# bool_mat.h – matrices over booleans

A `bool_mat_t` represents a dense matrix over the boolean semiring \(\langle \left\{0, 1\right\}, \vee, \wedge \rangle\), implemented as an array of entries of type `int`.

The dimension (number of rows and columns) of a matrix is fixed at initialization, and the user must ensure that inputs and outputs to an operation have compatible dimensions. The number of rows or columns in a matrix can be zero.

## Types, macros and constants

type bool_mat_struct

type bool_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 bool_mat_t is defined as an array of length one of type bool_mat_struct, permitting an bool_mat_t to be passed by reference.

int bool_mat_get_entry(const bool_mat_t mat, slong i, slong j)

  Returns the entry of matrix mat at row i and column j.

void bool_mat_set_entry(bool_mat_t mat, slong i, slong j, int x)

  Sets the entry of matrix mat at row i and column j to x.

bool_mat_nrows(mat)

  Returns the number of rows of the matrix.

bool_mat_ncols(mat)

  Returns the number of columns of the matrix.

## Memory management

void bool_mat_init(bool_mat_t mat, slong r, slong c)

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

void bool_mat_clear(bool_mat_t mat)

  Clears the matrix, deallocating all entries.

int bool_mat_is_empty(const bool_mat_t mat)

  Returns nonzero iff the number of rows or the number of columns in mat is zero. Note that this does not depend on the entry values of mat.

int bool_mat_is_square(const bool_mat_t mat)

  Returns nonzero iff the number of rows is equal to the number of columns in mat.

## Conversions

void bool_mat_set(bool_mat_t dest, const bool_mat_t src)

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

## Input and output

void bool_mat_print(const bool_mat_t mat)

  Prints each entry in the matrix.

void bool_mat_fprint(FILE *file, const bool_mat_t mat)

  Prints each entry in the matrix to the stream file.

## Value comparisons

int bool_mat_equal(const bool_mat_t mat1, const bool_mat_t mat2)

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

int bool_mat_any(const bool_mat_t mat)

  Returns nonzero iff mat has a nonzero entry.

int bool_mat_all(const bool_mat_t mat)

  Returns nonzero iff all entries of mat are nonzero.

int bool_mat_is_diagonal(const bool_mat_t A)

  Returns nonzero iff \(i \ne j \implies \bar{A_{ij}}\).

int bool_mat_is_lower_triangular(const bool_mat_t A)

  Returns nonzero iff \(i < j \implies \bar{A_{ij}}\).

int bool_mat_is_transitive(const bool_mat_t mat)

  Returns nonzero iff \(A_{ij} \wedge A_{jk} \implies A_{ik}\).

int bool_mat_is_nilpotent(const bool_mat_t A)

  Returns nonzero iff some positive matrix power of \(A\) is zero.

## Random generation

void bool_mat_randtest(bool_mat_t mat, flint_rand_t state)

  Sets mat to a random matrix.

void bool_mat_randtest_diagonal(bool_mat_t mat, flint_rand_t state)

  Sets mat to a random diagonal matrix.

void bool_mat_randtest_nilpotent(bool_mat_t mat, flint_rand_t state)

  Sets mat to a random nilpotent matrix.

## Special matrices

void bool_mat_zero(bool_mat_t mat)

  Sets all entries in mat to zero.

void bool_mat_one(bool_mat_t mat)

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

void bool_mat_directed_path(bool_mat_t A)

  Sets \(A_{ij}\) to \(j = i + 1\). Requires that \(A\) is a square matrix.

void bool_mat_directed_cycle(bool_mat_t A)

  Sets \(A_{ij}\) to \(j = (i + 1) \mod n\) where \(n\) is the order of the square matrix \(A\).

## Transpose

void bool_mat_transpose(bool_mat_t dest, const bool_mat_t src)

  Sets dest to the transpose of src. The operands must have compatible dimensions. Aliasing is allowed for square matrices.

## Arithmetic

void bool_mat_complement(bool_mat_t B, const bool_mat_t A)

  Sets B to the logical complement of A. That is \(B_{ij}\) is set to \(\bar{A_{ij}}\). The operands must have the same dimensions.

void bool_mat_add(bool_mat_t res, const bool_mat_t mat1, const bool_mat_t mat2)

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

void bool_mat_mul(bool_mat_t res, const bool_mat_t mat1, const bool_mat_t mat2)

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

void bool_mat_mul_entrywise(bool_mat_t res, const bool_mat_t mat1, const bool_mat_t mat2)

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

void bool_mat_sqr(bool_mat_t B, const bool_mat_t A)

  Sets B to the matrix square of A. The operands must both be square with the same dimensions.

void bool_mat_pow_ui(bool_mat_t B, const bool_mat_t A, ulong exp)

  Sets B to A raised to the power exp. Requires that A is a square matrix.

## Special functions

int bool_mat_trace(const bool_mat_t mat)

  Returns the trace of the matrix, i.e. the sum of entries on the main diagonal of mat. The matrix is required to be square. The sum is in the boolean semiring, so this function returns nonzero iff any entry on the diagonal of mat is nonzero.

slong bool_mat_nilpotency_degree(const bool_mat_t A)

  Returns the nilpotency degree of the \(n \times n\) matrix A. It returns the smallest positive \(k\) such that \(A^k = 0\). If no such \(k\) exists then the function returns \(-1\) if \(n\) is positive, and otherwise it returns \(0\).

void bool_mat_transitive_closure(bool_mat_t B, const bool_mat_t A)

  Sets B to the transitive closure \(\sum_{k=1}^\infty A^k\). The matrix A is required to be square.

slong bool_mat_get_strongly_connected_components(slong *p, const bool_mat_t A)

  Partitions the \(n\) row and column indices of the \(n \times n\) matrix A according to the strongly connected components (SCC) of the graph for which A is the adjacency matrix. If the graph has \(k\) SCCs then the function returns \(k\), and for each vertex \(i \in [0, n-1]\), \(p_i\) is set to the index of the SCC to which the vertex belongs. The SCCs themselves can be considered as nodes in a directed acyclic graph (DAG), and the SCCs are indexed in postorder with respect to that DAG.

slong bool_mat_all_pairs_longest_walk(fmpz_mat_t B, const bool_mat_t A)

  Sets \(B_{ij}\) to the length of the longest walk with endpoint vertices \(i\) and \(j\) in the graph whose adjacency matrix is A. The matrix A must be square. Empty walks with zero length which begin and end at the same vertex are allowed. If \(j\) is not reachable from \(i\) then no walk from \(i\) to \(j\) exists and \(B_{ij}\) is set to the special value \(-1\). If arbitrarily long walks from \(i\) to \(j\) exist then \(B_{ij}\) is set to the special value \(-2\).

  The function returns \(-2\) if any entry of \(B_{ij}\) is \(-2\), and otherwise it returns the maximum entry in \(B\), except if \(A\) is empty in which case \(-1\) is returned. Note that the returned value is one less than that of `nilpotency_degree()`.

  This function can help quantify entrywise errors in a truncated evaluation of a matrix power series. If A is an indicator matrix with the same sparsity pattern as a matrix \(M\) over the real or complex numbers, and if \(B_{ij}\) does not take the special value \(-2\), then the tail \(\left[ \sum_{k=N}^\infty a_k M^k \right]_{ij}\) vanishes when \(N > B_{ij}\).