661 lines
15 KiB
C
661 lines
15 KiB
C
/* Integer matrix math routines
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Copyright (C) 2003, 2004, 2005 Free Software Foundation, Inc.
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Contributed by Daniel Berlin <dberlin@dberlin.org>.
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This file is part of GCC.
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GCC is free software; you can redistribute it and/or modify it under
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the terms of the GNU General Public License as published by the Free
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Software Foundation; either version 2, or (at your option) any later
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version.
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GCC is distributed in the hope that it will be useful, but WITHOUT ANY
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WARRANTY; without even the implied warranty of MERCHANTABILITY or
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FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
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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 GCC; see the file COPYING. If not, write to the Free
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Software Foundation, 51 Franklin Street, Fifth Floor, Boston, MA
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02110-1301, USA. */
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#include "config.h"
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#include "system.h"
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#include "coretypes.h"
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#include "tm.h"
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#include "ggc.h"
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#include "tree.h"
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#include "lambda.h"
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static void lambda_matrix_get_column (lambda_matrix, int, int,
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lambda_vector);
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/* Allocate a matrix of M rows x N cols. */
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lambda_matrix
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lambda_matrix_new (int m, int n)
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{
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lambda_matrix mat;
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int i;
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mat = ggc_alloc (m * sizeof (lambda_vector));
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for (i = 0; i < m; i++)
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mat[i] = lambda_vector_new (n);
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return mat;
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}
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/* Copy the elements of M x N matrix MAT1 to MAT2. */
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void
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lambda_matrix_copy (lambda_matrix mat1, lambda_matrix mat2,
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int m, int n)
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{
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int i;
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for (i = 0; i < m; i++)
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lambda_vector_copy (mat1[i], mat2[i], n);
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}
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/* Store the N x N identity matrix in MAT. */
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void
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lambda_matrix_id (lambda_matrix mat, int size)
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{
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int i, j;
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for (i = 0; i < size; i++)
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for (j = 0; j < size; j++)
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mat[i][j] = (i == j) ? 1 : 0;
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}
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/* Return true if MAT is the identity matrix of SIZE */
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bool
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lambda_matrix_id_p (lambda_matrix mat, int size)
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{
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int i, j;
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for (i = 0; i < size; i++)
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for (j = 0; j < size; j++)
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{
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if (i == j)
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{
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if (mat[i][j] != 1)
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return false;
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}
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else
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{
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if (mat[i][j] != 0)
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return false;
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}
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}
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return true;
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}
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/* Negate the elements of the M x N matrix MAT1 and store it in MAT2. */
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void
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lambda_matrix_negate (lambda_matrix mat1, lambda_matrix mat2, int m, int n)
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{
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int i;
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for (i = 0; i < m; i++)
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lambda_vector_negate (mat1[i], mat2[i], n);
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}
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/* Take the transpose of matrix MAT1 and store it in MAT2.
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MAT1 is an M x N matrix, so MAT2 must be N x M. */
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void
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lambda_matrix_transpose (lambda_matrix mat1, lambda_matrix mat2, int m, int n)
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{
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int i, j;
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for (i = 0; i < n; i++)
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for (j = 0; j < m; j++)
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mat2[i][j] = mat1[j][i];
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}
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/* Add two M x N matrices together: MAT3 = MAT1+MAT2. */
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void
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lambda_matrix_add (lambda_matrix mat1, lambda_matrix mat2,
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lambda_matrix mat3, int m, int n)
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{
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int i;
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for (i = 0; i < m; i++)
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lambda_vector_add (mat1[i], mat2[i], mat3[i], n);
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}
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/* MAT3 = CONST1 * MAT1 + CONST2 * MAT2. All matrices are M x N. */
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void
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lambda_matrix_add_mc (lambda_matrix mat1, int const1,
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lambda_matrix mat2, int const2,
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lambda_matrix mat3, int m, int n)
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{
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int i;
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for (i = 0; i < m; i++)
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lambda_vector_add_mc (mat1[i], const1, mat2[i], const2, mat3[i], n);
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}
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/* Multiply two matrices: MAT3 = MAT1 * MAT2.
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MAT1 is an M x R matrix, and MAT2 is R x N. The resulting MAT2
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must therefore be M x N. */
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void
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lambda_matrix_mult (lambda_matrix mat1, lambda_matrix mat2,
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lambda_matrix mat3, int m, int r, int n)
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{
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int i, j, k;
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for (i = 0; i < m; i++)
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{
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for (j = 0; j < n; j++)
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{
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mat3[i][j] = 0;
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for (k = 0; k < r; k++)
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mat3[i][j] += mat1[i][k] * mat2[k][j];
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}
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}
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}
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/* Get column COL from the matrix MAT and store it in VEC. MAT has
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N rows, so the length of VEC must be N. */
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static void
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lambda_matrix_get_column (lambda_matrix mat, int n, int col,
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lambda_vector vec)
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{
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int i;
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for (i = 0; i < n; i++)
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vec[i] = mat[i][col];
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}
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/* Delete rows r1 to r2 (not including r2). */
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void
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lambda_matrix_delete_rows (lambda_matrix mat, int rows, int from, int to)
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{
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int i;
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int dist;
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dist = to - from;
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for (i = to; i < rows; i++)
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mat[i - dist] = mat[i];
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for (i = rows - dist; i < rows; i++)
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mat[i] = NULL;
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}
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/* Swap rows R1 and R2 in matrix MAT. */
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void
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lambda_matrix_row_exchange (lambda_matrix mat, int r1, int r2)
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{
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lambda_vector row;
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row = mat[r1];
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mat[r1] = mat[r2];
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mat[r2] = row;
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}
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/* Add a multiple of row R1 of matrix MAT with N columns to row R2:
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R2 = R2 + CONST1 * R1. */
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void
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lambda_matrix_row_add (lambda_matrix mat, int n, int r1, int r2, int const1)
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{
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int i;
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if (const1 == 0)
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return;
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for (i = 0; i < n; i++)
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mat[r2][i] += const1 * mat[r1][i];
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}
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/* Negate row R1 of matrix MAT which has N columns. */
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void
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lambda_matrix_row_negate (lambda_matrix mat, int n, int r1)
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{
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lambda_vector_negate (mat[r1], mat[r1], n);
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}
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/* Multiply row R1 of matrix MAT with N columns by CONST1. */
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void
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lambda_matrix_row_mc (lambda_matrix mat, int n, int r1, int const1)
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{
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int i;
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for (i = 0; i < n; i++)
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mat[r1][i] *= const1;
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}
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/* Exchange COL1 and COL2 in matrix MAT. M is the number of rows. */
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void
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lambda_matrix_col_exchange (lambda_matrix mat, int m, int col1, int col2)
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{
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int i;
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int tmp;
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for (i = 0; i < m; i++)
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{
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tmp = mat[i][col1];
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mat[i][col1] = mat[i][col2];
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mat[i][col2] = tmp;
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}
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}
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/* Add a multiple of column C1 of matrix MAT with M rows to column C2:
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C2 = C2 + CONST1 * C1. */
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void
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lambda_matrix_col_add (lambda_matrix mat, int m, int c1, int c2, int const1)
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{
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int i;
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if (const1 == 0)
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return;
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for (i = 0; i < m; i++)
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mat[i][c2] += const1 * mat[i][c1];
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}
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/* Negate column C1 of matrix MAT which has M rows. */
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void
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lambda_matrix_col_negate (lambda_matrix mat, int m, int c1)
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{
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int i;
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for (i = 0; i < m; i++)
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mat[i][c1] *= -1;
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}
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/* Multiply column C1 of matrix MAT with M rows by CONST1. */
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void
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lambda_matrix_col_mc (lambda_matrix mat, int m, int c1, int const1)
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{
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int i;
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for (i = 0; i < m; i++)
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mat[i][c1] *= const1;
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}
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/* Compute the inverse of the N x N matrix MAT and store it in INV.
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We don't _really_ compute the inverse of MAT. Instead we compute
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det(MAT)*inv(MAT), and we return det(MAT) to the caller as the function
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result. This is necessary to preserve accuracy, because we are dealing
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with integer matrices here.
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The algorithm used here is a column based Gauss-Jordan elimination on MAT
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and the identity matrix in parallel. The inverse is the result of applying
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the same operations on the identity matrix that reduce MAT to the identity
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matrix.
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When MAT is a 2 x 2 matrix, we don't go through the whole process, because
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it is easily inverted by inspection and it is a very common case. */
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static int lambda_matrix_inverse_hard (lambda_matrix, lambda_matrix, int);
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int
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lambda_matrix_inverse (lambda_matrix mat, lambda_matrix inv, int n)
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{
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if (n == 2)
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{
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int a, b, c, d, det;
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a = mat[0][0];
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b = mat[1][0];
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c = mat[0][1];
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d = mat[1][1];
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inv[0][0] = d;
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inv[0][1] = -c;
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inv[1][0] = -b;
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inv[1][1] = a;
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det = (a * d - b * c);
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if (det < 0)
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{
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det *= -1;
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inv[0][0] *= -1;
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inv[1][0] *= -1;
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inv[0][1] *= -1;
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inv[1][1] *= -1;
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}
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return det;
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}
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else
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return lambda_matrix_inverse_hard (mat, inv, n);
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}
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/* If MAT is not a special case, invert it the hard way. */
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static int
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lambda_matrix_inverse_hard (lambda_matrix mat, lambda_matrix inv, int n)
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{
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lambda_vector row;
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lambda_matrix temp;
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int i, j;
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int determinant;
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temp = lambda_matrix_new (n, n);
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lambda_matrix_copy (mat, temp, n, n);
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lambda_matrix_id (inv, n);
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/* Reduce TEMP to a lower triangular form, applying the same operations on
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INV which starts as the identity matrix. N is the number of rows and
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columns. */
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for (j = 0; j < n; j++)
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{
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row = temp[j];
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/* Make every element in the current row positive. */
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for (i = j; i < n; i++)
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if (row[i] < 0)
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{
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lambda_matrix_col_negate (temp, n, i);
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lambda_matrix_col_negate (inv, n, i);
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}
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/* Sweep the upper triangle. Stop when only the diagonal element in the
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current row is nonzero. */
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while (lambda_vector_first_nz (row, n, j + 1) < n)
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{
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int min_col = lambda_vector_min_nz (row, n, j);
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lambda_matrix_col_exchange (temp, n, j, min_col);
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lambda_matrix_col_exchange (inv, n, j, min_col);
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for (i = j + 1; i < n; i++)
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{
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int factor;
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factor = -1 * row[i];
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if (row[j] != 1)
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factor /= row[j];
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lambda_matrix_col_add (temp, n, j, i, factor);
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lambda_matrix_col_add (inv, n, j, i, factor);
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}
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}
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}
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/* Reduce TEMP from a lower triangular to the identity matrix. Also compute
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the determinant, which now is simply the product of the elements on the
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diagonal of TEMP. If one of these elements is 0, the matrix has 0 as an
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eigenvalue so it is singular and hence not invertible. */
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determinant = 1;
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for (j = n - 1; j >= 0; j--)
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{
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int diagonal;
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row = temp[j];
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diagonal = row[j];
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/* The matrix must not be singular. */
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gcc_assert (diagonal);
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determinant = determinant * diagonal;
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/* If the diagonal is not 1, then multiply the each row by the
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diagonal so that the middle number is now 1, rather than a
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rational. */
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if (diagonal != 1)
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{
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for (i = 0; i < j; i++)
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lambda_matrix_col_mc (inv, n, i, diagonal);
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for (i = j + 1; i < n; i++)
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lambda_matrix_col_mc (inv, n, i, diagonal);
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row[j] = diagonal = 1;
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}
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/* Sweep the lower triangle column wise. */
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for (i = j - 1; i >= 0; i--)
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{
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if (row[i])
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{
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int factor = -row[i];
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lambda_matrix_col_add (temp, n, j, i, factor);
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lambda_matrix_col_add (inv, n, j, i, factor);
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}
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}
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}
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return determinant;
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}
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/* Decompose a N x N matrix MAT to a product of a lower triangular H
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and a unimodular U matrix such that MAT = H.U. N is the size of
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the rows of MAT. */
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void
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lambda_matrix_hermite (lambda_matrix mat, int n,
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lambda_matrix H, lambda_matrix U)
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{
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lambda_vector row;
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int i, j, factor, minimum_col;
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lambda_matrix_copy (mat, H, n, n);
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lambda_matrix_id (U, n);
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for (j = 0; j < n; j++)
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{
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row = H[j];
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/* Make every element of H[j][j..n] positive. */
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for (i = j; i < n; i++)
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{
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if (row[i] < 0)
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{
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lambda_matrix_col_negate (H, n, i);
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lambda_vector_negate (U[i], U[i], n);
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}
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}
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/* Stop when only the diagonal element is nonzero. */
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while (lambda_vector_first_nz (row, n, j + 1) < n)
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{
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minimum_col = lambda_vector_min_nz (row, n, j);
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lambda_matrix_col_exchange (H, n, j, minimum_col);
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lambda_matrix_row_exchange (U, j, minimum_col);
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for (i = j + 1; i < n; i++)
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{
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factor = row[i] / row[j];
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lambda_matrix_col_add (H, n, j, i, -1 * factor);
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lambda_matrix_row_add (U, n, i, j, factor);
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}
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}
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}
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}
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/* Given an M x N integer matrix A, this function determines an M x
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M unimodular matrix U, and an M x N echelon matrix S such that
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"U.A = S". This decomposition is also known as "right Hermite".
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Ref: Algorithm 2.1 page 33 in "Loop Transformations for
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Restructuring Compilers" Utpal Banerjee. */
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void
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lambda_matrix_right_hermite (lambda_matrix A, int m, int n,
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lambda_matrix S, lambda_matrix U)
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{
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int i, j, i0 = 0;
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lambda_matrix_copy (A, S, m, n);
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lambda_matrix_id (U, m);
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for (j = 0; j < n; j++)
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{
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if (lambda_vector_first_nz (S[j], m, i0) < m)
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{
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++i0;
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for (i = m - 1; i >= i0; i--)
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{
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while (S[i][j] != 0)
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{
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int sigma, factor, a, b;
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a = S[i-1][j];
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b = S[i][j];
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sigma = (a * b < 0) ? -1: 1;
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a = abs (a);
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b = abs (b);
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factor = sigma * (a / b);
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lambda_matrix_row_add (S, n, i, i-1, -factor);
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lambda_matrix_row_exchange (S, i, i-1);
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lambda_matrix_row_add (U, m, i, i-1, -factor);
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lambda_matrix_row_exchange (U, i, i-1);
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}
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}
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}
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}
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}
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/* Given an M x N integer matrix A, this function determines an M x M
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unimodular matrix V, and an M x N echelon matrix S such that "A =
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V.S". This decomposition is also known as "left Hermite".
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Ref: Algorithm 2.2 page 36 in "Loop Transformations for
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Restructuring Compilers" Utpal Banerjee. */
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void
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lambda_matrix_left_hermite (lambda_matrix A, int m, int n,
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lambda_matrix S, lambda_matrix V)
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{
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int i, j, i0 = 0;
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lambda_matrix_copy (A, S, m, n);
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lambda_matrix_id (V, m);
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for (j = 0; j < n; j++)
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{
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if (lambda_vector_first_nz (S[j], m, i0) < m)
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{
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++i0;
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for (i = m - 1; i >= i0; i--)
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{
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while (S[i][j] != 0)
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{
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int sigma, factor, a, b;
|
|
|
|
a = S[i-1][j];
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|
b = S[i][j];
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|
sigma = (a * b < 0) ? -1: 1;
|
|
a = abs (a);
|
|
b = abs (b);
|
|
factor = sigma * (a / b);
|
|
|
|
lambda_matrix_row_add (S, n, i, i-1, -factor);
|
|
lambda_matrix_row_exchange (S, i, i-1);
|
|
|
|
lambda_matrix_col_add (V, m, i-1, i, factor);
|
|
lambda_matrix_col_exchange (V, m, i, i-1);
|
|
}
|
|
}
|
|
}
|
|
}
|
|
}
|
|
|
|
/* When it exists, return the first nonzero row in MAT after row
|
|
STARTROW. Otherwise return rowsize. */
|
|
|
|
int
|
|
lambda_matrix_first_nz_vec (lambda_matrix mat, int rowsize, int colsize,
|
|
int startrow)
|
|
{
|
|
int j;
|
|
bool found = false;
|
|
|
|
for (j = startrow; (j < rowsize) && !found; j++)
|
|
{
|
|
if ((mat[j] != NULL)
|
|
&& (lambda_vector_first_nz (mat[j], colsize, startrow) < colsize))
|
|
found = true;
|
|
}
|
|
|
|
if (found)
|
|
return j - 1;
|
|
return rowsize;
|
|
}
|
|
|
|
/* Calculate the projection of E sub k to the null space of B. */
|
|
|
|
void
|
|
lambda_matrix_project_to_null (lambda_matrix B, int rowsize,
|
|
int colsize, int k, lambda_vector x)
|
|
{
|
|
lambda_matrix M1, M2, M3, I;
|
|
int determinant;
|
|
|
|
/* Compute c(I-B^T inv(B B^T) B) e sub k. */
|
|
|
|
/* M1 is the transpose of B. */
|
|
M1 = lambda_matrix_new (colsize, colsize);
|
|
lambda_matrix_transpose (B, M1, rowsize, colsize);
|
|
|
|
/* M2 = B * B^T */
|
|
M2 = lambda_matrix_new (colsize, colsize);
|
|
lambda_matrix_mult (B, M1, M2, rowsize, colsize, rowsize);
|
|
|
|
/* M3 = inv(M2) */
|
|
M3 = lambda_matrix_new (colsize, colsize);
|
|
determinant = lambda_matrix_inverse (M2, M3, rowsize);
|
|
|
|
/* M2 = B^T (inv(B B^T)) */
|
|
lambda_matrix_mult (M1, M3, M2, colsize, rowsize, rowsize);
|
|
|
|
/* M1 = B^T (inv(B B^T)) B */
|
|
lambda_matrix_mult (M2, B, M1, colsize, rowsize, colsize);
|
|
lambda_matrix_negate (M1, M1, colsize, colsize);
|
|
|
|
I = lambda_matrix_new (colsize, colsize);
|
|
lambda_matrix_id (I, colsize);
|
|
|
|
lambda_matrix_add_mc (I, determinant, M1, 1, M2, colsize, colsize);
|
|
|
|
lambda_matrix_get_column (M2, colsize, k - 1, x);
|
|
|
|
}
|
|
|
|
/* Multiply a vector VEC by a matrix MAT.
|
|
MAT is an M*N matrix, and VEC is a vector with length N. The result
|
|
is stored in DEST which must be a vector of length M. */
|
|
|
|
void
|
|
lambda_matrix_vector_mult (lambda_matrix matrix, int m, int n,
|
|
lambda_vector vec, lambda_vector dest)
|
|
{
|
|
int i, j;
|
|
|
|
lambda_vector_clear (dest, m);
|
|
for (i = 0; i < m; i++)
|
|
for (j = 0; j < n; j++)
|
|
dest[i] += matrix[i][j] * vec[j];
|
|
}
|
|
|
|
/* Print out an M x N matrix MAT to OUTFILE. */
|
|
|
|
void
|
|
print_lambda_matrix (FILE * outfile, lambda_matrix matrix, int m, int n)
|
|
{
|
|
int i;
|
|
|
|
for (i = 0; i < m; i++)
|
|
print_lambda_vector (outfile, matrix[i], n);
|
|
fprintf (outfile, "\n");
|
|
}
|
|
|