443 lines
17 KiB
Plaintext
443 lines
17 KiB
Plaintext
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/*=============================================================================
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This file is part of FLINT.
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FLINT is free software; you can redistribute it and/or modify
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it under the terms of the GNU General Public License as published by
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the Free Software Foundation; either version 2 of the License, or
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(at your option) any later version.
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FLINT is distributed in the hope that it will be useful,
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but WITHOUT ANY WARRANTY; without even the implied warranty of
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MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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GNU General Public License for more details.
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You should have received a copy of the GNU General Public License
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along with FLINT; if not, write to the Free Software
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Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA
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=============================================================================*/
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/******************************************************************************
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Copyright (C) 2013 Mike Hansen
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******************************************************************************/
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*******************************************************************************
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Memory management
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*******************************************************************************
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void fq_zech_mat_init(fq_zech_mat_t mat, slong rows, slong cols,
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const fq_zech_ctx_t ctx)
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Initialises \code{mat} to a \code{rows}-by-\code{cols} matrix with
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coefficients in $\mathbb{F}_{q}$ given by \code{ctx}. All elements
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are set to zero.
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void fq_zech_mat_init_set(fq_zech_mat_t mat, fq_zech_mat_t src,
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const fq_zech_ctx_t ctx)
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Initialises \code{mat} and sets its dimensions and elements to
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those of \code{src}.
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void fq_zech_mat_clear(fq_zech_mat_t mat, const fq_zech_ctx_t ctx)
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Clears the matrix and releases any memory it used. The matrix
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cannot be used again until it is initialised. This function must be
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called exactly once when finished using an \code{fq_zech_mat_t} object.
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void fq_zech_mat_set(fq_zech_mat_t mat, fq_zech_mat_t src,
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const fq_zech_ctx_t ctx)
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Sets \code{mat} to a copy of \code{src}. It is assumed
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that \code{mat} and \code{src} have identical dimensions.
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*******************************************************************************
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Basic properties and manipulation
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*******************************************************************************
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fq_zech_struct * fq_zech_mat_entry(fq_zech_mat_t mat, slong i, slong j)
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Directly accesses the entry in \code{mat} in row $i$ and column $j$,
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indexed from zero. No bounds checking is performed.
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fq_zech_struct * fq_zech_mat_entry_set(fq_zech_mat_t mat, slong i, slong j,
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fq_zech_t x, const fq_zech_ctx_t ctx)
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Sets the entry in \code{mat} in row $i$ and column $j$ to \code{x}.
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slong fq_zech_mat_nrows(fq_zech_mat_t mat, const fq_zech_ctx_t ctx)
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Returns the number of rows in \code{mat}.
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slong fq_zech_mat_ncols(fq_zech_mat_t mat, const fq_zech_ctx_t ctx)
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Returns the number of columns in \code{mat}.
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void fq_zech_mat_swap(fq_zech_mat_t mat1, fq_zech_mat_t mat2,
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const fq_zech_ctx_t ctx)
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Swaps two matrices. The dimensions of \code{mat1} and \code{mat2}
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are allowed to be different.
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void fq_zech_mat_zero(fq_zech_mat_t mat, const fq_zech_ctx_t ctx)
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Sets all entries of \code{mat} to 0.
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*******************************************************************************
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Printing
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*******************************************************************************
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void fq_zech_mat_print_pretty(const fq_zech_mat_t mat, const fq_zech_ctx_t ctx)
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Pretty-prints \code{mat} to \code{stdout}. A header is printed
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followed by the rows enclosed in brackets.
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int fq_zech_mat_fprint_pretty(FILE * file, const fq_zech_mat_t mat,
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const fq_zech_ctx_t ctx)
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Pretty-prints \code{mat} to \code{file}. A header is printed
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followed by the rows enclosed in brackets.
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In case of success, returns a positive value. In case of failure,
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returns a non-positive value.
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void fq_zech_mat_print(const fq_zech_mat_t mat, const fq_zech_ctx_t ctx)
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Prints \code{mat} to \code{stdout}. A header is printed followed
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by the rows enclosed in brackets.
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int fq_zech_mat_fprint(FILE * file, const fq_zech_mat_t mat,
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const fq_zech_ctx_t ctx)
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Prints \code{mat} to \code{file}. A header is printed followed by
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the rows enclosed in brackets.
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In case of success, returns a positive value. In case of failure,
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returns a non-positive value.
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*******************************************************************************
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Window
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*******************************************************************************
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void fq_zech_mat_window_init(fq_zech_mat_t window, const fq_zech_mat_t mat,
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slong r1, slong c1, slong r2, slong c2,
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const fq_zech_ctx_t ctx)
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Initializes the matrix \code{window} to be an \code{r2 - r1} by
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\code{c2 - c1} submatrix of \code{mat} whose \code{(0,0)} entry
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is the \code{(r1, c1)} entry of \code{mat}. The memory for the
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elements of \code{window} is shared with \code{mat}.
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void fq_zech_mat_window_clear(fq_zech_mat_t window, const fq_zech_ctx_t ctx)
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Clears the matrix \code{window} and releases any memory that it
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uses. Note that the memory to the underlying matrix that
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\code{window} points to is not freed.
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*******************************************************************************
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Random matrix generation
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*******************************************************************************
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void fq_zech_mat_randtest(fq_zech_mat_t mat, flint_rand_t state,
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const fq_zech_ctx_t ctx)
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Sets the elements of \code{mat} to random elements of
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$\mathbb{F}_{q}$, given by \code{ctx}.
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int fq_zech_mat_randpermdiag(fq_zech_mat_t mat, fq_zech_struct * diag, slong n,
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flint_rand_t state, const fq_zech_ctx_t ctx)
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Sets \code{mat} to a random permutation of the diagonal matrix
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with $n$ leading entries given by the vector \code{diag}. It is
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assumed that the main diagonal of \code{mat} has room for at
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least $n$ entries.
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Returns $0$ or $1$, depending on whether the permutation is even
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or odd respectively.
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void fq_zech_mat_randrank(fq_zech_mat_t mat, slong rank, flint_rand_t state,
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const fq_zech_ctx_t ctx)
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Sets \code{mat} to a random sparse matrix with the given rank,
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having exactly as many non-zero elements as the rank, with the
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non-zero elements being uniformly random elements of
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$\mathbb{F}_{q}$.
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The matrix can be transformed into a dense matrix with unchanged
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rank by subsequently calling \code{fq_zech_mat_randops()}.
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void fq_zech_mat_randops(fq_zech_mat_t mat, slong count, flint_rand_t state,
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const fq_zech_ctx_t ctx)
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Randomises \code{mat} by performing elementary row or column
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operations. More precisely, at most \code{count} random additions
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or subtractions of distinct rows and columns will be performed.
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This leaves the rank (and for square matrices, determinant)
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unchanged.
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void fq_zech_mat_randtril(fq_zech_mat_t mat, flint_rand_t state, int unit,
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const fq_zech_ctx_t ctx)
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Sets \code{mat} to a random lower triangular matrix. If
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\code{unit} is 1, it will have ones on the main diagonal,
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otherwise it will have random nonzero entries on the main
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diagonal.
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void fq_zech_mat_randtriu(fq_zech_mat_t mat, flint_rand_t state, int unit,
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x const fq_zech_ctx_t ctx)
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Sets \code{mat} to a random upper triangular matrix. If
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\code{unit} is 1, it will have ones on the main diagonal,
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otherwise it will have random nonzero entries on the main
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diagonal.
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*******************************************************************************
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Comparison
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*******************************************************************************
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int fq_zech_mat_equal(fq_zech_mat_t mat1, fq_zech_mat_t mat2,
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const fq_zech_ctx_t ctx)
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Returns nonzero if mat1 and mat2 have the same dimensions and elements,
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and zero otherwise.
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int fq_zech_mat_is_zero(const fq_zech_mat_t mat, const fq_zech_ctx_t ctx)
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Returns a non-zero value if all entries \code{mat} are zero, and
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otherwise returns zero.
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int fq_zech_mat_is_empty(const fq_zech_mat_t mat, const fq_zech_ctx_t ctx)
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Returns a non-zero value if the number of rows or the number of
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columns in \code{mat} is zero, and otherwise returns zero.
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int fq_zech_mat_is_square(const fq_zech_mat_t mat, const fq_zech_ctx_t ctx)
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Returns a non-zero value if the number of rows is equal to the
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number of columns in \code{mat}, and otherwise returns zero.
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*******************************************************************************
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Addition and subtraction
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*******************************************************************************
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void fq_zech_mat_add(fq_zech_mat_t C, const fq_zech_mat_t A,
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const fq_zech_mat_t B, const fq_zech_ctx_t ctx)
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Computes $C = A + B$. Dimensions must be identical.
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void fq_zech_mat_sub(fq_zech_mat_t C, const fq_zech_mat_t A,
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const fq_zech_mat_t B, const fq_zech_ctx_t ctx)
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Computes $C = A - B$. Dimensions must be identical.
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void fq_zech_mat_neg(fq_zech_mat_t A, const fq_zech_mat_t B,
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const fq_zech_ctx_t ctx)
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Sets $B = -A$. Dimensions must be identical.
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*******************************************************************************
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Matrix multiplication
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*******************************************************************************
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void fq_zech_mat_mul(fq_zech_mat_t C, const fq_zech_mat_t A,
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const fq_zech_mat_t B, const fq_zech_ctx_t ctx)
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Sets $C = AB$. Dimensions must be compatible for matrix
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multiplication. $C$ is not allowed to be aliased with $A$ or
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$B$. This function automatically chooses between classical and
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KS multiplication.
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void fq_zech_mat_mul_classical(fq_zech_mat_t C, const fq_zech_mat_t A,
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const fq_zech_mat_t B, const fq_zech_ctx_t ctx)
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Sets $C = AB$. Dimensions must be compatible for matrix multiplication.
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$C$ is not allowed to be aliased with $A$ or $B$. Uses classical
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matrix multiplication.
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void fq_zech_mat_mul_KS(fq_zech_mat_t C, const fq_zech_mat_t A,
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const fq_zech_mat_t B, const fq_zech_ctx_t ctx)
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Sets $C = AB$. Dimensions must be compatible for matrix
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multiplication. $C$ is not allowed to be aliased with $A$ or
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$B$. Uses Kronecker substitution to perform the multiplication
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over the integers.
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void fq_zech_mat_submul(fq_zech_mat_t D, const fq_zech_mat_t C,
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const fq_zech_mat_t A, const fq_zech_mat_t B,
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const fq_zech_ctx_t ctx)
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Sets $D = C + AB$. $C$ and $D$ may be aliased with each other but
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not with $A$ or $B$.
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*******************************************************************************
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LU decomposition
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*******************************************************************************
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slong fq_zech_mat_lu(slong * P, fq_zech_mat_t A, int rank_check,
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const fq_zech_ctx_t ctx)
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Computes a generalised LU decomposition $LU = PA$ of a given
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matrix $A$, returning the rank of $A$.
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If $A$ is a nonsingular square matrix, it will be overwritten with
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a unit diagonal lower triangular matrix $L$ and an upper
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triangular matrix $U$ (the diagonal of $L$ will not be stored
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explicitly).
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If $A$ is an arbitrary matrix of rank $r$, $U$ will be in row
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echelon form having $r$ nonzero rows, and $L$ will be lower
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triangular but truncated to $r$ columns, having implicit ones on
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the $r$ first entries of the main diagonal. All other entries will
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be zero.
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If a nonzero value for \code{rank_check} is passed, the function
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will abandon the output matrix in an undefined state and return 0
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if $A$ is detected to be rank-deficient.
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This function calls \code{fq_zech_mat_lu_recursive}.
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slong fq_zech_mat_lu_classical(slong * P, fq_zech_mat_t A, int rank_check,
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const fq_zech_ctx_t ctx)
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Computes a generalised LU decomposition $LU = PA$ of a given
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matrix $A$, returning the rank of $A$. The behavior of this
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function is identical to that of \code{fq_zech_mat_lu}. Uses Gaussian
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elimination.
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slong fq_zech_mat_lu_recursive(slong * P, fq_zech_mat_t A, int rank_check,
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const fq_zech_ctx_t ctx)
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Computes a generalised LU decomposition $LU = PA$ of a given
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matrix $A$, returning the rank of $A$. The behavior of this
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function is identical to that of \code{fq_zech_mat_lu}. Uses recursive
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block decomposition, switching to classical Gaussian elimination
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for sufficiently small blocks.
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*******************************************************************************
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Reduced row echelon form
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*******************************************************************************
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slong fq_zech_mat_rref(fq_zech_mat_t A, const fq_zech_ctx_t ctx)
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Puts $A$ in reduced row echelon form and returns the rank of $A$.
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The rref is computed by first obtaining an unreduced row echelon
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form via LU decomposition and then solving an additional
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triangular system.
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*******************************************************************************
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Triangular solving
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*******************************************************************************
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void fq_zech_mat_solve_tril(fq_zech_mat_t X, const fq_zech_mat_t L,
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const fq_zech_mat_t B, int unit,
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const fq_zech_ctx_t ctx)
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Sets $X = L^{-1} B$ where $L$ is a full rank lower triangular
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square matrix. If \code{unit} = 1, $L$ is assumed to have ones on
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its main diagonal, and the main diagonal will not be read. $X$
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and $B$ are allowed to be the same matrix, but no other aliasing
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is allowed. Automatically chooses between the classical and
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recursive algorithms.
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void fq_zech_mat_solve_tril_classical(fq_zech_mat_t X, const fq_zech_mat_t L,
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const fq_zech_mat_t B, int unit,
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const fq_zech_ctx_t ctx)
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Sets $X = L^{-1} B$ where $L$ is a full rank lower triangular
|
||
|
|
square matrix. If \code{unit} = 1, $L$ is assumed to have ones on
|
||
|
|
its main diagonal, and the main diagonal will not be read. $X$
|
||
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|
and $B$ are allowed to be the same matrix, but no other aliasing
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is allowed. Uses forward substitution.
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void fq_zech_mat_solve_tril_recursive(fq_zech_mat_t X, const fq_zech_mat_t L,
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const fq_zech_mat_t B, int unit,
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const fq_zech_ctx_t ctx)
|
||
|
|
|
||
|
|
Sets $X = L^{-1} B$ where $L$ is a full rank lower triangular
|
||
|
|
square matrix. If \code{unit} = 1, $L$ is assumed to have ones on
|
||
|
|
its main diagonal, and the main diagonal will not be read. $X$
|
||
|
|
and $B$ are allowed to be the same matrix, but no other aliasing
|
||
|
|
is allowed.
|
||
|
|
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||
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Uses the block inversion formula
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|
|
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||
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$$
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|
|
\begin{pmatrix} A & 0 \\ C & D \end{pmatrix}^{-1}
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\begin{pmatrix} X \\ Y \end{pmatrix} =
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|
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\begin{pmatrix} A^{-1} X \\ D^{-1} ( Y - C A^{-1} X ) \end{pmatrix}
|
||
|
|
$$
|
||
|
|
|
||
|
|
to reduce the problem to matrix multiplication and triangular
|
||
|
|
solving of smaller systems.
|
||
|
|
|
||
|
|
void fq_zech_mat_solve_triu(fq_zech_mat_t X, const fq_zech_mat_t U,
|
||
|
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const fq_zech_mat_t B, int unit,
|
||
|
|
const fq_zech_ctx_t ctx)
|
||
|
|
|
||
|
|
Sets $X = U^{-1} B$ where $U$ is a full rank upper triangular
|
||
|
|
square matrix. If \code{unit} = 1, $U$ is assumed to have ones on
|
||
|
|
its main diagonal, and the main diagonal will not be read. $X$
|
||
|
|
and $B$ are allowed to be the same matrix, but no other aliasing
|
||
|
|
is allowed. Automatically chooses between the classical and
|
||
|
|
recursive algorithms.
|
||
|
|
|
||
|
|
void fq_zech_mat_solve_triu_classical(fq_zech_mat_t X, const fq_zech_mat_t U,
|
||
|
|
const fq_zech_mat_t B, int unit,
|
||
|
|
const fq_zech_ctx_t ctx)
|
||
|
|
|
||
|
|
Sets $X = U^{-1} B$ where $U$ is a full rank upper triangular
|
||
|
|
square matrix. If \code{unit} = 1, $U$ is assumed to have ones on
|
||
|
|
its main diagonal, and the main diagonal will not be read. $X$
|
||
|
|
and $B$ are allowed to be the same matrix, but no other aliasing
|
||
|
|
is allowed. Uses forward substitution.
|
||
|
|
|
||
|
|
void fq_zech_mat_solve_triu_recursive(fq_zech_mat_t X, const fq_zech_mat_t U,
|
||
|
|
const fq_zech_mat_t B, int unit,
|
||
|
|
const fq_zech_ctx_t ctx)
|
||
|
|
|
||
|
|
Sets $X = U^{-1} B$ where $U$ is a full rank upper triangular
|
||
|
|
square matrix. If \code{unit} = 1, $U$ is assumed to have ones on
|
||
|
|
its main diagonal, and the main diagonal will not be read. $X$
|
||
|
|
and $B$ are allowed to be the same matrix, but no other aliasing
|
||
|
|
is allowed.
|
||
|
|
|
||
|
|
Uses the block inversion formula
|
||
|
|
|
||
|
|
$$
|
||
|
|
\begin{pmatrix} A & B \\ 0 & D \end{pmatrix}^{-1}
|
||
|
|
\begin{pmatrix} X \\ Y \end{pmatrix} =
|
||
|
|
\begin{pmatrix} A^{-1} (X - B D^{-1} Y) \\ D^{-1} Y \end{pmatrix}
|
||
|
|
$$
|
||
|
|
|
||
|
|
to reduce the problem to matrix multiplication and triangular
|
||
|
|
solving of smaller systems.
|