.. _padic-mat: **padic_mat.h** -- matrices over p-adic numbers =============================================================================== Module documentation -------------------------------------------------------------------------------- We represent a matrix over `\mathbf{Q}_p` as a product `p^v M`, where `p` is a prime number, `v \in \mathbf{Z}` and `M` a matrix over `\mathbf{Z}`. We say this matrix is in *canonical form* if either `M` is zero, in which case we choose `v = 0`, too, or if `M` contains at least one `p`-adic unit. We say this matrix is *reduced* modulo `p^N` if it is canonical form and if all coefficients of `M` lie in the range `[0, p^{N-v})`. Macros -------------------------------------------------------------------------------- .. function:: fmpz_mat_struct * padic_mat(const padic_mat_t A) Returns a pointer to the unit part of the matrix, which is a matrix over `\mathbf{Z}`. The return value can be used as an argument to the functions in the ``fmpz_mat`` module. .. function:: fmpz * padic_mat_entry(const padic_mat_t A, slong i, slong j) Returns a pointer to unit part of the entry in position `(i, j)`. Note that this is not necessarily a unit. The return value can be used as an argument to the functions in the ``fmpz`` module. .. function:: slong padic_mat_val(const padic_mat_t A) Allow access (as L-value or R-value) to ``val`` field of `A`. This function is implemented as a macro. .. function:: slong padic_mat_prec(const padic_mat_t A) Allow access (as L-value or R-value) to ``prec`` field of `A`. This function is implemented as a macro. .. function:: slong padic_mat_get_val(const padic_mat_t A) Returns the valuation of the matrix. .. function:: slong padic_mat_get_prec(const padic_mat_t A) Returns the `p`-adic precision of the matrix. .. function:: slong padic_mat_nrows(const padic_mat_t A) Returns the number of rows of the matrix `A`. .. function:: slong padic_mat_ncols(const padic_mat_t A) Returns the number of columns of the matrix `A`. Memory management -------------------------------------------------------------------------------- .. function:: void padic_mat_init(padic_mat_t A, slong r, slong c) Initialises the matrix `A` as a zero matrix with the specified numbers of rows and columns and precision ``PADIC_DEFAULT_PREC``. .. function:: void padic_mat_init2(padic_mat_t A, slong r, slong c, slong prec) Initialises the matrix `A` as a zero matrix with the specified numbers of rows and columns and the given precision. .. function:: void padic_mat_clear(padic_mat_t A) Clears the matrix `A`. .. function:: void _padic_mat_canonicalise(padic_mat_t A, const padic_ctx_t ctx) Ensures that the matrix `A` is in canonical form. .. function:: void _padic_mat_reduce(padic_mat_t A, const padic_ctx_t ctx) Ensures that the matrix `A` is reduced modulo `p^N`, assuming that it is in canonical form already. .. function:: void padic_mat_reduce(padic_mat_t A, const padic_ctx_t ctx) Ensures that the matrix `A` is reduced modulo `p^N`, without assuming that it is necessarily in canonical form. .. function:: int padic_mat_is_empty(const padic_mat_t A) Returns whether the matrix `A` is empty, that is, whether it has zero rows or zero columns. .. function:: int padic_mat_is_square(const padic_mat_t A) Returns whether the matrix `A` is square. .. function:: int padic_mat_is_canonical(const padic_mat_t A, const padic_ctx_t p) Returns whether the matrix `A` is in canonical form. Basic assignment -------------------------------------------------------------------------------- .. function:: void padic_mat_set(padic_mat_t B, const padic_mat_t A, const padic_ctx_t p) Sets `B` to a copy of `A`, respecting the precision of `B`. .. function:: void padic_mat_swap(padic_mat_t A, padic_mat_t B) Swaps the two matrices `A` and `B`. This is done efficiently by swapping pointers. .. function:: void padic_mat_swap_entrywise(padic_mat_t mat1, padic_mat_t mat2) Swaps two matrices by swapping the individual entries rather than swapping the contents of the structs. .. function:: void padic_mat_zero(padic_mat_t A) Sets the matrix `A` to zero. .. function:: void padic_mat_one(padic_mat_t A) Sets the matrix `A` to the identity matrix. If the precision is negative then the matrix will be the zero matrix. Conversions -------------------------------------------------------------------------------- .. function:: void padic_mat_set_fmpq_mat(padic_mat_t B, const fmpq_mat_t A, const padic_ctx_t ctx) Sets the `p`-adic matrix `B` to the rational matrix `A`, reduced according to the given context. .. function:: void padic_mat_get_fmpq_mat(fmpq_mat_t B, const padic_mat_t A, const padic_ctx_t ctx) Sets the rational matrix `B` to the `p`-adic matrices `A`; no reduction takes place. Entries -------------------------------------------------------------------------------- Because of the choice of the data structure, representing the matrix as `p^v M`, setting an entry of the matrix might lead to changes in all entries in the matrix `M`. Also, a specific entry is not readily available as a `p`-adic number. Thus, there are separate functions available for getting and setting entries. .. function:: void padic_mat_get_entry_padic(padic_t rop, const padic_mat_t op, slong i, slong j, const padic_ctx_t ctx) Sets ``rop`` to the entry in position `(i, j)` in the matrix ``op``. .. function:: void padic_mat_set_entry_padic(padic_mat_t rop, slong i, slong j, const padic_t op, const padic_ctx_t ctx) Sets the entry in position `(i, j)` in the matrix to ``rop``. Comparison -------------------------------------------------------------------------------- .. function:: int padic_mat_equal(const padic_mat_t A, const padic_mat_t B) Returns whether the two matrices `A` and `B` are equal. .. function:: int padic_mat_is_zero(const padic_mat_t A) Returns whether the matrix `A` is zero. Input and output -------------------------------------------------------------------------------- .. function:: int padic_mat_fprint(FILE * file, const padic_mat_t A, const padic_ctx_t ctx) Prints a simple representation of the matrix `A` to the output stream ``file``. The format is the number of rows, a space, the number of columns, two spaces, followed by a list of all the entries, one row after the other. In the current implementation, always returns `1`. .. function:: int padic_mat_fprint_pretty(FILE * file, const padic_mat_t A, const padic_ctx_t ctx) Prints a *pretty* representation of the matrix `A` to the output stream ``file``. In the current implementation, always returns `1`. .. function:: int padic_mat_print(const padic_mat_t A, const padic_ctx_t ctx) int padic_mat_print_pretty(const padic_mat_t A, const padic_ctx_t ctx) Random matrix generation -------------------------------------------------------------------------------- .. function:: void padic_mat_randtest(padic_mat_t A, flint_rand_t state, const padic_ctx_t ctx) Sets `A` to a random matrix. The valuation will be in the range `[- \lceil N/10\rceil, N)`, `[N - \lceil -N/10\rceil, N)`, or `[-10, 0)` as `N` is positive, negative or zero. Transpose -------------------------------------------------------------------------------- .. function:: void padic_mat_transpose(padic_mat_t B, const padic_mat_t A) Sets `B` to `A^t`. Addition and subtraction -------------------------------------------------------------------------------- .. function:: void _padic_mat_add(padic_mat_t C, const padic_mat_t A, const padic_mat_t B, const padic_ctx_t ctx) Sets `C` to the exact sum `A + B`, ensuring that the result is in canonical form. .. function:: void padic_mat_add(padic_mat_t C, const padic_mat_t A, const padic_mat_t B, const padic_ctx_t ctx) Sets `C` to the sum `A + B` modulo `p^N`. .. function:: void _padic_mat_sub(padic_mat_t C, const padic_mat_t A, const padic_mat_t B, const padic_ctx_t ctx) Sets `C` to the exact difference `A - B`, ensuring that the result is in canonical form. .. function:: void padic_mat_sub(padic_mat_t C, const padic_mat_t A, const padic_mat_t B, const padic_ctx_t ctx) Sets `C` to `A - B`, ensuring that the result is reduced. .. function:: void _padic_mat_neg(padic_mat_t B, const padic_mat_t A) Sets `B` to `-A` in canonical form. .. function:: void padic_mat_neg(padic_mat_t B, const padic_mat_t A, const padic_ctx_t ctx) Sets `B` to `-A`, ensuring the result is reduced. Scalar operations -------------------------------------------------------------------------------- .. function:: void _padic_mat_scalar_mul_padic(padic_mat_t B, const padic_mat_t A, const padic_t c, const padic_ctx_t ctx) Sets `B` to `c A`, ensuring that the result is in canonical form. .. function:: void padic_mat_scalar_mul_padic(padic_mat_t B, const padic_mat_t A, const padic_t c, const padic_ctx_t ctx) Sets `B` to `c A`, ensuring that the result is reduced. .. function:: void _padic_mat_scalar_mul_fmpz(padic_mat_t B, const padic_mat_t A, const fmpz_t c, const padic_ctx_t ctx) Sets `B` to `c A`, ensuring that the result is in canonical form. .. function:: void padic_mat_scalar_mul_fmpz(padic_mat_t B, const padic_mat_t A, const fmpz_t c, const padic_ctx_t ctx) Sets `B` to `c A`, ensuring that the result is reduced. .. function:: void padic_mat_scalar_div_fmpz(padic_mat_t B, const padic_mat_t A, const fmpz_t c, const padic_ctx_t ctx) Sets `B` to `c^{-1} A`, assuming that `c \neq 0`. Ensures that the result `B` is reduced. Multiplication -------------------------------------------------------------------------------- .. function:: void _padic_mat_mul(padic_mat_t C, const padic_mat_t A, const padic_mat_t B, const padic_ctx_t ctx) Sets `C` to the product `A B` of the two matrices `A` and `B`, ensuring that `C` is in canonical form. .. function:: void padic_mat_mul(padic_mat_t C, const padic_mat_t A, const padic_mat_t B, const padic_ctx_t ctx) Sets `C` to the product `A B` of the two matrices `A` and `B`, ensuring that `C` is reduced.