.. _fmpz-poly-mat:

**fmpz_poly_mat.h** -- matrices of polynomials over the integers
===============================================================================

The :type:`fmpz_poly_mat_t` data type represents matrices whose
entries are integer polynomials.

The :type:`fmpz_poly_mat_t` type is defined as an array of
:type:`fmpz_poly_mat_struct`'s of length one. This permits passing
parameters of type :type:`fmpz_poly_mat_t` by reference.

An integer polynomial matrix internally consists of a single array of
:type:`fmpz_poly_struct`'s, representing a dense matrix in row-major
order. This array is only directly indexed during memory allocation
and deallocation. A separate array holds pointers to the start of each
row, and is used for all indexing. This allows the rows of a matrix to
be permuted quickly by swapping pointers.

Matrices having zero rows or columns are allowed.

The shape of a matrix is fixed upon initialisation. The user is
assumed to provide input and output variables whose dimensions are
compatible with the given operation.

Simple example
--------------

The following example constructs the matrix `\begin{pmatrix} 2x+1 & x
\\ 1-x & -1 \end{pmatrix}` and computes its determinant.

.. code:: c

   #include "fmpz_poly.h"
   #include "fmpz_poly_mat.h"
   int main()
   {
       fmpz_poly_mat_t A;
       fmpz_poly_t P;

       fmpz_poly_mat_init(A, 2, 2);
       fmpz_poly_init(P);

       fmpz_poly_set_str(fmpz_poly_mat_entry(A, 0, 0), "2  1 2");
       fmpz_poly_set_str(fmpz_poly_mat_entry(A, 0, 1), "2  0 1");
       fmpz_poly_set_str(fmpz_poly_mat_entry(A, 1, 0), "2  1 -1");
       fmpz_poly_set_str(fmpz_poly_mat_entry(A, 1, 1), "1  -1");

       fmpz_poly_mat_det(P, A);
       fmpz_poly_print_pretty(P, "x");

       fmpz_poly_clear(P);
       fmpz_poly_mat_clear(A);
   }

The output is:

::

   x^2-3*x-1

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

.. type:: fmpz_poly_mat_struct

.. type:: fmpz_poly_mat_t

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


.. function:: void fmpz_poly_mat_init(fmpz_poly_mat_t mat, slong rows, slong cols)

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

.. function:: void fmpz_poly_mat_init_set(fmpz_poly_mat_t mat, const fmpz_poly_mat_t src)

    Initialises a matrix ``mat`` of the same dimensions as ``src``,
    and sets it to a copy of ``src``.

.. function:: void fmpz_poly_mat_clear(fmpz_poly_mat_t mat)

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


Basic properties
--------------------------------------------------------------------------------


.. function:: slong fmpz_poly_mat_nrows(const fmpz_poly_mat_t mat)

    Returns the number of rows in ``mat``.

.. function:: slong fmpz_poly_mat_ncols(const fmpz_poly_mat_t mat)

    Returns the number of columns in ``mat``.



Basic assignment and manipulation
--------------------------------------------------------------------------------


.. function:: fmpz_poly_struct * fmpz_poly_mat_entry(const fmpz_poly_mat_t mat, slong i, slong j)

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

.. function:: void fmpz_poly_mat_set(fmpz_poly_mat_t mat1, const fmpz_poly_mat_t mat2)

    Sets ``mat1`` to a copy of ``mat2``.

.. function:: void fmpz_poly_mat_swap(fmpz_poly_mat_t mat1, fmpz_poly_mat_t mat2)

    Swaps ``mat1`` and ``mat2`` efficiently.

.. function:: void fmpz_poly_mat_swap_entrywise(fmpz_poly_mat_t mat1, fmpz_poly_mat_t mat2)

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


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


.. function:: void fmpz_poly_mat_print(const fmpz_poly_mat_t mat, const char * x)

    Prints the matrix ``mat`` to standard output, using the
    variable ``x``.


Random matrix generation
--------------------------------------------------------------------------------


.. function:: void fmpz_poly_mat_randtest(fmpz_poly_mat_t mat, flint_rand_t state, slong len, flint_bitcnt_t bits)

    This is equivalent to applying ``fmpz_poly_randtest`` to all entries
    in the matrix.

.. function:: void fmpz_poly_mat_randtest_unsigned(fmpz_poly_mat_t mat, flint_rand_t state, slong len, flint_bitcnt_t bits)

    This is equivalent to applying ``fmpz_poly_randtest_unsigned`` to
    all entries in the matrix.

.. function:: void fmpz_poly_mat_randtest_sparse(fmpz_poly_mat_t A, flint_rand_t state, slong len, flint_bitcnt_t bits, float density)

    Creates a random matrix with the amount of nonzero entries given
    approximately by the ``density`` variable, which should be a fraction
    between 0 (most sparse) and 1 (most dense).

    The nonzero entries will have random lengths between 1 and ``len``.


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


.. function:: void fmpz_poly_mat_zero(fmpz_poly_mat_t mat)

    Sets ``mat`` to the zero matrix.

.. function:: void fmpz_poly_mat_one(fmpz_poly_mat_t mat)

    Sets ``mat`` to the unit or identity matrix of given shape,
    having the element 1 on the main diagonal and zeros elsewhere.
    If ``mat`` is nonsquare, it is set to the truncation of a unit matrix.


Basic comparison and properties
--------------------------------------------------------------------------------


.. function:: int fmpz_poly_mat_equal(const fmpz_poly_mat_t mat1, const fmpz_poly_mat_t mat2)

    Returns nonzero if ``mat1`` and ``mat2`` have the same shape and
    all their entries agree, and returns zero otherwise.

.. function:: int fmpz_poly_mat_is_zero(const fmpz_poly_mat_t mat)

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

.. function:: int fmpz_poly_mat_is_one(const fmpz_poly_mat_t mat)

    Returns nonzero if all entries of ``mat`` on the main diagonal
    are the constant polynomial 1 and all remaining entries are zero,
    and returns zero otherwise. The matrix need not be square.

.. function:: int fmpz_poly_mat_is_empty(const fmpz_poly_mat_t mat)

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

.. function:: int fmpz_poly_mat_is_square(const fmpz_poly_mat_t mat)

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



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


.. function:: slong fmpz_poly_mat_max_bits(const fmpz_poly_mat_t A)

    Returns the maximum number of bits among the coefficients of the
    entries in ``A``, or the negative of that value if any
    coefficient is negative.

.. function:: slong fmpz_poly_mat_max_length(const fmpz_poly_mat_t A)

    Returns the maximum polynomial length among all the entries in ``A``.



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


.. function:: void fmpz_poly_mat_transpose(fmpz_poly_mat_t B, const fmpz_poly_mat_t A)

    Sets `B` to `A^t`.



Evaluation
--------------------------------------------------------------------------------


.. function:: void fmpz_poly_mat_evaluate_fmpz(fmpz_mat_t B, const fmpz_poly_mat_t A, const fmpz_t x)

    Sets the ``fmpz_mat_t`` ``B`` to ``A`` evaluated entrywise
    at the point ``x``.



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


.. function:: void fmpz_poly_mat_scalar_mul_fmpz_poly(fmpz_poly_mat_t B, const fmpz_poly_mat_t A, const fmpz_poly_t c)

    Sets ``B`` to ``A`` multiplied entrywise by the polynomial ``c``.

.. function:: void fmpz_poly_mat_scalar_mul_fmpz(fmpz_poly_mat_t B, const fmpz_poly_mat_t A, const fmpz_t c)

    Sets ``B`` to ``A`` multiplied entrywise by the integer ``c``.

.. function:: void fmpz_poly_mat_add(fmpz_poly_mat_t C, const fmpz_poly_mat_t A, const fmpz_poly_mat_t B)

    Sets ``C`` to the sum of ``A`` and ``B``.
    All matrices must have the same shape. Aliasing is allowed.

.. function:: void fmpz_poly_mat_sub(fmpz_poly_mat_t C, const fmpz_poly_mat_t A, const fmpz_poly_mat_t B)

    Sets ``C`` to the sum of ``A`` and ``B``.
    All matrices must have the same shape. Aliasing is allowed.

.. function:: void fmpz_poly_mat_neg(fmpz_poly_mat_t B, const fmpz_poly_mat_t A)

    Sets ``B`` to the negation of ``A``.
    The matrices must have the same shape. Aliasing is allowed.

.. function:: void fmpz_poly_mat_mul(fmpz_poly_mat_t C, const fmpz_poly_mat_t A, const fmpz_poly_mat_t B)

    Sets ``C`` to the matrix product of ``A`` and ``B``.
    The matrices must have compatible dimensions for matrix multiplication.
    Aliasing is allowed. This function automatically chooses between
    classical and KS multiplication.

.. function:: void fmpz_poly_mat_mul_classical(fmpz_poly_mat_t C, const fmpz_poly_mat_t A, const fmpz_poly_mat_t B)

    Sets ``C`` to the matrix product of ``A`` and ``B``,
    computed using the classical algorithm. The matrices must have
    compatible dimensions for matrix multiplication. Aliasing is allowed.

.. function:: void fmpz_poly_mat_mul_KS(fmpz_poly_mat_t C, const fmpz_poly_mat_t A, const fmpz_poly_mat_t B)

    Sets ``C`` to the matrix product of ``A`` and ``B``,
    computed using Kronecker segmentation. The matrices must have
    compatible dimensions for matrix multiplication. Aliasing is allowed.

.. function:: void fmpz_poly_mat_mullow(fmpz_poly_mat_t C, const fmpz_poly_mat_t A, const fmpz_poly_mat_t B, slong len)

    Sets ``C`` to the matrix product of ``A`` and ``B``,
    truncating each entry in the result to length ``len``.
    Uses classical matrix multiplication. The matrices must have
    compatible dimensions for matrix multiplication. Aliasing is allowed.

.. function:: void fmpz_poly_mat_sqr(fmpz_poly_mat_t B, const fmpz_poly_mat_t A)

    Sets ``B`` to the square of ``A``, which must be a square matrix.
    Aliasing is allowed. This function automatically chooses between
    classical and KS squaring.

.. function:: void fmpz_poly_mat_sqr_classical(fmpz_poly_mat_t B, const fmpz_poly_mat_t A)

    Sets ``B`` to the square of ``A``, which must be a square matrix.
    Aliasing is allowed. This function uses direct formulas for very small
    matrices, and otherwise classical matrix multiplication.

.. function:: void fmpz_poly_mat_sqr_KS(fmpz_poly_mat_t B, const fmpz_poly_mat_t A)

    Sets ``B`` to the square of ``A``, which must be a square matrix.
    Aliasing is allowed. This function uses Kronecker segmentation.

.. function:: void fmpz_poly_mat_sqrlow(fmpz_poly_mat_t B, const fmpz_poly_mat_t A, slong len)

    Sets ``B`` to the square of ``A``, which must be a square matrix,
    truncating all entries to length ``len``.
    Aliasing is allowed. This function uses direct formulas for very small
    matrices, and otherwise classical matrix multiplication.

.. function:: void fmpz_poly_mat_pow(fmpz_poly_mat_t B, const fmpz_poly_mat_t A, ulong exp)

    Sets ``B`` to ``A`` raised to the power ``exp``, where ``A``
    is a square matrix. Uses exponentiation by squaring. Aliasing is allowed.

.. function:: void fmpz_poly_mat_pow_trunc(fmpz_poly_mat_t B, const fmpz_poly_mat_t A, ulong exp, slong len)

    Sets ``B`` to ``A`` raised to the power ``exp``, truncating
    all entries to length ``len``, where ``A`` is a square matrix.
    Uses exponentiation by squaring. Aliasing is allowed.

.. function:: void fmpz_poly_mat_prod(fmpz_poly_mat_t res, fmpz_poly_mat_t * const factors, slong n)

    Sets ``res`` to the product of the ``n`` matrices given in
    the vector ``factors``, all of which must be square and of the
    same size. Uses binary splitting.


Row reduction
--------------------------------------------------------------------------------


.. function:: slong fmpz_poly_mat_find_pivot_any(const fmpz_poly_mat_t mat, slong start_row, slong end_row, slong c)

    Attempts to find a pivot entry for row reduction.
    Returns a row index `r` between ``start_row`` (inclusive) and
    ``stop_row`` (exclusive) such that column `c` in ``mat`` has
    a nonzero entry on row `r`, or returns -1 if no such entry exists.

    This implementation simply chooses the first nonzero entry
    it encounters. This is likely to be a nearly optimal choice if all
    entries in the matrix have roughly the same size, but can lead to
    unnecessary coefficient growth if the entries vary in size.

.. function:: slong fmpz_poly_mat_find_pivot_partial(const fmpz_poly_mat_t mat, slong start_row, slong end_row, slong c)

    Attempts to find a pivot entry for row reduction.
    Returns a row index `r` between ``start_row`` (inclusive) and
    ``stop_row`` (exclusive) such that column `c` in ``mat`` has
    a nonzero entry on row `r`, or returns -1 if no such entry exists.

    This implementation searches all the rows in the column and
    chooses the nonzero entry of smallest degree. If there are several
    entries with the same minimal degree, it chooses the entry with
    the smallest coefficient bit bound. This heuristic typically reduces
    coefficient growth when the matrix entries vary in size.

.. function:: slong fmpz_poly_mat_fflu(fmpz_poly_mat_t B, fmpz_poly_t den, slong * perm, const fmpz_poly_mat_t A, int rank_check)

    Uses fraction-free Gaussian elimination to set (``B``, ``den``) to a
    fraction-free LU decomposition of ``A`` and returns the
    rank of ``A``. Aliasing of ``A`` and ``B`` is allowed.

    Pivot elements are chosen with ``fmpz_poly_mat_find_pivot_partial``.
    If ``perm`` is non-``NULL``, the permutation of
    rows in the matrix will also be applied to ``perm``.

    If ``rank_check`` is set, the function aborts and returns 0 if the
    matrix is detected not to have full rank without completing the
    elimination.

    The denominator ``den`` is set to `\pm \operatorname{det}(A)`, where
    the sign is decided by the parity of the permutation. Note that the
    determinant is not generally the minimal denominator.

.. function:: slong fmpz_poly_mat_rref(fmpz_poly_mat_t B, fmpz_poly_t den, const fmpz_poly_mat_t A)

    Sets (``B``, ``den``) to the reduced row echelon form of
    ``A`` and returns the rank of ``A``. Aliasing of ``A`` and
    ``B`` is allowed.

    The denominator ``den`` is set to `\pm \operatorname{det}(A)`.
    Note that the determinant is not generally the minimal denominator.


Trace
--------------------------------------------------------------------------------


.. function:: void fmpz_poly_mat_trace(fmpz_poly_t trace, const fmpz_poly_mat_t mat)

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


Determinant and rank
--------------------------------------------------------------------------------


.. function:: void fmpz_poly_mat_det(fmpz_poly_t det, const fmpz_poly_mat_t A)

    Sets ``det`` to the determinant of the square matrix ``A``. Uses
    a direct formula, fraction-free LU decomposition, or interpolation,
    depending on the size of the matrix.

.. function:: void fmpz_poly_mat_det_fflu(fmpz_poly_t det, const fmpz_poly_mat_t A)

    Sets ``det`` to the determinant of the square matrix ``A``.
    The determinant is computed by performing a fraction-free LU
    decomposition on a copy of ``A``.

.. function:: void fmpz_poly_mat_det_interpolate(fmpz_poly_t det, const fmpz_poly_mat_t A)

    Sets ``det`` to the determinant of the square matrix ``A``.
    The determinant is computed by determining a bound `n` for its length,
    evaluating the matrix at `n` distinct points, computing the determinant
    of each integer matrix, and forming the interpolating polynomial.

.. function:: slong fmpz_poly_mat_rank(const fmpz_poly_mat_t A)

    Returns the rank of ``A``. Performs fraction-free LU decomposition
    on a copy of ``A``.



Inverse
--------------------------------------------------------------------------------


.. function:: int fmpz_poly_mat_inv(fmpz_poly_mat_t Ainv, fmpz_poly_t den, const fmpz_poly_mat_t A)

    Sets (``Ainv``, ``den``) to the inverse matrix of ``A``.
    Returns 1 if ``A`` is nonsingular and 0 if ``A`` is singular.
    Aliasing of ``Ainv`` and ``A`` is allowed.

    More precisely, ``det`` will be set to the determinant of ``A``
    and ``Ainv`` will be set to the adjugate matrix of ``A``.
    Note that the determinant is not necessarily the minimal denominator.

    Uses fraction-free LU decomposition, followed by solving for
    the identity matrix.



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


.. function:: slong fmpz_poly_mat_nullspace(fmpz_poly_mat_t res, const fmpz_poly_mat_t mat)

    Computes the right rational nullspace of the matrix ``mat`` and
    returns the nullity.

    More precisely, assume that ``mat`` has rank `r` and nullity `n`.
    Then this function sets the first `n` columns of ``res``
    to linearly independent vectors spanning the nullspace of ``mat``.
    As a result, we always have rank(``res``) `= n`, and
    ``mat`` `\times` ``res`` is the zero matrix.

    The computed basis vectors will not generally be in a reduced form.
    In general, the polynomials in each column vector in the result
    will have a nontrivial common GCD.


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


.. function:: int fmpz_poly_mat_solve(fmpz_poly_mat_t X, fmpz_poly_t den, const fmpz_poly_mat_t A, const fmpz_poly_mat_t B)

    Solves the equation `AX = B` for nonsingular `A`. More precisely, computes
    (``X``, ``den``) such that `AX = B \times \operatorname{den}`.
    Returns 1 if `A` is nonsingular and 0 if `A` is singular.
    The computed denominator will not generally be minimal.

    Uses fraction-free LU decomposition followed by fraction-free
    forward and back substitution.

.. function:: int fmpz_poly_mat_solve_fflu(fmpz_poly_mat_t X, fmpz_poly_t den, const fmpz_poly_mat_t A, const fmpz_poly_mat_t B)

    Solves the equation `AX = B` for nonsingular `A`. More precisely, computes
    (``X``, ``den``) such that `AX = B \times \operatorname{den}`.
    Returns 1 if `A` is nonsingular and 0 if `A` is singular.
    The computed denominator will not generally be minimal.

    Uses fraction-free LU decomposition followed by fraction-free
    forward and back substitution.

.. function:: void fmpz_poly_mat_solve_fflu_precomp(fmpz_poly_mat_t X, const slong * perm, const fmpz_poly_mat_t FFLU, const fmpz_poly_mat_t B)

    Performs fraction-free forward and back substitution given a precomputed
    fraction-free LU decomposition and corresponding permutation.
