmpz_mat(gr_mat_t res, const fmpz_mat_t mat, gr_ctx_t ctx)
              int gr_mat_set_fmpq_mat(gr_mat_t res, const fmpq_mat_t mat, gr_ctx_t ctx)
              int gr_mat_set_gr_mat_other(gr_mat_t res, const gr_mat_t mat, gr_ctx_t mat_ctx, gr_ctx_t ctx)

    Sets *res* to the value of *mat*.

.. function:: int gr_mat_set_scalar(gr_mat_t res, gr_srcptr c, gr_ctx_t ctx)
              int gr_mat_set_ui(gr_mat_t res, ulong c, gr_ctx_t ctx)
              int gr_mat_set_si(gr_mat_t res, slong c, gr_ctx_t ctx)
              int gr_mat_set_fmpz(gr_mat_t res, const fmpz_t c, gr_ctx_t ctx)
              int gr_mat_set_fmpq(gr_mat_t res, const fmpq_t c, gr_ctx_t ctx)

    Set *res* to the scalar matrix with *c* on the main diagonal
    and zero elsewhere.

Basic row, column and entry operations
-------------------------------------------------------------------------------

.. function:: int gr_mat_concat_horizontal(gr_mat_t res, const gr_mat_t mat1, const gr_mat_t mat2, gr_ctx_t ctx)

.. function:: int gr_mat_concat_vertical(gr_mat_t res, const gr_mat_t mat1, const gr_mat_t mat2, gr_ctx_t ctx)

.. function:: int gr_mat_transpose(gr_mat_t B, const gr_mat_t A, gr_ctx_t ctx)

    Sets *B* to the transpose of *A*.

.. function:: int gr_mat_swap_rows(gr_mat_t mat, slong * perm, slong r, slong s, gr_ctx_t ctx)

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

.. function:: int gr_mat_swap_cols(gr_mat_t mat, slong * perm, slong r, slong s, gr_ctx_t ctx)

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

.. function:: int gr_mat_invert_rows(gr_mat_t mat, slong * perm, gr_ctx_t ctx)

    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``.

.. function:: int gr_mat_invert_cols(gr_mat_t mat, slong * perm, gr_ctx_t ctx)

    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``.

.. function:: truth_t gr_mat_is_empty(const gr_mat_t mat, gr_ctx_t ctx)

    Returns whether *mat* is an empty matrix, having either zero
    rows or zero column. This predicate is always decidable (even if
    the underlying ring is not computable), returning
    ``T_TRUE`` or ``T_FALSE``.

.. function:: truth_t gr_mat_is_square(const gr_mat_t mat, gr_ctx_t ctx)

    Returns whether *mat* is a square matrix, having the same number
    of rows as columns (not the same thing as being a perfect square!).
    This predicate is always decidable (even if the underlying ring
    is not computable), returning ``T_TRUE`` or ``T_FALSE``.

Entrywise operations
-------------------------------------------------------------------------------

.. function:: int gr_mat_entrywise_unary_op(gr_mat_t res, gr_method_unary_op f, const gr_mat_t mat, gr_ctx_t ctx)

    Sets *res* to the application of the function *f* to the
    entries of matrix *mat*. Returns ``GR_DOMAIN`` if the matrix dimensions do not match.

.. function:: int gr_mat_entrywise_binary_op(gr_mat_t res, gr_method_binary_op f, const gr_mat_t mat1, const gr_mat_t mat2, gr_ctx_t ctx)

    Sets *res* to the application of the function *f*
    to the entries of *mat1* as first argument and the entries of *mat2*
    as second argument.
    Returns ``GR_DOMAIN`` if the matrix dimensions do not match.

.. function:: int gr_mat_entrywise_binary_op_scalar(gr_mat_t res, gr_method_binary_op f, const gr_mat_t mat, gr_srcptr c, gr_ctx_t ctx)

    Sets *res* to the application of the function *f*
    to the entries of *mat* as first argument and the scalar *c*
    as second argument.
    Returns ``GR_DOMAIN`` if the matrix dimensions do not match.

.. function:: truth_t gr_mat_entrywise_unary_predicate_all(gr_method_unary_predicate f, const gr_mat_t mat, gr_ctx_t ctx)
              truth_t gr_mat_entrywise_unary_predicate_any(gr_method_unary_predicate f, const gr_mat_t mat, gr_ctx_t ctx)

    Returns whether the predicate *f* is true for all entries,
    respectively for any entry, in the matrix *mat*.

.. function:: truth_t gr_mat_entrywise_binary_predicate_all(gr_method_binary_predicate f, const gr_mat_t mat1, const gr_mat_t mat2, gr_ctx_t ctx)

    Returns whether the binary predicate *f* is true for all entries
    in *mat1* paired with the corresponding entries in *mat2*.
    Returns ``T_FALSE`` if the matrix dimensions are not compatible.

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

.. function:: int gr_mat_norm_max(gr_ptr res, const gr_mat_t mat, gr_ctx_t ctx)

    Max norm: `\max_{i,j} |a_{i,j}|`.

.. function:: int gr_mat_norm_1(gr_ptr res, const gr_mat_t mat, gr_ctx_t ctx)

    1-norm (largest absolute column sum): `\max_{1\le j \le n} \sum_{i=1}^m |a_{i,j}|`.

.. function:: int gr_mat_norm_inf(gr_ptr res, const gr_mat_t mat, gr_ctx_t ctx)

    Infinity-norm (largest absolute row sum): `\max_{1\le i \le m} \sum_{j=1}^n |a_{i,j}|`.

.. function:: int gr_mat_norm_frobenius(gr_ptr res, const gr_mat_t mat, gr_ctx_t ctx)

    Frobenius norm: `\sqrt{\sum_{i,j} |a_{i,j}|^2}`.

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

.. function:: int gr_mat_neg(gr_mat_t res, const gr_mat_t mat, gr_ctx_t ctx)

.. function:: int gr_mat_add(gr_mat_t res, const gr_mat_t mat1, const gr_mat_t mat2, gr_ctx_t ctx)

.. function:: int gr_mat_sub(gr_mat_t res, const gr_mat_t mat1, const gr_mat_t mat2, gr_ctx_t ctx)

.. function:: int gr_mat_mul_classical(gr_mat_t res, const gr_mat_t mat1, const gr_mat_t mat2, gr_ctx_t ctx)
              int gr_mat_mul_strassen(gr_mat_t C, const gr_mat_t A, const gr_mat_t B, gr_ctx_t ctx)
              int gr_mat_mul_waksman(gr_mat_t C, const gr_mat_t A, const gr_mat_t B, gr_ctx_t ctx)
              int gr_mat_mul_generic(gr_mat_t C, const gr_mat_t A, const gr_mat_t B, gr_ctx_t ctx)
              int gr_mat_mul(gr_mat_t res, const gr_mat_t mat1, const gr_mat_t mat2, gr_ctx_t ctx)

    Matrix multiplication. The default function can be overloaded by specific rings;
    otherwise, it falls back to :func:`gr_mat_mul_generic` which currently
    only performs classical multiplication.

    The *Waksman* algorithm assumes a commutative base ring which supports
    exact division by two.

.. function:: int gr_mat_sqr(gr_mat_t res, const gr_mat_t mat, gr_ctx_t ctx)

.. function:: int gr_mat_add_scalar(gr_mat_t res, const gr_mat_t mat, gr_srcptr x, gr_ctx_t ctx)
              int gr_mat_scalar_add(gr_mat_t res, gr_srcptr x, const gr_mat_t mat, gr_ctx_t ctx)
              int gr_mat_add_ui(gr_mat_t res, const gr_mat_t mat, ulong x, gr_ctx_t ctx)
              int gr_mat_add_si(gr_mat_t res, const gr_mat_t mat, slong x, gr_ctx_t ctx)
              int gr_mat_add_fmpz(gr_mat_t res, const gr_mat_t mat, const fmpz_t x, gr_ctx_t ctx)
              int gr_mat_add_fmpq(gr_mat_t res, const gr_mat_t mat, const fmpq_t x, gr_ctx_t ctx)
              int gr_mat_add_scalar_other(gr_mat_t res, const gr_mat_t mat, gr_srcptr x, gr_ctx_t x_ctx, gr_ctx_t ctx)
              int gr_mat_scalar_other_add(gr_mat_t res, gr_srcptr x, gr_ctx_t x_ctx, const gr_mat_t mat, gr_ctx_t ctx)

    Perform the matrix-scalar or scalar-matrix operation `A + Ix` or `Ix + A`.

.. function:: int gr_mat_sub_scalar(gr_mat_t res, const gr_mat_t mat, gr_srcptr x, gr_ctx_t ctx)
              int gr_mat_scalar_sub(gr_mat_t res, gr_srcptr x, const gr_mat_t mat, gr_ctx_t ctx)
              int gr_mat_sub_ui(gr_mat_t res, const gr_mat_t mat, ulong x, gr_ctx_t ctx)
              int gr_mat_sub_si(gr_mat_t res, const gr_mat_t mat, slong x, gr_ctx_t ctx)
              int gr_mat_sub_fmpz(gr_mat_t res, const gr_mat_t mat, const fmpz_t x, gr_ctx_t ctx)
              int gr_mat_sub_fmpq(gr_mat_t res, const gr_mat_t mat, const fmpq_t x, gr_ctx_t ctx)
              int gr_mat_sub_scalar_other(gr_mat_t res, const gr_mat_t mat, gr_srcptr x, gr_ctx_t x_ctx, gr_ctx_t ctx)
              int gr_mat_scalar_other_sub(gr_mat_t res, gr_srcptr x, gr_ctx_t x_ctx, const gr_mat_t mat, gr_ctx_t ctx)

    Perform the matrix-scalar or scalar-matrix operation `A - Ix` or `Ix - A`.

.. function:: int gr_mat_mul_scalar(gr_mat_t res, const gr_mat_t mat, gr_srcptr x, gr_ctx_t ctx)
              int gr_mat_scalar_mul(gr_mat_t res, gr_srcptr x, const gr_mat_t mat, gr_ctx_t ctx)
              int gr_mat_mul_ui(gr_mat_t res, const gr_mat_t mat, ulong x, gr_ctx_t ctx)
              int gr_mat_mul_si(gr_mat_t res, const gr_mat_t mat, slong x, gr_ctx_t ctx)
              int gr_mat_mul_fmpz(gr_mat_t res, const gr_mat_t mat, const fmpz_t x, gr_ctx_t ctx)
              int gr_mat_mul_fmpq(gr_mat_t res, const gr_mat_t mat, const fmpq_t x, gr_ctx_t ctx)
              int gr_mat_mul_scalar_other(gr_mat_t res, const gr_mat_t mat, gr_srcptr x, gr_ctx_t x_ctx, gr_ctx_t ctx)
              int gr_mat_scalar_other_mul(gr_mat_t res, gr_srcptr x, gr_ctx_t x_ctx, const gr_mat_t mat, gr_ctx_t ctx)

    Perform the matrix-scalar or scalar-matrix operation `A x` or `x A`.

.. function:: int gr_mat_div_scalar(gr_mat_t res, const gr_mat_t mat, gr_srcptr x, gr_ctx_t ctx)
              int gr_mat_div_scalar_other(gr_mat_t res, const gr_mat_t mat, gr_srcptr x, gr_ctx_t x_ctx, gr_ctx_t ctx)
              int gr_mat_div_ui(gr_mat_t res, const gr_mat_t mat, ulong x, gr_ctx_t ctx)
              int gr_mat_div_si(gr_mat_t res, const gr_mat_t mat, slong x, gr_ctx_t ctx)
              int gr_mat_div_fmpz(gr_mat_t res, const gr_mat_t mat, const fmpz_t x, gr_ctx_t ctx)
              int gr_mat_div_fmpq(gr_mat_t res, const gr_mat_t mat, const fmpq_t x, gr_ctx_t ctx)

    Perform the matrix-scalar operation `A / x`.

.. function:: int gr_mat_addmul_scalar(gr_mat_t res, const gr_mat_t mat, gr_srcptr c, gr_ctx_t ctx)
              int gr_mat_submul_scalar(gr_mat_t res, const gr_mat_t mat, gr_srcptr c, gr_ctx_t ctx)

.. function:: int _gr_mat_gr_poly_evaluate(gr_mat_t res, gr_srcptr poly, slong len, const gr_mat_t mat, gr_ctx_t ctx)
              int gr_mat_gr_poly_evaluate(gr_mat_t res, const gr_poly_t poly, const gr_mat_t mat, gr_ctx_t ctx)

    Sets *res* to the matrix obtained by evaluating the
    scalar polynomial *poly* with matrix argument *mat*.

Diagonal and triangular matrices
-------------------------------------------------------------------------------

.. function:: truth_t gr_mat_is_upper_triangular(const gr_mat_t mat, gr_ctx_t ctx)
              truth_t gr_mat_is_lower_triangular(const gr_mat_t mat, gr_ctx_t ctx)

    Returns whether *mat* is upper (respectively lower) triangular, having
    zeros everywhere below (respectively above) the main diagonal.
    The matrix need not be square.

.. function:: truth_t gr_mat_is_diagonal(const gr_mat_t mat, gr_ctx_t ctx)

    Returns whether *mat* is a diagonal matrix, having zeros everywhere
    except on the main diagonal.
    The matrix need not be square.

.. function:: int gr_mat_mul_diag(gr_mat_t res, const gr_mat_t A, const gr_vec_t D, gr_ctx_t ctx)
              int gr_mat_diag_mul(gr_mat_t res, const gr_vec_t D, const gr_mat_t A, gr_ctx_t ctx)

    Set *res* to the product `AD` or `DA` respectively, where `D` is
    a diagonal matrix represented as a vector of entries.

Gaussian elimination
-------------------------------------------------------------------------------

.. function:: int gr_mat_find_nonzero_pivot_large_abs(slong * pivot_row, gr_mat_t mat, slong start_row, slong end_row, slong column, gr_ctx_t ctx)
              int gr_mat_find_nonzero_pivot_generic(slong * pivot_row, gr_mat_t mat, slong start_row, slong end_row, slong column, gr_ctx_t ctx)
              int gr_mat_find_nonzero_pivot(slong * pivot_row, gr_mat_t mat, slong start_row, slong end_row, slong column, gr_ctx_t ctx)

    Attempts to find a nonzero element in column number *column*
    of the matrix *mat* in a row between *start_row* (inclusive)
    and *end_row* (exclusive).
    On success, sets ``pivot_row`` to the row index and returns
    ``GR_SUCCESS``. If no nonzero pivot element exists, returns ``GR_DOMAIN``.
    If no nonzero pivot element exists and zero-testing fails for some
    element, returns the flag ``GR_UNABLE``.

    This function may be destructive: any elements that are nontrivially
    zero but can be certified zero may be overwritten by exact zeros.

.. function:: int gr_mat_lu_classical(slong * rank, slong * P, gr_mat_t LU, const gr_mat_t A, int rank_check, gr_ctx_t ctx)
              int gr_mat_lu_recursive(slong * rank, slong * P, gr_mat_t LU, const gr_mat_t A, int rank_check, gr_ctx_t ctx)
              int gr_mat_lu_generic(slong * rank, slong * P, gr_mat_t LU, const gr_mat_t A, int rank_check, gr_ctx_t ctx)
              int gr_mat_lu(slong * rank, slong * P, gr_mat_t LU, const gr_mat_t A, int rank_check, gr_ctx_t ctx)

    Computes a generalized LU decomposition `A = PLU` of a given
    matrix *A*, writing the rank of *A* to *rank*.

    If *A* is a nonsingular square matrix, *LU* will be set to
    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 set
    the rank to 0 if *A* is detected to be rank-deficient.
    This currently only works as expected for square matrices.

    The algorithm can fail if it fails to certify that a pivot
    element is zero or nonzero, in which case the correct rank
    cannot be determined. It can also fail if a pivot element
    is not invertible. In these cases the ``GR_UNABLE`` and/or
    ``GR_DOMAIN`` flags will be returned. On failure,
    the data in the output variables
    ``rank``, ``P`` and ``LU`` will be meaningless.

    The *classical* version uses iterative Gaussian elimination.
    The *recursive* version uses a block recursive algorithm
    to take advantage of fast matrix multiplication.
    The *generic* version calls the recursive algorithm with a
    default cutoff.

.. function:: int gr_mat_fflu(slong * rank, slong * P, gr_mat_t LU, gr_ptr den, const gr_mat_t A, int rank_check, gr_ctx_t ctx)

    Similar to :func:`gr_mat_lu`, but computes a fraction-free
    LU decomposition using the Bareiss algorithm.
    The denominator is written to *den*.

Solving
-------------------------------------------------------------------------------

.. function:: int gr_mat_nonsingular_solve_tril_classical(gr_mat_t X, const gr_mat_t L, const gr_mat_t B, int unit, gr_ctx_t ctx)
              int gr_mat_nonsingular_solve_tril_recursive(gr_mat_t X, const gr_mat_t L, const gr_mat_t B, int unit, gr_ctx_t ctx)
              int gr_mat_nonsingular_solve_tril_generic(gr_mat_t X, const gr_mat_t L, const gr_mat_t B, int unit, gr_ctx_t ctx)
              int gr_mat_nonsingular_solve_tril(gr_mat_t X, const gr_mat_t L, const gr_mat_t B, int unit, gr_ctx_t ctx)
              int gr_mat_nonsingular_solve_triu_classical(gr_mat_t X, const gr_mat_t U, const gr_mat_t B, int unit, gr_ctx_t ctx)
              int gr_mat_nonsingular_solve_triu_recursive(gr_mat_t X, const gr_mat_t U, const gr_mat_t B, int unit, gr_ctx_t ctx)
              int gr_mat_nonsingular_solve_triu_generic(gr_mat_t X, const gr_mat_t U, const gr_mat_t B, int unit, gr_ctx_t ctx)
              int gr_mat_nonsingular_solve_triu(gr_mat_t X, const gr_mat_t U, const gr_mat_t B, int unit, gr_ctx_t ctx)

    Solves the lower triangular system `LX = B` or the upper triangular system
    `UX = B`, respectively. Division by the the diagonal entries must
    be possible; if not a division fails, ``GR_DOMAIN`` is returned
    even if the system is solvable.
    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:: int gr_mat_nonsingular_solve_fflu(gr_mat_t X, const gr_mat_t A, const gr_mat_t B, gr_ctx_t ctx)
              int gr_mat_nonsingular_solve_lu(gr_mat_t X, const gr_mat_t A, const gr_mat_t B, gr_ctx_t ctx)
              int gr_mat_nonsingular_solve(gr_mat_t X, const gr_mat_t A, const gr_mat_t B, gr_ctx_t ctx)

    Solves `AX = B`. If *A* is not invertible,
    returns ``GR_DOMAIN`` even if the system has a solution.

.. function:: int gr_mat_nonsingular_solve_fflu_precomp(gr_mat_t X, const slong * perm, const gr_mat_t LU, const gr_mat_t B, gr_ctx_t ctx)
              int gr_mat_nonsingular_solve_lu_precomp(gr_mat_t X, const slong * perm, const gr_mat_t LU, const gr_mat_t B, gr_ctx_t ctx)

    Solves `AX = B` given a precomputed FFLU or LU factorization of *A*.

.. function:: int gr_mat_nonsingular_solve_den_fflu(gr_mat_t X, gr_ptr den, const gr_mat_t A, const gr_mat_t B, gr_ctx_t ctx)
              int gr_mat_nonsingular_solve_den(gr_mat_t X, gr_ptr den, const gr_mat_t A, const gr_mat_t B, gr_ctx_t ctx)

    Solves `AX = B` over the fraction field of the present ring
    (assumed to be an integral domain), returning `X` with
    an implied denominator *den*.
    If *A* is not invertible over the fraction field, returns
    ``GR_DOMAIN`` even if the system has a solution.

.. function:: int gr_mat_solve_field(gr_mat_t X, const gr_mat_t A, const gr_mat_t B, gr_ctx_t ctx)

    Solves `AX = B` where *A* is not necessarily square and not necessarily
    invertible. Assuming that the ring is a field, a return value of
    ``GR_DOMAIN`` indicates that the system has no solution.
    If there are multiple solutions, an arbitrary solution is returned.

Determinant and trace
-------------------------------------------------------------------------------

.. function:: int gr_mat_det_fflu(gr_ptr res, const gr_mat_t mat, gr_ctx_t ctx)
              int gr_mat_det_berkowitz(gr_ptr res, const gr_mat_t mat, gr_ctx_t ctx)
              int gr_mat_det_lu(gr_ptr res, const gr_mat_t mat, gr_ctx_t ctx)
              int gr_mat_det_cofactor(gr_ptr res, const gr_mat_t mat, gr_ctx_t ctx)
              int gr_mat_det_generic_field(gr_ptr res, const gr_mat_t A, gr_ctx_t ctx)
              int gr_mat_det_generic_integral_domain(gr_ptr res, const gr_mat_t A, gr_ctx_t ctx)
              int gr_mat_det_generic(gr_ptr res, const gr_mat_t A, gr_ctx_t ctx)
              int gr_mat_det(gr_ptr res, const gr_mat_t mat, gr_ctx_t ctx)

    Sets *res* to the determinant of the square matrix *mat*.
    Various algorithms are available:

    * The *berkowitz* version uses the division-free Berkowitz algorithm
      performing `O(n^4)` operations. Since no zero tests are required, it
      is guaranteed to succeed if the ring arithmetic succeeds.

    * The *cofactor* version performs cofactor expansion. This is currently
      only supported for matrices up to size 4, and for larger
      matrices returns the ``GR_UNABLE`` flag.

    * The *lu* and *fflu* versions use rational LU decomposition
      and fraction-free LU decomposition (Bareiss algorithm) respectively,
      requiring `O(n^3)` operations. These algorithms can fail if zero
      certification or inversion fails, in which case the ``GR_UNABLE``
      flag is returned.

    * The *generic*, *generic_field* and *generic_integral_domain*
      versions choose an appropriate algorithm for a generic ring
      depending on the availability of division.

    * The *default* method can be overloaded.

    If the matrix is not square, ``GR_DOMAIN`` is returned.

.. function:: int gr_mat_trace(gr_ptr res, const gr_mat_t mat, gr_ctx_t ctx)

    Sets *res* to the trace (sum of entries on the main diagonal) of
    the square matrix *mat*.
    If the matrix is not square, ``GR_DOMAIN`` is returned.


Rank
-------------------------------------------------------------------------------

.. function:: int gr_mat_rank_fflu(slong * rank, const gr_mat_t mat, gr_ctx_t ctx)
              int gr_mat_rank_lu(slong * rank, const gr_mat_t mat, gr_ctx_t ctx)
              int gr_mat_rank(slong * rank, const gr_mat_t mat, gr_ctx_t ctx)

    Sets *res* to the rank of *mat*.
    The default method returns ``GR_DOMAIN`` if the element ring
    is not an integral domain, in which case the usual rank is
    not well-defined. The *fflu* and *lu* variants currently do
    not check the element domain, and simply return this flag if they
    encounter an impossible inverse in the execution of the
    respective algorithms.

Row echelon form
-------------------------------------------------------------------------------

.. function:: int gr_mat_rref_lu(slong * rank, gr_mat_t R, const gr_mat_t A, gr_ctx_t ctx)
              int gr_mat_rref_fflu(slong * rank, gr_mat_t R, const gr_mat_t A, gr_ctx_t ctx)
              int gr_mat_rref(slong * rank, gr_mat_t R, const gr_mat_t A, gr_ctx_t ctx)

    Sets *R* to the reduced row echelon form of *A*, also setting
    *rank* to its rank.

.. function:: int gr_mat_rref_den_fflu(slong * rank, gr_mat_t R, gr_ptr den, const gr_mat_t A, gr_ctx_t ctx)
              int gr_mat_rref_den(slong * rank, gr_mat_t R, gr_ptr den, const gr_mat_t A, gr_ctx_t ctx)

    Like *rref*, but computes the reduced row echelon multiplied
    by a common (not necessarily minimal) denominator which is written
    to *den*. This can be used to compute the rref over an integral
    domain which is not a field.

Nullspace
-------------------------------------------------------------------------------

.. function:: int gr_mat_nullspace(gr_mat_t X, const gr_mat_t A, gr_ctx_t ctx)

    Sets *X* to a basis for the (right) nullspace of *A*.
    On success, the output matrix will be resized to the correct
    number of columns.

    The basis is not guaranteed to be presented in a
    canonical or minimal form.

    If the ring is not a field, this is implied to compute a nullspace
    basis over the fraction field. The result may be meaningless
    if the ring is not an integral domain.

Inverse and adjugate
-------------------------------------------------------------------------------

.. function:: int gr_mat_inv(gr_mat_t res, const gr_mat_t mat, gr_ctx_t ctx)

    Sets *res* to the inverse of *mat*, computed by solving
    `A A^{-1} = I`.

    Returns ``GR_DOMAIN`` if it can be determined that *mat* is not
    invertible over the present ring (warning: this may not work
    over non-integral domains). If invertibility cannot be proved,
    returns ``GR_UNABLE``.

    To compute the inverse over the fraction field, one may use
    :func:`gr_mat_nonsingular_solve_den` or :func:`gr_mat_adjugate`.

.. function:: int gr_mat_adjugate_charpoly(gr_mat_t adj, gr_ptr det, const gr_mat_t mat, gr_ctx_t ctx)
              int gr_mat_adjugate_cofactor(gr_mat_t adj, gr_ptr det, const gr_mat_t mat, gr_ctx_t ctx)
              int gr_mat_adjugate(gr_mat_t adj, gr_ptr det, const gr_mat_t mat, gr_ctx_t ctx)

    Sets *adj* to the adjugate matrix of *mat*, simultaneously
    setting *det* to the determinant of *mat*. We have
    `\operatorname{adj}(A) A = A \operatorname{adj}(A) = \det(A) I`,
    and `A^{-1} = \operatorname{adj}(A) / \det(A)` when *A*
    is invertible.

    The *cofactor* version uses cofactor expansion, requiring the
    evaluation of `n^2` determinants.
    The *charpoly* version computes and then evaluates the
    characteristic polynomial, requiring `O(n^{1/2})`
    matrix multiplications plus `O(n^3)` or `O(n^4)` operations
    for the characteristic polynomial itself depending on the
    algorithm used.

Characteristic polynomial
-------------------------------------------------------------------------------

.. function:: int _gr_mat_charpoly(gr_ptr res, const gr_mat_t mat, gr_ctx_t ctx)
              int gr_mat_charpoly(gr_poly_t res, const gr_mat_t mat, gr_ctx_t ctx)

    Computes the characteristic polynomial using a default
    algorithm choice. The
    underscore method assumes that *res* is a preallocated
    array of `n + 1` coefficients.

.. function:: int _gr_mat_charpoly_berkowitz(gr_ptr res, const gr_mat_t mat, gr_ctx_t ctx)
              int gr_mat_charpoly_berkowitz(gr_poly_t res, const gr_mat_t mat, gr_ctx_t ctx)

    Sets *res* to the characteristic polynomial of the square matrix
    *mat*, computed using the division-free Berkowitz algorithm.
    The number of operations is `O(n^4)` where *n* is the
    size of the matrix.

.. function:: int _gr_mat_charpoly_danilevsky_inplace(gr_ptr res, gr_mat_t mat, gr_ctx_t ctx)
              int _gr_mat_charpoly_danilevsky(gr_ptr res, const gr_mat_t mat, gr_ctx_t ctx)
              int gr_mat_charpoly_danilevsky(gr_poly_t res, const gr_mat_t mat, gr_ctx_t ctx)
              int _gr_mat_charpoly_gauss(gr_ptr res, const gr_mat_t mat, gr_ctx_t ctx)
              int gr_mat_charpoly_gauss(gr_poly_t res, const gr_mat_t mat, gr_ctx_t ctx)
              int _gr_mat_charpoly_householder(gr_ptr res, const gr_mat_t mat, gr_ctx_t ctx)
              int gr_mat_charpoly_householder(gr_poly_t res, const gr_mat_t mat, gr_ctx_t ctx)

    Sets *res* to the characteristic polynomial of the square matrix
    *mat*, computed using the Danilevsky algorithm,
    Hessenberg reduction using Gaussian elimination,
    and Hessenberg reduction using Householder reflections.
    The number of operations of each method is `O(n^3)` where *n* is the
    size of the matrix. The *inplace* version overwrites the input matrix.

    These methods require divisions and can therefore fail when the
    ring is not a field. They also require zero tests.
    The *householder* version also requires square roots.
    The flags ``GR_UNABLE`` or ``GR_DOMAIN`` are returned when
    an impossible division or square root
    is encountered or when a comparison cannot be performed.

.. function:: int _gr_mat_charpoly_faddeev(gr_ptr res, gr_mat_t adj, const gr_mat_t mat, gr_ctx_t ctx)
              int gr_mat_charpoly_faddeev(gr_poly_t res, gr_mat_t adj, const gr_mat_t mat, gr_ctx_t ctx)
              int _gr_mat_charpoly_faddeev_bsgs(gr_ptr res, gr_mat_t adj, const gr_mat_t mat, gr_ctx_t ctx)
              int gr_mat_charpoly_faddeev_bsgs(gr_poly_t res, gr_mat_t adj, const gr_mat_t mat, gr_ctx_t ctx)

    Sets *res* to the characteristic polynomial of the square matrix
    *mat*, computed using the Faddeev-LeVerrier algorithm.
    If the optional output argument *adj* is not *NULL*, it is
    set to the adjugate matrix, which is computed free of charge.

    The *bsgs* version uses a baby-step giant-step strategy,
    also known as the Preparata-Sarwate algorithm.
    This reduces the complexity from `O(n^4)` to `O(n^{3.5})` operations
    at the cost of requiring `n^{0.5}` temporary matrices to be
    stored.

    This method requires divisions by small integers and can
    therefore fail (returning the ``GR_UNABLE`` or ``GR_DOMAIN`` flags)
    in finite characteristic or when the underlying ring does
    not implement a division algorithm.

.. function:: int _gr_mat_charpoly_from_hessenberg(gr_ptr res, const gr_mat_t mat, gr_ctx_t ctx)
              int gr_mat_charpoly_from_hessenberg(gr_poly_t res, const gr_mat_t mat, gr_ctx_t ctx)

    Sets *res* to the characteristic polynomial of the square matrix
    *mat*, which is assumed to be in Hessenberg form (this is
    currently not checked).

Minimal polynomial
-------------------------------------------------------------------------------

.. function:: int gr_mat_minpoly_field(gr_poly_t res, const gr_mat_t mat, gr_ctx_t ctx)

    Compute the minimal polynomial of the matrix *mat*.
    The algorithm assumes that the coefficient ring is a field.

Similarity transformations
-------------------------------------------------------------------------------

.. function:: int gr_mat_apply_row_similarity(gr_mat_t M, slong r, gr_ptr d, gr_ctx_t ctx)

    Applies an elementary 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.

Eigenvalues
-------------------------------------------------------------------------------

.. function:: int gr_mat_eigenvalues(gr_vec_t lambda, gr_vec_t mult, const gr_mat_t mat, int flags, gr_ctx_t ctx)
              int gr_mat_eigenvalues_other(gr_vec_t lambda, gr_vec_t mult, const gr_mat_t mat, gr_ctx_t mat_ctx, int flags, gr_ctx_t ctx)

    Finds all eigenvalues of the given matrix in the ring defined by *ctx*,
    storing the eigenvalues without duplication in *lambda* (a vector with
    elements of type ``ctx``) and the corresponding multiplicities in
    *mult* (a vector with elements of type ``fmpz``).

    The interface is essentially the same as that of
    :func:`gr_poly_roots`; see its documentation for details.

.. function:: int gr_mat_diagonalization_precomp(gr_vec_t D, gr_mat_t L, gr_mat_t R, const gr_mat_t A, const gr_vec_t eigenvalues, const gr_vec_t mult, gr_ctx_t ctx)
              int gr_mat_diagonalization_generic(gr_vec_t D, gr_mat_t L, gr_mat_t R, const gr_mat_t A, int flags, gr_ctx_t ctx)
              int gr_mat_diagonalization(gr_vec_t D, gr_mat_t L, gr_mat_t R, const gr_mat_t A, int flags, gr_ctx_t ctx)

    Computes a diagonalization `LAR = D` given a square matrix `A`,
    where `D` is a diagonal matrix (returned as a vector) of the eigenvalues
    repeated according to their multiplicities,
    `L` is a matrix of left eigenvectors,
    and `R` is a matrix of right eigenvectors,
    normalized such that `L = R^{-1}`.
    This implies that `A = RDL = RDR^{-1}`.
    Either `L` or `R` (or both) can be set to ``NULL`` to omit computing
    the respective matrix.

    If the matrix has entries in a field then a return flag
    of ``GR_DOMAIN`` indicates that the matrix is non-diagonalizable
    over this field.

    The *precomp* version requires as input a precomputed set of eigenvalues
    with corresponding multiplicities, which can be computed
    with :func:`gr_mat_eigenvalues`.

Jordan decomposition
-------------------------------------------------------------------------------

.. function:: int gr_mat_set_jordan_blocks(gr_mat_t mat, const gr_vec_t lambda, slong num_blocks, slong * block_lambda, slong * block_size, gr_ctx_t ctx)
              int gr_mat_jordan_blocks(gr_vec_t lambda, slong * num_blocks, slong * block_lambda, slong * block_size, const gr_mat_t A, gr_ctx_t ctx)
              int gr_mat_jordan_transformation(gr_mat_t mat, const gr_vec_t lambda, slong num_blocks, slong * block_lambda, slong * block_size, const gr_mat_t A, gr_ctx_t ctx)
              int gr_mat_jordan_form(gr_mat_t J, gr_mat_t P, const gr_mat_t A, gr_ctx_t ctx)

Matrix functions
-------------------------------------------------------------------------------

.. function:: int gr_mat_exp_jordan(gr_mat_t res, const gr_mat_t A, gr_ctx_t ctx)
              int gr_mat_exp(gr_mat_t res, const gr_mat_t A, gr_ctx_t ctx)

.. function:: int gr_mat_log_jordan(gr_mat_t res, const gr_mat_t A, gr_ctx_t ctx)
              int gr_mat_log(gr_mat_t res, const gr_mat_t A, gr_ctx_t ctx)

Hessenberg form
-------------------------------------------------------------------------------

.. function:: truth_t gr_mat_is_hessenberg(const gr_mat_t mat, gr_ctx_t ctx)

    Returns whether *mat* is in upper Hessenberg form.

.. function:: int gr_mat_hessenberg_gauss(gr_mat_t res, const gr_mat_t mat, gr_ctx_t ctx)
              int gr_mat_hessenberg_householder(gr_mat_t res, const gr_mat_t mat, gr_ctx_t ctx)
              int gr_mat_hessenberg(gr_mat_t res, const gr_mat_t mat, gr_ctx_t ctx)

    Sets *res* to an upper Hessenberg form of *mat*.
    The *gauss* version uses Gaussian elimination.
    The *householder* version uses Householder reflections.

    These methods require divisions and zero testing
    and can therefore fail (returning ``GR_UNABLE`` or ``GR_DOMAIN``)
    when the ring is not a field.
    The *householder* version additionally requires complex
    conjugation and the ability to compute square roots.

Random matrices
-------------------------------------------------------------------------------

.. function:: int gr_mat_randtest(gr_mat_t res, flint_rand_t state, gr_ctx_t ctx)

    Sets *res* to a random matrix. The distribution is nonuniform.

.. function:: int gr_mat_randops(gr_mat_t mat, flint_rand_t state, slong count, gr_ctx_t ctx)

    Randomises *mat* in-place by performing elementary row or column
    operations. More precisely, at most *count* random additions or
    subtractions of distinct rows and columns will be performed.

.. function:: int gr_mat_randpermdiag(int * parity, gr_mat_t mat, flint_rand_t state, gr_ptr diag, slong n, gr_ctx_t ctx)

    Sets mat to a random permutation of the diagonal matrix with *n* leading entries given by
    the vector ``diag``. Returns ``GR_DOMAIN`` if the main diagonal of ``mat``
    does not have room for at least *n* entries.
    The parity (0 or 1) of the permutation is written to ``parity``.

.. function:: int gr_mat_randrank(gr_mat_t mat, flint_rand_t state, slong rank, gr_ctx_t ctx)

    Sets ``mat`` to a random sparse matrix with the given rank, having exactly as many
    non-zero elements as the rank. The matrix can be transformed into a dense matrix
    with unchanged rank by subsequently calling :func:`gr_mat_randops`.

    This operation only makes sense over integral domains (currently not checked).

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

For the following functions, the user supplies an output matrix
with the intended number of rows and columns.

.. function:: int gr_mat_ones(gr_mat_t res, gr_ctx_t ctx)

    Sets all entries in *res* to one.

.. function:: int gr_mat_pascal(gr_mat_t res, int triangular, gr_ctx_t ctx)

    Sets *res* 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.

.. function:: int gr_mat_stirling(gr_mat_t res, int kind, gr_ctx_t ctx)

    Sets *res* 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.

.. function:: int gr_mat_hilbert(gr_mat_t res, gr_ctx_t ctx)

    Sets *res* to the Hilbert matrix, which has entries `1/(i+j+1)`
    for `i, j \ge 0`.

.. function:: int gr_mat_hadamard(gr_mat_t res, gr_ctx_t ctx)

    If possible, sets *res* to a Hadamard matrix of the provided size
    and returns ``GR_SUCCESS``. Returns ``GR_DOMAIN``
    if no Hadamard matrix of the given size exists,
    and ``GR_UNABLE`` if the implementation does
    not know how to construct a Hadamard matrix of the given
    size.

    A Hadamard matrix of size *n* can only exist if *n* is 0, 1, 2,
    or a multiple of 4. It is not known whether a
    Hadamard matrix exists for every size that is a multiple of 4.
    This function uses the Paley construction, which
    succeeds for all *n* of the form `n = 2^e` or `n = 2^e (q + 1)` where
    *q* is an odd prime power. Orders *n* for which Hadamard matrices are
    known to exist but for which this construction fails are
    92, 116, 156, ... (OEIS A046116).

Helper functions for reduction
-------------------------------------------------------------------------------

.. function:: int gr_mat_reduce_row(slong * column, gr_mat_t A, slong * P, slong * L, slong m, gr_ctx_t 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 sets *column* to 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.


Test functions
-------------------------------------------------------------------------------

The following functions run *iters* test iterations, generating matrices
up to size *maxn*. If *ctx* is set to ``NULL``, a random ring is generated
on each test iteration, otherwise the given ring is tested.

.. function:: void gr_mat_test_mul(gr_method_mat_binary_op mul_impl, flint_rand_t state, slong iters, slong maxn, gr_ctx_t ctx)

    Tests the given function ``mul_impl`` for correctness as an implementation
    of :func:`gr_mat_mul`.

.. function:: void gr_mat_test_lu(gr_method_mat_lu_op lu_impl, flint_rand_t state, slong iters, slong maxn, gr_ctx_t ctx)

    Tests the given function ``mul_impl`` for correctness as an implementation
    of :func:`gr_mat_lu`.

.. function:: void gr_mat_test_det(gr_method_mat_unary_op_get_scalar det_impl, flint_rand_t state, slong iters, slong maxn, gr_ctx_t ctx)

    Tests the given function ``det_impl`` for correctness as an implementation
    of :func:`gr_mat_det`.

.. function:: void gr_mat_test_nonsingular_solve_tril(gr_method_mat_binary_op_with_flag solve_impl, flint_rand_t state, slong iters, slong maxn, gr_ctx_t ctx)
              void gr_mat_test_nonsingular_solve_triu(gr_method_mat_binary_op_with_flag solve_impl, flint_rand_t state, slong iters, slong maxn, gr_ctx_t ctx)

    Tests the given function ``solve_impl`` for correctness as an implementation
    of :func:`gr_mat_nonsingular_solve_tril` / :func:`gr_mat_nonsingular_solve_triu`.

.. function:: void gr_mat_test_approx_mul_max_norm(gr_method_mat_binary_op mul_impl, gr_srcptr rel_tol, flint_rand_t state, slong iters, slong maxn, gr_ctx_t ctx)

    Tests the given implementation of matrix multiplication for accuracy
    over an approximate numerical ring by checking that
    `|C-AB| \le |A||B| rel\_tol` holds in the max norm,
    using classical multiplication for reference.

.. function:: void gr_mat_test_approx_mul_pos_entrywise_accurate(gr_method_mat_binary_op mul_impl, gr_srcptr rel_tol, flint_rand_t state, slong iters, slong maxn, gr_ctx_t ctx)

    Tests the given implementation of matrix multiplication for accuracy
    over an approximate numerical ring by generating nonnegative matrices
    and checking that the entrywise relative error compared to
    classical multiplication does not exceed *rel_tol*.

.. raw:: latex

    \newpage
