# nmod_poly_mat.h – matrices of univariate polynomials over integers mod n (word-size n)

The `nmod_poly_mat_t` data type represents matrices whose entries are polynomials having coefficients in \(\mathbb{Z}/n\mathbb{Z}\). We generally assume that \(n\) is a prime number.

The `nmod_poly_mat_t` type is defined as an array of `nmod_poly_mat_struct`’s of length one. This permits passing parameters of type `nmod_poly_mat_t` by reference.

A matrix internally consists of a single array of `nmod_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.

## Types, macros and constants

type nmod_poly_mat_struct

type nmod_poly_mat_t

## Memory management

void nmod_poly_mat_init(nmod_poly_mat_t mat, slong rows, slong cols, ulong n)

  Initialises a matrix with the given number of rows and columns for use. The modulus is set to \(n\).

void nmod_poly_mat_init_set(nmod_poly_mat_t mat, const nmod_poly_mat_t src)

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

void nmod_poly_mat_clear(nmod_poly_mat_t mat)

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

## Truncate, shift

void nmod_poly_mat_set_trunc(nmod_poly_mat_t res, const nmod_poly_mat_t pmat, long len)

  Set `res` to the truncation of `pmat` to length `len`. Entries of `res` are normalized.

void nmod_poly_mat_truncate(nmod_poly_mat_t pmat, long len)

  Truncates `pmat` to the given length `len`, and normalize its entries. If `len` is greater than the maximum length of the entries of `pmat`, then nothing happens.

void nmod_poly_mat_shift_left(nmod_poly_mat_t res, const nmod_poly_mat_t pmat, slong k)

  Sets `res` to `pmat` shifted left by `k` coefficients, that is, multiplied by \(x^k\).

void nmod_poly_mat_shift_right(nmod_poly_mat_t res, const nmod_poly_mat_t pmat, slong k)

  Sets `res` to `pmat` shifted right by `k` coefficients, that is, divide by \(x^k\) and throw away the remainder. If `k` is greater than or equal to the length of `pmat`, the result is the zero polynomial matrix.

## Basic properties

slong nmod_poly_mat_nrows(const nmod_poly_mat_t mat)

  Returns the number of rows in `mat`.

slong nmod_poly_mat_ncols(const nmod_poly_mat_t mat)

  Returns the number of columns in `mat`.

ulong nmod_poly_mat_modulus(const nmod_poly_mat_t mat)

  Returns the modulus of `mat`.

## Basic assignment and manipulation

nmod_poly_struct *nmod_poly_mat_entry(const nmod_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 `nmod_poly` function for direct manipulation of the matrix element. No bounds checking is performed.

void nmod_poly_mat_set(nmod_poly_mat_t mat1, const nmod_poly_mat_t mat2)

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

void nmod_poly_mat_set_nmod_mat(nmod_poly_mat_t pmat, const nmod_mat_t cmat)

  Sets the already-initialized polynomial matrix `pmat` to a constant matrix with the same entries as `cmat`. Both input matrices must have the same dimensions and modulus.

void nmod_poly_mat_swap(nmod_poly_mat_t mat1, nmod_poly_mat_t mat2)

  Swaps `mat1` and `mat2` efficiently.

void nmod_poly_mat_swap_entrywise(nmod_poly_mat_t mat1, nmod_poly_mat_t mat2)

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

## Window

void nmod_poly_mat_window_init(nmod_poly_mat_t window, const nmod_poly_mat_t mat, slong r1, slong c1, slong r2, slong c2)

  Initializes the matrix `window` to be an `r2 - r1` by `c2 - c1` submatrix of `mat` whose `(0,0)` entry is the `(r1, c1)` entry of `mat`. The memory for the elements of `window` is shared with `mat`.

void nmod_poly_mat_window_clear(nmod_poly_mat_t window)

  Clears the matrix `window` and releases any memory that it uses. Note that the memory of the underlying matrix that `window` points to is not freed.

## Concatenate

void nmod_poly_mat_concat_vertical(nmod_poly_mat_t res, const nmod_poly_mat_t mat1, const nmod_poly_mat_t mat2)

  Sets `res` to vertical concatenation of (`mat1`, `mat2`) in that order. Matrix dimensions : `mat1` : \(m \times n\), `mat2` : \(k \times n\), `res` : \((m + k) \times n\).

void nmod_poly_mat_concat_horizontal(nmod_poly_mat_t res, const nmod_poly_mat_t mat1, const nmod_poly_mat_t mat2)

  Sets `res` to horizontal concatenation of (`mat1`, `mat2`) in that order. Matrix dimensions : `mat1` : \(m \times n\), `mat2` : \(m \times k\), `res` : \(m \times (n + k)\).

## Input and output

void nmod_poly_mat_print(const nmod_poly_mat_t mat, const char *x)

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

## Random matrix generation

void nmod_poly_mat_randtest(nmod_poly_mat_t mat, flint_rand_t state, slong len)

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

void nmod_poly_mat_randtest_sparse(nmod_poly_mat_t A, flint_rand_t state, slong len, 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

void nmod_poly_mat_zero(nmod_poly_mat_t mat)

  Sets `mat` to the zero matrix.

void nmod_poly_mat_one(nmod_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

int nmod_poly_mat_equal(const nmod_poly_mat_t mat1, const nmod_poly_mat_t mat2)

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

int nmod_poly_mat_equal_nmod_mat(const nmod_poly_mat_t pmat, const nmod_mat_t cmat)

  Returns nonzero if `pmat` is a constant matrix with the same dimensions and entries as `cmat`; returns zero otherwise.

int nmod_poly_mat_is_zero(const nmod_poly_mat_t mat)

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

int nmod_poly_mat_is_one(const nmod_poly_mat_t mat)

  Returns nonzero if all entry 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.

int nmod_poly_mat_is_empty(const nmod_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.

int nmod_poly_mat_is_square(const nmod_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.

void nmod_poly_mat_get_coeff_mat(nmod_mat_t coeff, const nmod_poly_mat_t pmat, slong deg)

  Sets `coeff` to be the coefficient of `pmat` of degree `deg`, where `pmat` is seen as a polynomial with matrix coefficients and coefficients are numbered from zero. `coeff` must be already initialized with the right dimensions and modulus. For entries of `pmat` of degree less than `deg`, the corresponding entry of `coeff` is zero.

void nmod_poly_mat_set_coeff_mat(nmod_poly_mat_t pmat, const nmod_mat_t coeff, slong deg)

  Sets the coefficient of `pmat` of degree `deg` to `coeff`, where `pmat` is seen as a polynomial with matrix coefficients and coefficients are numbered from zero. For each entry of `pmat`, if `deg` is larger than its degree, this entry is first resized to the appropriate length, with intervening coefficients being set to zero.

## Transposition

void nmod_poly_mat_transpose(nmod_poly_mat_t B, const nmod_poly_mat_t A)

  Sets \(B\) to the transpose of \(A\). Dimensions must be compatible. Aliasing is allowed for square matrices.

## Norms

slong nmod_poly_mat_max_length(const nmod_poly_mat_t A)

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

slong nmod_poly_mat_degree(const nmod_poly_mat_t pmat)

  Returns the degree of the polynomial matrix `pmat`. The zero matrix is deemed to have degree \(-1\).

## Evaluation

void nmod_poly_mat_evaluate_nmod(nmod_mat_t B, const nmod_poly_mat_t A, ulong x)

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

## Arithmetic

void nmod_poly_mat_scalar_mul_nmod_poly(nmod_poly_mat_t B, const nmod_poly_mat_t A, const nmod_poly_t c)

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

void nmod_poly_mat_scalar_mul_nmod(nmod_poly_mat_t B, const nmod_poly_mat_t A, ulong c)

  Sets `B` to `A` multiplied entrywise by the coefficient `c`, which is assumed to be reduced modulo the modulus.

void nmod_poly_mat_add(nmod_poly_mat_t C, const nmod_poly_mat_t A, const nmod_poly_mat_t B)

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

void nmod_poly_mat_sub(nmod_poly_mat_t C, const nmod_poly_mat_t A, const nmod_poly_mat_t B)

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

void nmod_poly_mat_neg(nmod_poly_mat_t B, const nmod_poly_mat_t A)

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

void nmod_poly_mat_mul(nmod_poly_mat_t C, const nmod_poly_mat_t A, const nmod_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, KS and evaluation-interpolation multiplication.

void nmod_poly_mat_mul_classical(nmod_poly_mat_t C, const nmod_poly_mat_t A, const nmod_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.

void nmod_poly_mat_mul_KS(nmod_poly_mat_t C, const nmod_poly_mat_t A, const nmod_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.

void nmod_poly_mat_mul_interpolate(nmod_poly_mat_t C, const nmod_poly_mat_t A, const nmod_poly_mat_t B)

  Sets `C` to the matrix product of `A` and `B`, computed through evaluation and interpolation. The matrices must have compatible dimensions for matrix multiplication. For interpolation to be well-defined, we require that the modulus is a prime at least as large as \(m + n - 1\) where \(m\) and \(n\) are the maximum lengths of polynomials in the input matrices. Aliasing is allowed.

void nmod_poly_mat_sqr(nmod_poly_mat_t B, const nmod_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.

void nmod_poly_mat_sqr_classical(nmod_poly_mat_t B, const nmod_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.

void nmod_poly_mat_sqr_KS(nmod_poly_mat_t B, const nmod_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.

void nmod_poly_mat_sqr_interpolate(nmod_poly_mat_t B, const nmod_poly_mat_t A)

  Sets `B` to the square of `A`, which must be a square matrix, computed through evaluation and interpolation. For interpolation to be well-defined, we require that the modulus is a prime at least as large as \(2n - 1\) where \(n\) is the maximum length of polynomials in the input matrix. Aliasing is allowed.

void nmod_poly_mat_pow(nmod_poly_mat_t B, const nmod_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.

## Row reduction

slong nmod_poly_mat_find_pivot_any(const nmod_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 from 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.

slong nmod_poly_mat_find_pivot_partial(const nmod_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. This heuristic typically reduces coefficient growth when the matrix entries vary in size.

slong nmod_poly_mat_fflu(nmod_poly_mat_t B, nmod_poly_t den, slong *perm, const nmod_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 `nmod_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.

slong nmod_poly_mat_rref(nmod_poly_mat_t B, nmod_poly_t den, const nmod_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

void nmod_poly_mat_trace(nmod_poly_t trace, const nmod_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

void nmod_poly_mat_det(nmod_poly_t det, const nmod_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.

void nmod_poly_mat_det_fflu(nmod_poly_t det, const nmod_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`.

void nmod_poly_mat_det_interpolate(nmod_poly_t det, const nmod_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 coefficient matrix, and forming the interpolating polynomial.

  If the coefficient ring does not contain \(n\) distinct points (that is, if working over \(\mathbf{Z}/p\mathbf{Z}\) where \(p < n\)), this function automatically falls back to `nmod_poly_mat_det_fflu`.

slong nmod_poly_mat_rank(const nmod_poly_mat_t A)

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

## Inverse

int nmod_poly_mat_inv(nmod_poly_mat_t Ainv, nmod_poly_t den, const nmod_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

slong nmod_poly_mat_nullspace(nmod_poly_mat_t res, const nmod_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

int nmod_poly_mat_solve(nmod_poly_mat_t X, nmod_poly_t den, const nmod_poly_mat_t A, const nmod_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.

int nmod_poly_mat_solve_fflu(nmod_poly_mat_t X, nmod_poly_t den, const nmod_poly_mat_t A, const nmod_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.

void nmod_poly_mat_solve_fflu_precomp(nmod_poly_mat_t X, const slong *perm, const nmod_poly_mat_t FFLU, const nmod_poly_mat_t B)

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