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 emph{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 emph{reduced} modulo \(p^N\) if it is canonical form and if all coefficients of \(M\) lie in the range \([0, p^{N-v})\).

Macros¶

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.

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.

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.

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.

slong padic_mat_get_val(const padic_mat_t A)¶: Returns the valuation of the matrix.

slong padic_mat_get_prec(const padic_mat_t A)¶: Returns the \(p\)-adic precision of the matrix.

slong padic_mat_val(const padic_mat_t A)

Returns the valuation of the matrix.

This is implemented as a macro and can be used as an emph{lvalue} as well as an emph{rvalue}.

slong padic_mat_nrows(const padic_mat_t A)¶: Returns the number of rows of the matrix \(A\).

slong padic_mat_ncols(const padic_mat_t A)¶: Returns the number of columns of the matrix \(A\).

Memory management¶

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.

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.

void padic_mat_clear(padic_mat_t A)¶: Clears the matrix \(A\).

void _padic_mat_canonicalise(padic_mat_t A, const padic_ctx_t ctx)¶: Ensures that the matrix \(A\) is in canonical form.

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.

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.

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.

int padic_mat_is_square(const padic_mat_t A)¶: Returns whether the matrix \(A\) is square.

int padic_mat_is_canonical(const padic_mat_t A, const fmpz_t p)¶: Returns whether the matrix \(A\) is in canonical form.

Basic assignment¶

void padic_mat_set(padic_mat_t B, const padic_mat_t A)¶: Sets \(B\) to a copy of \(A\), respecting the precision of \(B\).

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.

void padic_mat_zero(padic_mat_t A)¶: Sets the matrix \(A\) to zero.

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¶

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.

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.

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.

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¶

int padic_mat_equal(const padic_mat_t A, const padic_mat_t B)¶: Returns whether the two matrices \(A\) and \(B\) are equal.

int padic_mat_is_zero(const padic_mat_t A)¶: Returns whether the matrix \(A\) is zero.

Input and output¶

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 \(1\).

int padic_mat_fprint_pretty(FILE * file, const padic_mat_t A, const padic_ctx_t ctx)¶

Prints a emph{pretty} representation of the matrix \(A\) to the output stream file.

In the current implementation, always returns \(1\).

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¶

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 \([- \ceil{N/10}, N)\), \([N - \ceil{-N/10}, N)\), or \([-10, 0)\) as \(N\) is positive, negative or zero.

Transpose¶

void padic_mat_transpose(padic_mat_t B, const padic_mat_t A)¶: Sets \(B\) to \(A^t\).

Addition and subtraction¶

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.

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 \(p^N\).

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.

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.

void _padic_mat_neg(padic_mat_t B, const padic_mat_t A)¶: Sets \(B\) to \(-A\) in canonical form.

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¶

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.

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.

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.

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.

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¶

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.

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.