diff -r d59bea55db9b -r c445c931472f src/glpluf.c --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/src/glpluf.c Mon Dec 06 13:09:21 2010 +0100 @@ -0,0 +1,1846 @@ +/* glpluf.c (LU-factorization) */ + +/*********************************************************************** +* This code is part of GLPK (GNU Linear Programming Kit). +* +* Copyright (C) 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, +* 2009, 2010 Andrew Makhorin, Department for Applied Informatics, +* Moscow Aviation Institute, Moscow, Russia. All rights reserved. +* E-mail: . +* +* GLPK is free software: you can redistribute it and/or modify it +* under the terms of the GNU General Public License as published by +* the Free Software Foundation, either version 3 of the License, or +* (at your option) any later version. +* +* GLPK is distributed in the hope that it will be useful, but WITHOUT +* ANY WARRANTY; without even the implied warranty of MERCHANTABILITY +* or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public +* License for more details. +* +* You should have received a copy of the GNU General Public License +* along with GLPK. If not, see . +***********************************************************************/ + +#include "glpenv.h" +#include "glpluf.h" +#define xfault xerror + +/* CAUTION: DO NOT CHANGE THE LIMIT BELOW */ + +#define N_MAX 100000000 /* = 100*10^6 */ +/* maximal order of the original matrix */ + +/*********************************************************************** +* NAME +* +* luf_create_it - create LU-factorization +* +* SYNOPSIS +* +* #include "glpluf.h" +* LUF *luf_create_it(void); +* +* DESCRIPTION +* +* The routine luf_create_it creates a program object, which represents +* LU-factorization of a square matrix. +* +* RETURNS +* +* The routine luf_create_it returns a pointer to the object created. */ + +LUF *luf_create_it(void) +{ LUF *luf; + luf = xmalloc(sizeof(LUF)); + luf->n_max = luf->n = 0; + luf->valid = 0; + luf->fr_ptr = luf->fr_len = NULL; + luf->fc_ptr = luf->fc_len = NULL; + luf->vr_ptr = luf->vr_len = luf->vr_cap = NULL; + luf->vr_piv = NULL; + luf->vc_ptr = luf->vc_len = luf->vc_cap = NULL; + luf->pp_row = luf->pp_col = NULL; + luf->qq_row = luf->qq_col = NULL; + luf->sv_size = 0; + luf->sv_beg = luf->sv_end = 0; + luf->sv_ind = NULL; + luf->sv_val = NULL; + luf->sv_head = luf->sv_tail = 0; + luf->sv_prev = luf->sv_next = NULL; + luf->vr_max = NULL; + luf->rs_head = luf->rs_prev = luf->rs_next = NULL; + luf->cs_head = luf->cs_prev = luf->cs_next = NULL; + luf->flag = NULL; + luf->work = NULL; + luf->new_sva = 0; + luf->piv_tol = 0.10; + luf->piv_lim = 4; + luf->suhl = 1; + luf->eps_tol = 1e-15; + luf->max_gro = 1e+10; + luf->nnz_a = luf->nnz_f = luf->nnz_v = 0; + luf->max_a = luf->big_v = 0.0; + luf->rank = 0; + return luf; +} + +/*********************************************************************** +* NAME +* +* luf_defrag_sva - defragment the sparse vector area +* +* SYNOPSIS +* +* #include "glpluf.h" +* void luf_defrag_sva(LUF *luf); +* +* DESCRIPTION +* +* The routine luf_defrag_sva defragments the sparse vector area (SVA) +* gathering all unused locations in one continuous extent. In order to +* do that the routine moves all unused locations from the left part of +* SVA (which contains rows and columns of the matrix V) to the middle +* part (which contains free locations). This is attained by relocating +* elements of rows and columns of the matrix V toward the beginning of +* the left part. +* +* NOTE that this "garbage collection" involves changing row and column +* pointers of the matrix V. */ + +void luf_defrag_sva(LUF *luf) +{ int n = luf->n; + int *vr_ptr = luf->vr_ptr; + int *vr_len = luf->vr_len; + int *vr_cap = luf->vr_cap; + int *vc_ptr = luf->vc_ptr; + int *vc_len = luf->vc_len; + int *vc_cap = luf->vc_cap; + int *sv_ind = luf->sv_ind; + double *sv_val = luf->sv_val; + int *sv_next = luf->sv_next; + int sv_beg = 1; + int i, j, k; + /* skip rows and columns, which do not need to be relocated */ + for (k = luf->sv_head; k != 0; k = sv_next[k]) + { if (k <= n) + { /* i-th row of the matrix V */ + i = k; + if (vr_ptr[i] != sv_beg) break; + vr_cap[i] = vr_len[i]; + sv_beg += vr_cap[i]; + } + else + { /* j-th column of the matrix V */ + j = k - n; + if (vc_ptr[j] != sv_beg) break; + vc_cap[j] = vc_len[j]; + sv_beg += vc_cap[j]; + } + } + /* relocate other rows and columns in order to gather all unused + locations in one continuous extent */ + for (k = k; k != 0; k = sv_next[k]) + { if (k <= n) + { /* i-th row of the matrix V */ + i = k; + memmove(&sv_ind[sv_beg], &sv_ind[vr_ptr[i]], + vr_len[i] * sizeof(int)); + memmove(&sv_val[sv_beg], &sv_val[vr_ptr[i]], + vr_len[i] * sizeof(double)); + vr_ptr[i] = sv_beg; + vr_cap[i] = vr_len[i]; + sv_beg += vr_cap[i]; + } + else + { /* j-th column of the matrix V */ + j = k - n; + memmove(&sv_ind[sv_beg], &sv_ind[vc_ptr[j]], + vc_len[j] * sizeof(int)); + memmove(&sv_val[sv_beg], &sv_val[vc_ptr[j]], + vc_len[j] * sizeof(double)); + vc_ptr[j] = sv_beg; + vc_cap[j] = vc_len[j]; + sv_beg += vc_cap[j]; + } + } + /* set new pointer to the beginning of the free part */ + luf->sv_beg = sv_beg; + return; +} + +/*********************************************************************** +* NAME +* +* luf_enlarge_row - enlarge row capacity +* +* SYNOPSIS +* +* #include "glpluf.h" +* int luf_enlarge_row(LUF *luf, int i, int cap); +* +* DESCRIPTION +* +* The routine luf_enlarge_row enlarges capacity of the i-th row of the +* matrix V to cap locations (assuming that its current capacity is less +* than cap). In order to do that the routine relocates elements of the +* i-th row to the end of the left part of SVA (which contains rows and +* columns of the matrix V) and then expands the left part by allocating +* cap free locations from the free part. If there are less than cap +* free locations, the routine defragments the sparse vector area. +* +* Due to "garbage collection" this operation may change row and column +* pointers of the matrix V. +* +* RETURNS +* +* If no error occured, the routine returns zero. Otherwise, in case of +* overflow of the sparse vector area, the routine returns non-zero. */ + +int luf_enlarge_row(LUF *luf, int i, int cap) +{ int n = luf->n; + int *vr_ptr = luf->vr_ptr; + int *vr_len = luf->vr_len; + int *vr_cap = luf->vr_cap; + int *vc_cap = luf->vc_cap; + int *sv_ind = luf->sv_ind; + double *sv_val = luf->sv_val; + int *sv_prev = luf->sv_prev; + int *sv_next = luf->sv_next; + int ret = 0; + int cur, k, kk; + xassert(1 <= i && i <= n); + xassert(vr_cap[i] < cap); + /* if there are less than cap free locations, defragment SVA */ + if (luf->sv_end - luf->sv_beg < cap) + { luf_defrag_sva(luf); + if (luf->sv_end - luf->sv_beg < cap) + { ret = 1; + goto done; + } + } + /* save current capacity of the i-th row */ + cur = vr_cap[i]; + /* copy existing elements to the beginning of the free part */ + memmove(&sv_ind[luf->sv_beg], &sv_ind[vr_ptr[i]], + vr_len[i] * sizeof(int)); + memmove(&sv_val[luf->sv_beg], &sv_val[vr_ptr[i]], + vr_len[i] * sizeof(double)); + /* set new pointer and new capacity of the i-th row */ + vr_ptr[i] = luf->sv_beg; + vr_cap[i] = cap; + /* set new pointer to the beginning of the free part */ + luf->sv_beg += cap; + /* now the i-th row starts in the rightmost location among other + rows and columns of the matrix V, so its node should be moved + to the end of the row/column linked list */ + k = i; + /* remove the i-th row node from the linked list */ + if (sv_prev[k] == 0) + luf->sv_head = sv_next[k]; + else + { /* capacity of the previous row/column can be increased at the + expense of old locations of the i-th row */ + kk = sv_prev[k]; + if (kk <= n) vr_cap[kk] += cur; else vc_cap[kk-n] += cur; + sv_next[sv_prev[k]] = sv_next[k]; + } + if (sv_next[k] == 0) + luf->sv_tail = sv_prev[k]; + else + sv_prev[sv_next[k]] = sv_prev[k]; + /* insert the i-th row node to the end of the linked list */ + sv_prev[k] = luf->sv_tail; + sv_next[k] = 0; + if (sv_prev[k] == 0) + luf->sv_head = k; + else + sv_next[sv_prev[k]] = k; + luf->sv_tail = k; +done: return ret; +} + +/*********************************************************************** +* NAME +* +* luf_enlarge_col - enlarge column capacity +* +* SYNOPSIS +* +* #include "glpluf.h" +* int luf_enlarge_col(LUF *luf, int j, int cap); +* +* DESCRIPTION +* +* The routine luf_enlarge_col enlarges capacity of the j-th column of +* the matrix V to cap locations (assuming that its current capacity is +* less than cap). In order to do that the routine relocates elements +* of the j-th column to the end of the left part of SVA (which contains +* rows and columns of the matrix V) and then expands the left part by +* allocating cap free locations from the free part. If there are less +* than cap free locations, the routine defragments the sparse vector +* area. +* +* Due to "garbage collection" this operation may change row and column +* pointers of the matrix V. +* +* RETURNS +* +* If no error occured, the routine returns zero. Otherwise, in case of +* overflow of the sparse vector area, the routine returns non-zero. */ + +int luf_enlarge_col(LUF *luf, int j, int cap) +{ int n = luf->n; + int *vr_cap = luf->vr_cap; + int *vc_ptr = luf->vc_ptr; + int *vc_len = luf->vc_len; + int *vc_cap = luf->vc_cap; + int *sv_ind = luf->sv_ind; + double *sv_val = luf->sv_val; + int *sv_prev = luf->sv_prev; + int *sv_next = luf->sv_next; + int ret = 0; + int cur, k, kk; + xassert(1 <= j && j <= n); + xassert(vc_cap[j] < cap); + /* if there are less than cap free locations, defragment SVA */ + if (luf->sv_end - luf->sv_beg < cap) + { luf_defrag_sva(luf); + if (luf->sv_end - luf->sv_beg < cap) + { ret = 1; + goto done; + } + } + /* save current capacity of the j-th column */ + cur = vc_cap[j]; + /* copy existing elements to the beginning of the free part */ + memmove(&sv_ind[luf->sv_beg], &sv_ind[vc_ptr[j]], + vc_len[j] * sizeof(int)); + memmove(&sv_val[luf->sv_beg], &sv_val[vc_ptr[j]], + vc_len[j] * sizeof(double)); + /* set new pointer and new capacity of the j-th column */ + vc_ptr[j] = luf->sv_beg; + vc_cap[j] = cap; + /* set new pointer to the beginning of the free part */ + luf->sv_beg += cap; + /* now the j-th column starts in the rightmost location among + other rows and columns of the matrix V, so its node should be + moved to the end of the row/column linked list */ + k = n + j; + /* remove the j-th column node from the linked list */ + if (sv_prev[k] == 0) + luf->sv_head = sv_next[k]; + else + { /* capacity of the previous row/column can be increased at the + expense of old locations of the j-th column */ + kk = sv_prev[k]; + if (kk <= n) vr_cap[kk] += cur; else vc_cap[kk-n] += cur; + sv_next[sv_prev[k]] = sv_next[k]; + } + if (sv_next[k] == 0) + luf->sv_tail = sv_prev[k]; + else + sv_prev[sv_next[k]] = sv_prev[k]; + /* insert the j-th column node to the end of the linked list */ + sv_prev[k] = luf->sv_tail; + sv_next[k] = 0; + if (sv_prev[k] == 0) + luf->sv_head = k; + else + sv_next[sv_prev[k]] = k; + luf->sv_tail = k; +done: return ret; +} + +/*********************************************************************** +* reallocate - reallocate LU-factorization arrays +* +* This routine reallocates arrays, whose size depends of n, the order +* of the matrix A to be factorized. */ + +static void reallocate(LUF *luf, int n) +{ int n_max = luf->n_max; + luf->n = n; + if (n <= n_max) goto done; + if (luf->fr_ptr != NULL) xfree(luf->fr_ptr); + if (luf->fr_len != NULL) xfree(luf->fr_len); + if (luf->fc_ptr != NULL) xfree(luf->fc_ptr); + if (luf->fc_len != NULL) xfree(luf->fc_len); + if (luf->vr_ptr != NULL) xfree(luf->vr_ptr); + if (luf->vr_len != NULL) xfree(luf->vr_len); + if (luf->vr_cap != NULL) xfree(luf->vr_cap); + if (luf->vr_piv != NULL) xfree(luf->vr_piv); + if (luf->vc_ptr != NULL) xfree(luf->vc_ptr); + if (luf->vc_len != NULL) xfree(luf->vc_len); + if (luf->vc_cap != NULL) xfree(luf->vc_cap); + if (luf->pp_row != NULL) xfree(luf->pp_row); + if (luf->pp_col != NULL) xfree(luf->pp_col); + if (luf->qq_row != NULL) xfree(luf->qq_row); + if (luf->qq_col != NULL) xfree(luf->qq_col); + if (luf->sv_prev != NULL) xfree(luf->sv_prev); + if (luf->sv_next != NULL) xfree(luf->sv_next); + if (luf->vr_max != NULL) xfree(luf->vr_max); + if (luf->rs_head != NULL) xfree(luf->rs_head); + if (luf->rs_prev != NULL) xfree(luf->rs_prev); + if (luf->rs_next != NULL) xfree(luf->rs_next); + if (luf->cs_head != NULL) xfree(luf->cs_head); + if (luf->cs_prev != NULL) xfree(luf->cs_prev); + if (luf->cs_next != NULL) xfree(luf->cs_next); + if (luf->flag != NULL) xfree(luf->flag); + if (luf->work != NULL) xfree(luf->work); + luf->n_max = n_max = n + 100; + luf->fr_ptr = xcalloc(1+n_max, sizeof(int)); + luf->fr_len = xcalloc(1+n_max, sizeof(int)); + luf->fc_ptr = xcalloc(1+n_max, sizeof(int)); + luf->fc_len = xcalloc(1+n_max, sizeof(int)); + luf->vr_ptr = xcalloc(1+n_max, sizeof(int)); + luf->vr_len = xcalloc(1+n_max, sizeof(int)); + luf->vr_cap = xcalloc(1+n_max, sizeof(int)); + luf->vr_piv = xcalloc(1+n_max, sizeof(double)); + luf->vc_ptr = xcalloc(1+n_max, sizeof(int)); + luf->vc_len = xcalloc(1+n_max, sizeof(int)); + luf->vc_cap = xcalloc(1+n_max, sizeof(int)); + luf->pp_row = xcalloc(1+n_max, sizeof(int)); + luf->pp_col = xcalloc(1+n_max, sizeof(int)); + luf->qq_row = xcalloc(1+n_max, sizeof(int)); + luf->qq_col = xcalloc(1+n_max, sizeof(int)); + luf->sv_prev = xcalloc(1+n_max+n_max, sizeof(int)); + luf->sv_next = xcalloc(1+n_max+n_max, sizeof(int)); + luf->vr_max = xcalloc(1+n_max, sizeof(double)); + luf->rs_head = xcalloc(1+n_max, sizeof(int)); + luf->rs_prev = xcalloc(1+n_max, sizeof(int)); + luf->rs_next = xcalloc(1+n_max, sizeof(int)); + luf->cs_head = xcalloc(1+n_max, sizeof(int)); + luf->cs_prev = xcalloc(1+n_max, sizeof(int)); + luf->cs_next = xcalloc(1+n_max, sizeof(int)); + luf->flag = xcalloc(1+n_max, sizeof(int)); + luf->work = xcalloc(1+n_max, sizeof(double)); +done: return; +} + +/*********************************************************************** +* initialize - initialize LU-factorization data structures +* +* This routine initializes data structures for subsequent computing +* the LU-factorization of a given matrix A, which is specified by the +* formal routine col. On exit V = A and F = P = Q = I, where I is the +* unity matrix. (Row-wise representation of the matrix F is not used +* at the factorization stage and therefore is not initialized.) +* +* If no error occured, the routine returns zero. Otherwise, in case of +* overflow of the sparse vector area, the routine returns non-zero. */ + +static int initialize(LUF *luf, int (*col)(void *info, int j, int rn[], + double aj[]), void *info) +{ int n = luf->n; + int *fc_ptr = luf->fc_ptr; + int *fc_len = luf->fc_len; + int *vr_ptr = luf->vr_ptr; + int *vr_len = luf->vr_len; + int *vr_cap = luf->vr_cap; + int *vc_ptr = luf->vc_ptr; + int *vc_len = luf->vc_len; + int *vc_cap = luf->vc_cap; + int *pp_row = luf->pp_row; + int *pp_col = luf->pp_col; + int *qq_row = luf->qq_row; + int *qq_col = luf->qq_col; + int *sv_ind = luf->sv_ind; + double *sv_val = luf->sv_val; + int *sv_prev = luf->sv_prev; + int *sv_next = luf->sv_next; + double *vr_max = luf->vr_max; + int *rs_head = luf->rs_head; + int *rs_prev = luf->rs_prev; + int *rs_next = luf->rs_next; + int *cs_head = luf->cs_head; + int *cs_prev = luf->cs_prev; + int *cs_next = luf->cs_next; + int *flag = luf->flag; + double *work = luf->work; + int ret = 0; + int i, i_ptr, j, j_beg, j_end, k, len, nnz, sv_beg, sv_end, ptr; + double big, val; + /* free all locations of the sparse vector area */ + sv_beg = 1; + sv_end = luf->sv_size + 1; + /* (row-wise representation of the matrix F is not initialized, + because it is not used at the factorization stage) */ + /* build the matrix F in column-wise format (initially F = I) */ + for (j = 1; j <= n; j++) + { fc_ptr[j] = sv_end; + fc_len[j] = 0; + } + /* clear rows of the matrix V; clear the flag array */ + for (i = 1; i <= n; i++) + vr_len[i] = vr_cap[i] = 0, flag[i] = 0; + /* build the matrix V in column-wise format (initially V = A); + count non-zeros in rows of this matrix; count total number of + non-zeros; compute largest of absolute values of elements */ + nnz = 0; + big = 0.0; + for (j = 1; j <= n; j++) + { int *rn = pp_row; + double *aj = work; + /* obtain j-th column of the matrix A */ + len = col(info, j, rn, aj); + if (!(0 <= len && len <= n)) + xfault("luf_factorize: j = %d; len = %d; invalid column len" + "gth\n", j, len); + /* check for free locations */ + if (sv_end - sv_beg < len) + { /* overflow of the sparse vector area */ + ret = 1; + goto done; + } + /* set pointer to the j-th column */ + vc_ptr[j] = sv_beg; + /* set length of the j-th column */ + vc_len[j] = vc_cap[j] = len; + /* count total number of non-zeros */ + nnz += len; + /* walk through elements of the j-th column */ + for (ptr = 1; ptr <= len; ptr++) + { /* get row index and numerical value of a[i,j] */ + i = rn[ptr]; + val = aj[ptr]; + if (!(1 <= i && i <= n)) + xfault("luf_factorize: i = %d; j = %d; invalid row index" + "\n", i, j); + if (flag[i]) + xfault("luf_factorize: i = %d; j = %d; duplicate element" + " not allowed\n", i, j); + if (val == 0.0) + xfault("luf_factorize: i = %d; j = %d; zero element not " + "allowed\n", i, j); + /* add new element v[i,j] = a[i,j] to j-th column */ + sv_ind[sv_beg] = i; + sv_val[sv_beg] = val; + sv_beg++; + /* big := max(big, |a[i,j]|) */ + if (val < 0.0) val = - val; + if (big < val) big = val; + /* mark non-zero in the i-th position of the j-th column */ + flag[i] = 1; + /* increase length of the i-th row */ + vr_cap[i]++; + } + /* reset all non-zero marks */ + for (ptr = 1; ptr <= len; ptr++) flag[rn[ptr]] = 0; + } + /* allocate rows of the matrix V */ + for (i = 1; i <= n; i++) + { /* get length of the i-th row */ + len = vr_cap[i]; + /* check for free locations */ + if (sv_end - sv_beg < len) + { /* overflow of the sparse vector area */ + ret = 1; + goto done; + } + /* set pointer to the i-th row */ + vr_ptr[i] = sv_beg; + /* reserve locations for the i-th row */ + sv_beg += len; + } + /* build the matrix V in row-wise format using representation of + this matrix in column-wise format */ + for (j = 1; j <= n; j++) + { /* walk through elements of the j-th column */ + j_beg = vc_ptr[j]; + j_end = j_beg + vc_len[j] - 1; + for (k = j_beg; k <= j_end; k++) + { /* get row index and numerical value of v[i,j] */ + i = sv_ind[k]; + val = sv_val[k]; + /* store element in the i-th row */ + i_ptr = vr_ptr[i] + vr_len[i]; + sv_ind[i_ptr] = j; + sv_val[i_ptr] = val; + /* increase count of the i-th row */ + vr_len[i]++; + } + } + /* initialize the matrices P and Q (initially P = Q = I) */ + for (k = 1; k <= n; k++) + pp_row[k] = pp_col[k] = qq_row[k] = qq_col[k] = k; + /* set sva partitioning pointers */ + luf->sv_beg = sv_beg; + luf->sv_end = sv_end; + /* the initial physical order of rows and columns of the matrix V + is n+1, ..., n+n, 1, ..., n (firstly columns, then rows) */ + luf->sv_head = n+1; + luf->sv_tail = n; + for (i = 1; i <= n; i++) + { sv_prev[i] = i-1; + sv_next[i] = i+1; + } + sv_prev[1] = n+n; + sv_next[n] = 0; + for (j = 1; j <= n; j++) + { sv_prev[n+j] = n+j-1; + sv_next[n+j] = n+j+1; + } + sv_prev[n+1] = 0; + sv_next[n+n] = 1; + /* clear working arrays */ + for (k = 1; k <= n; k++) + { flag[k] = 0; + work[k] = 0.0; + } + /* initialize some statistics */ + luf->nnz_a = nnz; + luf->nnz_f = 0; + luf->nnz_v = nnz; + luf->max_a = big; + luf->big_v = big; + luf->rank = -1; + /* initially the active submatrix is the entire matrix V */ + /* largest of absolute values of elements in each active row is + unknown yet */ + for (i = 1; i <= n; i++) vr_max[i] = -1.0; + /* build linked lists of active rows */ + for (len = 0; len <= n; len++) rs_head[len] = 0; + for (i = 1; i <= n; i++) + { len = vr_len[i]; + rs_prev[i] = 0; + rs_next[i] = rs_head[len]; + if (rs_next[i] != 0) rs_prev[rs_next[i]] = i; + rs_head[len] = i; + } + /* build linked lists of active columns */ + for (len = 0; len <= n; len++) cs_head[len] = 0; + for (j = 1; j <= n; j++) + { len = vc_len[j]; + cs_prev[j] = 0; + cs_next[j] = cs_head[len]; + if (cs_next[j] != 0) cs_prev[cs_next[j]] = j; + cs_head[len] = j; + } +done: /* return to the factorizing routine */ + return ret; +} + +/*********************************************************************** +* find_pivot - choose a pivot element +* +* This routine chooses a pivot element in the active submatrix of the +* matrix U = P*V*Q. +* +* It is assumed that on entry the matrix U has the following partially +* triangularized form: +* +* 1 k n +* 1 x x x x x x x x x x +* . x x x x x x x x x +* . . x x x x x x x x +* . . . x x x x x x x +* k . . . . * * * * * * +* . . . . * * * * * * +* . . . . * * * * * * +* . . . . * * * * * * +* . . . . * * * * * * +* n . . . . * * * * * * +* +* where rows and columns k, k+1, ..., n belong to the active submatrix +* (elements of the active submatrix are marked by '*'). +* +* Since the matrix U = P*V*Q is not stored, the routine works with the +* matrix V. It is assumed that the row-wise representation corresponds +* to the matrix V, but the column-wise representation corresponds to +* the active submatrix of the matrix V, i.e. elements of the matrix V, +* which doesn't belong to the active submatrix, are missing from the +* column linked lists. It is also assumed that each active row of the +* matrix V is in the set R[len], where len is number of non-zeros in +* the row, and each active column of the matrix V is in the set C[len], +* where len is number of non-zeros in the column (in the latter case +* only elements of the active submatrix are counted; such elements are +* marked by '*' on the figure above). +* +* For the reason of numerical stability the routine applies so called +* threshold pivoting proposed by J.Reid. It is assumed that an element +* v[i,j] can be selected as a pivot candidate if it is not very small +* (in absolute value) among other elements in the same row, i.e. if it +* satisfies to the stability condition |v[i,j]| >= tol * max|v[i,*]|, +* where 0 < tol < 1 is a given tolerance. +* +* In order to keep sparsity of the matrix V the routine uses Markowitz +* strategy, trying to choose such element v[p,q], which satisfies to +* the stability condition (see above) and has smallest Markowitz cost +* (nr[p]-1) * (nc[q]-1), where nr[p] and nc[q] are numbers of non-zero +* elements, respectively, in the p-th row and in the q-th column of the +* active submatrix. +* +* In order to reduce the search, i.e. not to walk through all elements +* of the active submatrix, the routine exploits a technique proposed by +* I.Duff. This technique is based on using the sets R[len] and C[len] +* of active rows and columns. +* +* If the pivot element v[p,q] has been chosen, the routine stores its +* indices to the locations *p and *q and returns zero. Otherwise, if +* the active submatrix is empty and therefore the pivot element can't +* be chosen, the routine returns non-zero. */ + +static int find_pivot(LUF *luf, int *_p, int *_q) +{ int n = luf->n; + int *vr_ptr = luf->vr_ptr; + int *vr_len = luf->vr_len; + int *vc_ptr = luf->vc_ptr; + int *vc_len = luf->vc_len; + int *sv_ind = luf->sv_ind; + double *sv_val = luf->sv_val; + double *vr_max = luf->vr_max; + int *rs_head = luf->rs_head; + int *rs_next = luf->rs_next; + int *cs_head = luf->cs_head; + int *cs_prev = luf->cs_prev; + int *cs_next = luf->cs_next; + double piv_tol = luf->piv_tol; + int piv_lim = luf->piv_lim; + int suhl = luf->suhl; + int p, q, len, i, i_beg, i_end, i_ptr, j, j_beg, j_end, j_ptr, + ncand, next_j, min_p, min_q, min_len; + double best, cost, big, temp; + /* initially no pivot candidates have been found so far */ + p = q = 0, best = DBL_MAX, ncand = 0; + /* if in the active submatrix there is a column that has the only + non-zero (column singleton), choose it as pivot */ + j = cs_head[1]; + if (j != 0) + { xassert(vc_len[j] == 1); + p = sv_ind[vc_ptr[j]], q = j; + goto done; + } + /* if in the active submatrix there is a row that has the only + non-zero (row singleton), choose it as pivot */ + i = rs_head[1]; + if (i != 0) + { xassert(vr_len[i] == 1); + p = i, q = sv_ind[vr_ptr[i]]; + goto done; + } + /* there are no singletons in the active submatrix; walk through + other non-empty rows and columns */ + for (len = 2; len <= n; len++) + { /* consider active columns that have len non-zeros */ + for (j = cs_head[len]; j != 0; j = next_j) + { /* the j-th column has len non-zeros */ + j_beg = vc_ptr[j]; + j_end = j_beg + vc_len[j] - 1; + /* save pointer to the next column with the same length */ + next_j = cs_next[j]; + /* find an element in the j-th column, which is placed in a + row with minimal number of non-zeros and satisfies to the + stability condition (such element may not exist) */ + min_p = min_q = 0, min_len = INT_MAX; + for (j_ptr = j_beg; j_ptr <= j_end; j_ptr++) + { /* get row index of v[i,j] */ + i = sv_ind[j_ptr]; + i_beg = vr_ptr[i]; + i_end = i_beg + vr_len[i] - 1; + /* if the i-th row is not shorter than that one, where + minimal element is currently placed, skip v[i,j] */ + if (vr_len[i] >= min_len) continue; + /* determine the largest of absolute values of elements + in the i-th row */ + big = vr_max[i]; + if (big < 0.0) + { /* the largest value is unknown yet; compute it */ + for (i_ptr = i_beg; i_ptr <= i_end; i_ptr++) + { temp = sv_val[i_ptr]; + if (temp < 0.0) temp = - temp; + if (big < temp) big = temp; + } + vr_max[i] = big; + } + /* find v[i,j] in the i-th row */ + for (i_ptr = vr_ptr[i]; sv_ind[i_ptr] != j; i_ptr++); + xassert(i_ptr <= i_end); + /* if v[i,j] doesn't satisfy to the stability condition, + skip it */ + temp = sv_val[i_ptr]; + if (temp < 0.0) temp = - temp; + if (temp < piv_tol * big) continue; + /* v[i,j] is better than the current minimal element */ + min_p = i, min_q = j, min_len = vr_len[i]; + /* if Markowitz cost of the current minimal element is + not greater than (len-1)**2, it can be chosen right + now; this heuristic reduces the search and works well + in many cases */ + if (min_len <= len) + { p = min_p, q = min_q; + goto done; + } + } + /* the j-th column has been scanned */ + if (min_p != 0) + { /* the minimal element is a next pivot candidate */ + ncand++; + /* compute its Markowitz cost */ + cost = (double)(min_len - 1) * (double)(len - 1); + /* choose between the minimal element and the current + candidate */ + if (cost < best) p = min_p, q = min_q, best = cost; + /* if piv_lim candidates have been considered, there are + doubts that a much better candidate exists; therefore + it's time to terminate the search */ + if (ncand == piv_lim) goto done; + } + else + { /* the j-th column has no elements, which satisfy to the + stability condition; Uwe Suhl suggests to exclude such + column from the further consideration until it becomes + a column singleton; in hard cases this significantly + reduces a time needed for pivot searching */ + if (suhl) + { /* remove the j-th column from the active set */ + if (cs_prev[j] == 0) + cs_head[len] = cs_next[j]; + else + cs_next[cs_prev[j]] = cs_next[j]; + if (cs_next[j] == 0) + /* nop */; + else + cs_prev[cs_next[j]] = cs_prev[j]; + /* the following assignment is used to avoid an error + when the routine eliminate (see below) will try to + remove the j-th column from the active set */ + cs_prev[j] = cs_next[j] = j; + } + } + } + /* consider active rows that have len non-zeros */ + for (i = rs_head[len]; i != 0; i = rs_next[i]) + { /* the i-th row has len non-zeros */ + i_beg = vr_ptr[i]; + i_end = i_beg + vr_len[i] - 1; + /* determine the largest of absolute values of elements in + the i-th row */ + big = vr_max[i]; + if (big < 0.0) + { /* the largest value is unknown yet; compute it */ + for (i_ptr = i_beg; i_ptr <= i_end; i_ptr++) + { temp = sv_val[i_ptr]; + if (temp < 0.0) temp = - temp; + if (big < temp) big = temp; + } + vr_max[i] = big; + } + /* find an element in the i-th row, which is placed in a + column with minimal number of non-zeros and satisfies to + the stability condition (such element always exists) */ + min_p = min_q = 0, min_len = INT_MAX; + for (i_ptr = i_beg; i_ptr <= i_end; i_ptr++) + { /* get column index of v[i,j] */ + j = sv_ind[i_ptr]; + /* if the j-th column is not shorter than that one, where + minimal element is currently placed, skip v[i,j] */ + if (vc_len[j] >= min_len) continue; + /* if v[i,j] doesn't satisfy to the stability condition, + skip it */ + temp = sv_val[i_ptr]; + if (temp < 0.0) temp = - temp; + if (temp < piv_tol * big) continue; + /* v[i,j] is better than the current minimal element */ + min_p = i, min_q = j, min_len = vc_len[j]; + /* if Markowitz cost of the current minimal element is + not greater than (len-1)**2, it can be chosen right + now; this heuristic reduces the search and works well + in many cases */ + if (min_len <= len) + { p = min_p, q = min_q; + goto done; + } + } + /* the i-th row has been scanned */ + if (min_p != 0) + { /* the minimal element is a next pivot candidate */ + ncand++; + /* compute its Markowitz cost */ + cost = (double)(len - 1) * (double)(min_len - 1); + /* choose between the minimal element and the current + candidate */ + if (cost < best) p = min_p, q = min_q, best = cost; + /* if piv_lim candidates have been considered, there are + doubts that a much better candidate exists; therefore + it's time to terminate the search */ + if (ncand == piv_lim) goto done; + } + else + { /* this can't be because this can never be */ + xassert(min_p != min_p); + } + } + } +done: /* bring the pivot to the factorizing routine */ + *_p = p, *_q = q; + return (p == 0); +} + +/*********************************************************************** +* eliminate - perform gaussian elimination. +* +* This routine performs elementary gaussian transformations in order +* to eliminate subdiagonal elements in the k-th column of the matrix +* U = P*V*Q using the pivot element u[k,k], where k is the number of +* the current elimination step. +* +* The parameters p and q are, respectively, row and column indices of +* the element v[p,q], which corresponds to the element u[k,k]. +* +* Each time when the routine applies the elementary transformation to +* a non-pivot row of the matrix V, it stores the corresponding element +* to the matrix F in order to keep the main equality A = F*V. +* +* The routine assumes that on entry the matrices L = P*F*inv(P) and +* U = P*V*Q are the following: +* +* 1 k 1 k n +* 1 1 . . . . . . . . . 1 x x x x x x x x x x +* x 1 . . . . . . . . . x x x x x x x x x +* x x 1 . . . . . . . . . x x x x x x x x +* x x x 1 . . . . . . . . . x x x x x x x +* k x x x x 1 . . . . . k . . . . * * * * * * +* x x x x _ 1 . . . . . . . . # * * * * * +* x x x x _ . 1 . . . . . . . # * * * * * +* x x x x _ . . 1 . . . . . . # * * * * * +* x x x x _ . . . 1 . . . . . # * * * * * +* n x x x x _ . . . . 1 n . . . . # * * * * * +* +* matrix L matrix U +* +* where rows and columns of the matrix U with numbers k, k+1, ..., n +* form the active submatrix (eliminated elements are marked by '#' and +* other elements of the active submatrix are marked by '*'). Note that +* each eliminated non-zero element u[i,k] of the matrix U gives the +* corresponding element l[i,k] of the matrix L (marked by '_'). +* +* Actually all operations are performed on the matrix V. Should note +* that the row-wise representation corresponds to the matrix V, but the +* column-wise representation corresponds to the active submatrix of the +* matrix V, i.e. elements of the matrix V, which doesn't belong to the +* active submatrix, are missing from the column linked lists. +* +* Let u[k,k] = v[p,q] be the pivot. In order to eliminate subdiagonal +* elements u[i',k] = v[i,q], i' = k+1, k+2, ..., n, the routine applies +* the following elementary gaussian transformations: +* +* (i-th row of V) := (i-th row of V) - f[i,p] * (p-th row of V), +* +* where f[i,p] = v[i,q] / v[p,q] is a gaussian multiplier. +* +* Additionally, in order to keep the main equality A = F*V, each time +* when the routine applies the transformation to i-th row of the matrix +* V, it also adds f[i,p] as a new element to the matrix F. +* +* IMPORTANT: On entry the working arrays flag and work should contain +* zeros. This status is provided by the routine on exit. +* +* If no error occured, the routine returns zero. Otherwise, in case of +* overflow of the sparse vector area, the routine returns non-zero. */ + +static int eliminate(LUF *luf, int p, int q) +{ int n = luf->n; + int *fc_ptr = luf->fc_ptr; + int *fc_len = luf->fc_len; + int *vr_ptr = luf->vr_ptr; + int *vr_len = luf->vr_len; + int *vr_cap = luf->vr_cap; + double *vr_piv = luf->vr_piv; + int *vc_ptr = luf->vc_ptr; + int *vc_len = luf->vc_len; + int *vc_cap = luf->vc_cap; + int *sv_ind = luf->sv_ind; + double *sv_val = luf->sv_val; + int *sv_prev = luf->sv_prev; + int *sv_next = luf->sv_next; + double *vr_max = luf->vr_max; + int *rs_head = luf->rs_head; + int *rs_prev = luf->rs_prev; + int *rs_next = luf->rs_next; + int *cs_head = luf->cs_head; + int *cs_prev = luf->cs_prev; + int *cs_next = luf->cs_next; + int *flag = luf->flag; + double *work = luf->work; + double eps_tol = luf->eps_tol; + /* at this stage the row-wise representation of the matrix F is + not used, so fr_len can be used as a working array */ + int *ndx = luf->fr_len; + int ret = 0; + int len, fill, i, i_beg, i_end, i_ptr, j, j_beg, j_end, j_ptr, k, + p_beg, p_end, p_ptr, q_beg, q_end, q_ptr; + double fip, val, vpq, temp; + xassert(1 <= p && p <= n); + xassert(1 <= q && q <= n); + /* remove the p-th (pivot) row from the active set; this row will + never return there */ + if (rs_prev[p] == 0) + rs_head[vr_len[p]] = rs_next[p]; + else + rs_next[rs_prev[p]] = rs_next[p]; + if (rs_next[p] == 0) + ; + else + rs_prev[rs_next[p]] = rs_prev[p]; + /* remove the q-th (pivot) column from the active set; this column + will never return there */ + if (cs_prev[q] == 0) + cs_head[vc_len[q]] = cs_next[q]; + else + cs_next[cs_prev[q]] = cs_next[q]; + if (cs_next[q] == 0) + ; + else + cs_prev[cs_next[q]] = cs_prev[q]; + /* find the pivot v[p,q] = u[k,k] in the p-th row */ + p_beg = vr_ptr[p]; + p_end = p_beg + vr_len[p] - 1; + for (p_ptr = p_beg; sv_ind[p_ptr] != q; p_ptr++) /* nop */; + xassert(p_ptr <= p_end); + /* store value of the pivot */ + vpq = (vr_piv[p] = sv_val[p_ptr]); + /* remove the pivot from the p-th row */ + sv_ind[p_ptr] = sv_ind[p_end]; + sv_val[p_ptr] = sv_val[p_end]; + vr_len[p]--; + p_end--; + /* find the pivot v[p,q] = u[k,k] in the q-th column */ + q_beg = vc_ptr[q]; + q_end = q_beg + vc_len[q] - 1; + for (q_ptr = q_beg; sv_ind[q_ptr] != p; q_ptr++) /* nop */; + xassert(q_ptr <= q_end); + /* remove the pivot from the q-th column */ + sv_ind[q_ptr] = sv_ind[q_end]; + vc_len[q]--; + q_end--; + /* walk through the p-th (pivot) row, which doesn't contain the + pivot v[p,q] already, and do the following... */ + for (p_ptr = p_beg; p_ptr <= p_end; p_ptr++) + { /* get column index of v[p,j] */ + j = sv_ind[p_ptr]; + /* store v[p,j] to the working array */ + flag[j] = 1; + work[j] = sv_val[p_ptr]; + /* remove the j-th column from the active set; this column will + return there later with new length */ + if (cs_prev[j] == 0) + cs_head[vc_len[j]] = cs_next[j]; + else + cs_next[cs_prev[j]] = cs_next[j]; + if (cs_next[j] == 0) + ; + else + cs_prev[cs_next[j]] = cs_prev[j]; + /* find v[p,j] in the j-th column */ + j_beg = vc_ptr[j]; + j_end = j_beg + vc_len[j] - 1; + for (j_ptr = j_beg; sv_ind[j_ptr] != p; j_ptr++) /* nop */; + xassert(j_ptr <= j_end); + /* since v[p,j] leaves the active submatrix, remove it from the + j-th column; however, v[p,j] is kept in the p-th row */ + sv_ind[j_ptr] = sv_ind[j_end]; + vc_len[j]--; + } + /* walk through the q-th (pivot) column, which doesn't contain the + pivot v[p,q] already, and perform gaussian elimination */ + while (q_beg <= q_end) + { /* element v[i,q] should be eliminated */ + /* get row index of v[i,q] */ + i = sv_ind[q_beg]; + /* remove the i-th row from the active set; later this row will + return there with new length */ + if (rs_prev[i] == 0) + rs_head[vr_len[i]] = rs_next[i]; + else + rs_next[rs_prev[i]] = rs_next[i]; + if (rs_next[i] == 0) + ; + else + rs_prev[rs_next[i]] = rs_prev[i]; + /* find v[i,q] in the i-th row */ + i_beg = vr_ptr[i]; + i_end = i_beg + vr_len[i] - 1; + for (i_ptr = i_beg; sv_ind[i_ptr] != q; i_ptr++) /* nop */; + xassert(i_ptr <= i_end); + /* compute gaussian multiplier f[i,p] = v[i,q] / v[p,q] */ + fip = sv_val[i_ptr] / vpq; + /* since v[i,q] should be eliminated, remove it from the i-th + row */ + sv_ind[i_ptr] = sv_ind[i_end]; + sv_val[i_ptr] = sv_val[i_end]; + vr_len[i]--; + i_end--; + /* and from the q-th column */ + sv_ind[q_beg] = sv_ind[q_end]; + vc_len[q]--; + q_end--; + /* perform gaussian transformation: + (i-th row) := (i-th row) - f[i,p] * (p-th row) + note that now the p-th row, which is in the working array, + doesn't contain the pivot v[p,q], and the i-th row doesn't + contain the eliminated element v[i,q] */ + /* walk through the i-th row and transform existing non-zero + elements */ + fill = vr_len[p]; + for (i_ptr = i_beg; i_ptr <= i_end; i_ptr++) + { /* get column index of v[i,j] */ + j = sv_ind[i_ptr]; + /* v[i,j] := v[i,j] - f[i,p] * v[p,j] */ + if (flag[j]) + { /* v[p,j] != 0 */ + temp = (sv_val[i_ptr] -= fip * work[j]); + if (temp < 0.0) temp = - temp; + flag[j] = 0; + fill--; /* since both v[i,j] and v[p,j] exist */ + if (temp == 0.0 || temp < eps_tol) + { /* new v[i,j] is closer to zero; replace it by exact + zero, i.e. remove it from the active submatrix */ + /* remove v[i,j] from the i-th row */ + sv_ind[i_ptr] = sv_ind[i_end]; + sv_val[i_ptr] = sv_val[i_end]; + vr_len[i]--; + i_ptr--; + i_end--; + /* find v[i,j] in the j-th column */ + j_beg = vc_ptr[j]; + j_end = j_beg + vc_len[j] - 1; + for (j_ptr = j_beg; sv_ind[j_ptr] != i; j_ptr++); + xassert(j_ptr <= j_end); + /* remove v[i,j] from the j-th column */ + sv_ind[j_ptr] = sv_ind[j_end]; + vc_len[j]--; + } + else + { /* v_big := max(v_big, |v[i,j]|) */ + if (luf->big_v < temp) luf->big_v = temp; + } + } + } + /* now flag is the pattern of the set v[p,*] \ v[i,*], and fill + is number of non-zeros in this set; therefore up to fill new + non-zeros may appear in the i-th row */ + if (vr_len[i] + fill > vr_cap[i]) + { /* enlarge the i-th row */ + if (luf_enlarge_row(luf, i, vr_len[i] + fill)) + { /* overflow of the sparse vector area */ + ret = 1; + goto done; + } + /* defragmentation may change row and column pointers of the + matrix V */ + p_beg = vr_ptr[p]; + p_end = p_beg + vr_len[p] - 1; + q_beg = vc_ptr[q]; + q_end = q_beg + vc_len[q] - 1; + } + /* walk through the p-th (pivot) row and create new elements + of the i-th row that appear due to fill-in; column indices + of these new elements are accumulated in the array ndx */ + len = 0; + for (p_ptr = p_beg; p_ptr <= p_end; p_ptr++) + { /* get column index of v[p,j], which may cause fill-in */ + j = sv_ind[p_ptr]; + if (flag[j]) + { /* compute new non-zero v[i,j] = 0 - f[i,p] * v[p,j] */ + temp = (val = - fip * work[j]); + if (temp < 0.0) temp = - temp; + if (temp == 0.0 || temp < eps_tol) + /* if v[i,j] is closer to zero; just ignore it */; + else + { /* add v[i,j] to the i-th row */ + i_ptr = vr_ptr[i] + vr_len[i]; + sv_ind[i_ptr] = j; + sv_val[i_ptr] = val; + vr_len[i]++; + /* remember column index of v[i,j] */ + ndx[++len] = j; + /* big_v := max(big_v, |v[i,j]|) */ + if (luf->big_v < temp) luf->big_v = temp; + } + } + else + { /* there is no fill-in, because v[i,j] already exists in + the i-th row; restore the flag of the element v[p,j], + which was reset before */ + flag[j] = 1; + } + } + /* add new non-zeros v[i,j] to the corresponding columns */ + for (k = 1; k <= len; k++) + { /* get column index of new non-zero v[i,j] */ + j = ndx[k]; + /* one free location is needed in the j-th column */ + if (vc_len[j] + 1 > vc_cap[j]) + { /* enlarge the j-th column */ + if (luf_enlarge_col(luf, j, vc_len[j] + 10)) + { /* overflow of the sparse vector area */ + ret = 1; + goto done; + } + /* defragmentation may change row and column pointers of + the matrix V */ + p_beg = vr_ptr[p]; + p_end = p_beg + vr_len[p] - 1; + q_beg = vc_ptr[q]; + q_end = q_beg + vc_len[q] - 1; + } + /* add new non-zero v[i,j] to the j-th column */ + j_ptr = vc_ptr[j] + vc_len[j]; + sv_ind[j_ptr] = i; + vc_len[j]++; + } + /* now the i-th row has been completely transformed, therefore + it can return to the active set with new length */ + rs_prev[i] = 0; + rs_next[i] = rs_head[vr_len[i]]; + if (rs_next[i] != 0) rs_prev[rs_next[i]] = i; + rs_head[vr_len[i]] = i; + /* the largest of absolute values of elements in the i-th row + is currently unknown */ + vr_max[i] = -1.0; + /* at least one free location is needed to store the gaussian + multiplier */ + if (luf->sv_end - luf->sv_beg < 1) + { /* there are no free locations at all; defragment SVA */ + luf_defrag_sva(luf); + if (luf->sv_end - luf->sv_beg < 1) + { /* overflow of the sparse vector area */ + ret = 1; + goto done; + } + /* defragmentation may change row and column pointers of the + matrix V */ + p_beg = vr_ptr[p]; + p_end = p_beg + vr_len[p] - 1; + q_beg = vc_ptr[q]; + q_end = q_beg + vc_len[q] - 1; + } + /* add the element f[i,p], which is the gaussian multiplier, + to the matrix F */ + luf->sv_end--; + sv_ind[luf->sv_end] = i; + sv_val[luf->sv_end] = fip; + fc_len[p]++; + /* end of elimination loop */ + } + /* at this point the q-th (pivot) column should be empty */ + xassert(vc_len[q] == 0); + /* reset capacity of the q-th column */ + vc_cap[q] = 0; + /* remove node of the q-th column from the addressing list */ + k = n + q; + if (sv_prev[k] == 0) + luf->sv_head = sv_next[k]; + else + sv_next[sv_prev[k]] = sv_next[k]; + if (sv_next[k] == 0) + luf->sv_tail = sv_prev[k]; + else + sv_prev[sv_next[k]] = sv_prev[k]; + /* the p-th column of the matrix F has been completely built; set + its pointer */ + fc_ptr[p] = luf->sv_end; + /* walk through the p-th (pivot) row and do the following... */ + for (p_ptr = p_beg; p_ptr <= p_end; p_ptr++) + { /* get column index of v[p,j] */ + j = sv_ind[p_ptr]; + /* erase v[p,j] from the working array */ + flag[j] = 0; + work[j] = 0.0; + /* the j-th column has been completely transformed, therefore + it can return to the active set with new length; however + the special case c_prev[j] = c_next[j] = j means that the + routine find_pivot excluded the j-th column from the active + set due to Uwe Suhl's rule, and therefore in this case the + column can return to the active set only if it is a column + singleton */ + if (!(vc_len[j] != 1 && cs_prev[j] == j && cs_next[j] == j)) + { cs_prev[j] = 0; + cs_next[j] = cs_head[vc_len[j]]; + if (cs_next[j] != 0) cs_prev[cs_next[j]] = j; + cs_head[vc_len[j]] = j; + } + } +done: /* return to the factorizing routine */ + return ret; +} + +/*********************************************************************** +* build_v_cols - build the matrix V in column-wise format +* +* This routine builds the column-wise representation of the matrix V +* using its row-wise representation. +* +* If no error occured, the routine returns zero. Otherwise, in case of +* overflow of the sparse vector area, the routine returns non-zero. */ + +static int build_v_cols(LUF *luf) +{ int n = luf->n; + int *vr_ptr = luf->vr_ptr; + int *vr_len = luf->vr_len; + int *vc_ptr = luf->vc_ptr; + int *vc_len = luf->vc_len; + int *vc_cap = luf->vc_cap; + int *sv_ind = luf->sv_ind; + double *sv_val = luf->sv_val; + int *sv_prev = luf->sv_prev; + int *sv_next = luf->sv_next; + int ret = 0; + int i, i_beg, i_end, i_ptr, j, j_ptr, k, nnz; + /* it is assumed that on entry all columns of the matrix V are + empty, i.e. vc_len[j] = vc_cap[j] = 0 for all j = 1, ..., n, + and have been removed from the addressing list */ + /* count non-zeros in columns of the matrix V; count total number + of non-zeros in this matrix */ + nnz = 0; + for (i = 1; i <= n; i++) + { /* walk through elements of the i-th row and count non-zeros + in the corresponding columns */ + i_beg = vr_ptr[i]; + i_end = i_beg + vr_len[i] - 1; + for (i_ptr = i_beg; i_ptr <= i_end; i_ptr++) + vc_cap[sv_ind[i_ptr]]++; + /* count total number of non-zeros */ + nnz += vr_len[i]; + } + /* store total number of non-zeros */ + luf->nnz_v = nnz; + /* check for free locations */ + if (luf->sv_end - luf->sv_beg < nnz) + { /* overflow of the sparse vector area */ + ret = 1; + goto done; + } + /* allocate columns of the matrix V */ + for (j = 1; j <= n; j++) + { /* set pointer to the j-th column */ + vc_ptr[j] = luf->sv_beg; + /* reserve locations for the j-th column */ + luf->sv_beg += vc_cap[j]; + } + /* build the matrix V in column-wise format using this matrix in + row-wise format */ + for (i = 1; i <= n; i++) + { /* walk through elements of the i-th row */ + i_beg = vr_ptr[i]; + i_end = i_beg + vr_len[i] - 1; + for (i_ptr = i_beg; i_ptr <= i_end; i_ptr++) + { /* get column index */ + j = sv_ind[i_ptr]; + /* store element in the j-th column */ + j_ptr = vc_ptr[j] + vc_len[j]; + sv_ind[j_ptr] = i; + sv_val[j_ptr] = sv_val[i_ptr]; + /* increase length of the j-th column */ + vc_len[j]++; + } + } + /* now columns are placed in the sparse vector area behind rows + in the order n+1, n+2, ..., n+n; so insert column nodes in the + addressing list using this order */ + for (k = n+1; k <= n+n; k++) + { sv_prev[k] = k-1; + sv_next[k] = k+1; + } + sv_prev[n+1] = luf->sv_tail; + sv_next[luf->sv_tail] = n+1; + sv_next[n+n] = 0; + luf->sv_tail = n+n; +done: /* return to the factorizing routine */ + return ret; +} + +/*********************************************************************** +* build_f_rows - build the matrix F in row-wise format +* +* This routine builds the row-wise representation of the matrix F using +* its column-wise representation. +* +* If no error occured, the routine returns zero. Otherwise, in case of +* overflow of the sparse vector area, the routine returns non-zero. */ + +static int build_f_rows(LUF *luf) +{ int n = luf->n; + int *fr_ptr = luf->fr_ptr; + int *fr_len = luf->fr_len; + int *fc_ptr = luf->fc_ptr; + int *fc_len = luf->fc_len; + int *sv_ind = luf->sv_ind; + double *sv_val = luf->sv_val; + int ret = 0; + int i, j, j_beg, j_end, j_ptr, ptr, nnz; + /* clear rows of the matrix F */ + for (i = 1; i <= n; i++) fr_len[i] = 0; + /* count non-zeros in rows of the matrix F; count total number of + non-zeros in this matrix */ + nnz = 0; + for (j = 1; j <= n; j++) + { /* walk through elements of the j-th column and count non-zeros + in the corresponding rows */ + j_beg = fc_ptr[j]; + j_end = j_beg + fc_len[j] - 1; + for (j_ptr = j_beg; j_ptr <= j_end; j_ptr++) + fr_len[sv_ind[j_ptr]]++; + /* increase total number of non-zeros */ + nnz += fc_len[j]; + } + /* store total number of non-zeros */ + luf->nnz_f = nnz; + /* check for free locations */ + if (luf->sv_end - luf->sv_beg < nnz) + { /* overflow of the sparse vector area */ + ret = 1; + goto done; + } + /* allocate rows of the matrix F */ + for (i = 1; i <= n; i++) + { /* set pointer to the end of the i-th row; later this pointer + will be set to the beginning of the i-th row */ + fr_ptr[i] = luf->sv_end; + /* reserve locations for the i-th row */ + luf->sv_end -= fr_len[i]; + } + /* build the matrix F in row-wise format using this matrix in + column-wise format */ + for (j = 1; j <= n; j++) + { /* walk through elements of the j-th column */ + j_beg = fc_ptr[j]; + j_end = j_beg + fc_len[j] - 1; + for (j_ptr = j_beg; j_ptr <= j_end; j_ptr++) + { /* get row index */ + i = sv_ind[j_ptr]; + /* store element in the i-th row */ + ptr = --fr_ptr[i]; + sv_ind[ptr] = j; + sv_val[ptr] = sv_val[j_ptr]; + } + } +done: /* return to the factorizing routine */ + return ret; +} + +/*********************************************************************** +* NAME +* +* luf_factorize - compute LU-factorization +* +* SYNOPSIS +* +* #include "glpluf.h" +* int luf_factorize(LUF *luf, int n, int (*col)(void *info, int j, +* int ind[], double val[]), void *info); +* +* DESCRIPTION +* +* The routine luf_factorize computes LU-factorization of a specified +* square matrix A. +* +* The parameter luf specifies LU-factorization program object created +* by the routine luf_create_it. +* +* The parameter n specifies the order of A, n > 0. +* +* The formal routine col specifies the matrix A to be factorized. To +* obtain j-th column of A the routine luf_factorize calls the routine +* col with the parameter j (1 <= j <= n). In response the routine col +* should store row indices and numerical values of non-zero elements +* of j-th column of A to locations ind[1,...,len] and val[1,...,len], +* respectively, where len is the number of non-zeros in j-th column +* returned on exit. Neither zero nor duplicate elements are allowed. +* +* The parameter info is a transit pointer passed to the routine col. +* +* RETURNS +* +* 0 LU-factorization has been successfully computed. +* +* LUF_ESING +* The specified matrix is singular within the working precision. +* (On some elimination step the active submatrix is exactly zero, +* so no pivot can be chosen.) +* +* LUF_ECOND +* The specified matrix is ill-conditioned. +* (On some elimination step too intensive growth of elements of the +* active submatix has been detected.) +* +* If matrix A is well scaled, the return code LUF_ECOND may also mean +* that the threshold pivoting tolerance piv_tol should be increased. +* +* In case of non-zero return code the factorization becomes invalid. +* It should not be used in other operations until the cause of failure +* has been eliminated and the factorization has been recomputed again +* with the routine luf_factorize. +* +* REPAIRING SINGULAR MATRIX +* +* If the routine luf_factorize returns non-zero code, it provides all +* necessary information that can be used for "repairing" the matrix A, +* where "repairing" means replacing linearly dependent columns of the +* matrix A by appropriate columns of the unity matrix. This feature is +* needed when this routine is used for factorizing the basis matrix +* within the simplex method procedure. +* +* On exit linearly dependent columns of the (partially transformed) +* matrix U have numbers rank+1, rank+2, ..., n, where rank is estimated +* rank of the matrix A stored by the routine to the member luf->rank. +* The correspondence between columns of A and U is the same as between +* columns of V and U. Thus, linearly dependent columns of the matrix A +* have numbers qq_col[rank+1], qq_col[rank+2], ..., qq_col[n], where +* qq_col is the column-like representation of the permutation matrix Q. +* It is understood that each j-th linearly dependent column of the +* matrix U should be replaced by the unity vector, where all elements +* are zero except the unity diagonal element u[j,j]. On the other hand +* j-th row of the matrix U corresponds to the row of the matrix V (and +* therefore of the matrix A) with the number pp_row[j], where pp_row is +* the row-like representation of the permutation matrix P. Thus, each +* j-th linearly dependent column of the matrix U should be replaced by +* column of the unity matrix with the number pp_row[j]. +* +* The code that repairs the matrix A may look like follows: +* +* for (j = rank+1; j <= n; j++) +* { replace the column qq_col[j] of the matrix A by the column +* pp_row[j] of the unity matrix; +* } +* +* where rank, pp_row, and qq_col are members of the structure LUF. */ + +int luf_factorize(LUF *luf, int n, int (*col)(void *info, int j, + int ind[], double val[]), void *info) +{ int *pp_row, *pp_col, *qq_row, *qq_col; + double max_gro = luf->max_gro; + int i, j, k, p, q, t, ret; + if (n < 1) + xfault("luf_factorize: n = %d; invalid parameter\n", n); + if (n > N_MAX) + xfault("luf_factorize: n = %d; matrix too big\n", n); + /* invalidate the factorization */ + luf->valid = 0; + /* reallocate arrays, if necessary */ + reallocate(luf, n); + pp_row = luf->pp_row; + pp_col = luf->pp_col; + qq_row = luf->qq_row; + qq_col = luf->qq_col; + /* estimate initial size of the SVA, if not specified */ + if (luf->sv_size == 0 && luf->new_sva == 0) + luf->new_sva = 5 * (n + 10); +more: /* reallocate the sparse vector area, if required */ + if (luf->new_sva > 0) + { if (luf->sv_ind != NULL) xfree(luf->sv_ind); + if (luf->sv_val != NULL) xfree(luf->sv_val); + luf->sv_size = luf->new_sva; + luf->sv_ind = xcalloc(1+luf->sv_size, sizeof(int)); + luf->sv_val = xcalloc(1+luf->sv_size, sizeof(double)); + luf->new_sva = 0; + } + /* initialize LU-factorization data structures */ + if (initialize(luf, col, info)) + { /* overflow of the sparse vector area */ + luf->new_sva = luf->sv_size + luf->sv_size; + xassert(luf->new_sva > luf->sv_size); + goto more; + } + /* main elimination loop */ + for (k = 1; k <= n; k++) + { /* choose a pivot element v[p,q] */ + if (find_pivot(luf, &p, &q)) + { /* no pivot can be chosen, because the active submatrix is + exactly zero */ + luf->rank = k - 1; + ret = LUF_ESING; + goto done; + } + /* let v[p,q] correspond to u[i',j']; permute k-th and i'-th + rows and k-th and j'-th columns of the matrix U = P*V*Q to + move the element u[i',j'] to the position u[k,k] */ + i = pp_col[p], j = qq_row[q]; + xassert(k <= i && i <= n && k <= j && j <= n); + /* permute k-th and i-th rows of the matrix U */ + t = pp_row[k]; + pp_row[i] = t, pp_col[t] = i; + pp_row[k] = p, pp_col[p] = k; + /* permute k-th and j-th columns of the matrix U */ + t = qq_col[k]; + qq_col[j] = t, qq_row[t] = j; + qq_col[k] = q, qq_row[q] = k; + /* eliminate subdiagonal elements of k-th column of the matrix + U = P*V*Q using the pivot element u[k,k] = v[p,q] */ + if (eliminate(luf, p, q)) + { /* overflow of the sparse vector area */ + luf->new_sva = luf->sv_size + luf->sv_size; + xassert(luf->new_sva > luf->sv_size); + goto more; + } + /* check relative growth of elements of the matrix V */ + if (luf->big_v > max_gro * luf->max_a) + { /* the growth is too intensive, therefore most probably the + matrix A is ill-conditioned */ + luf->rank = k - 1; + ret = LUF_ECOND; + goto done; + } + } + /* now the matrix U = P*V*Q is upper triangular, the matrix V has + been built in row-wise format, and the matrix F has been built + in column-wise format */ + /* defragment the sparse vector area in order to merge all free + locations in one continuous extent */ + luf_defrag_sva(luf); + /* build the matrix V in column-wise format */ + if (build_v_cols(luf)) + { /* overflow of the sparse vector area */ + luf->new_sva = luf->sv_size + luf->sv_size; + xassert(luf->new_sva > luf->sv_size); + goto more; + } + /* build the matrix F in row-wise format */ + if (build_f_rows(luf)) + { /* overflow of the sparse vector area */ + luf->new_sva = luf->sv_size + luf->sv_size; + xassert(luf->new_sva > luf->sv_size); + goto more; + } + /* the LU-factorization has been successfully computed */ + luf->valid = 1; + luf->rank = n; + ret = 0; + /* if there are few free locations in the sparse vector area, try + increasing its size in the future */ + t = 3 * (n + luf->nnz_v) + 2 * luf->nnz_f; + if (luf->sv_size < t) + { luf->new_sva = luf->sv_size; + while (luf->new_sva < t) + { k = luf->new_sva; + luf->new_sva = k + k; + xassert(luf->new_sva > k); + } + } +done: /* return to the calling program */ + return ret; +} + +/*********************************************************************** +* NAME +* +* luf_f_solve - solve system F*x = b or F'*x = b +* +* SYNOPSIS +* +* #include "glpluf.h" +* void luf_f_solve(LUF *luf, int tr, double x[]); +* +* DESCRIPTION +* +* The routine luf_f_solve solves either the system F*x = b (if the +* flag tr is zero) or the system F'*x = b (if the flag tr is non-zero), +* where the matrix F is a component of LU-factorization specified by +* the parameter luf, F' is a matrix transposed to F. +* +* On entry the array x should contain elements of the right-hand side +* vector b in locations x[1], ..., x[n], where n is the order of the +* matrix F. On exit this array will contain elements of the solution +* vector x in the same locations. */ + +void luf_f_solve(LUF *luf, int tr, double x[]) +{ int n = luf->n; + int *fr_ptr = luf->fr_ptr; + int *fr_len = luf->fr_len; + int *fc_ptr = luf->fc_ptr; + int *fc_len = luf->fc_len; + int *pp_row = luf->pp_row; + int *sv_ind = luf->sv_ind; + double *sv_val = luf->sv_val; + int i, j, k, beg, end, ptr; + double xk; + if (!luf->valid) + xfault("luf_f_solve: LU-factorization is not valid\n"); + if (!tr) + { /* solve the system F*x = b */ + for (j = 1; j <= n; j++) + { k = pp_row[j]; + xk = x[k]; + if (xk != 0.0) + { beg = fc_ptr[k]; + end = beg + fc_len[k] - 1; + for (ptr = beg; ptr <= end; ptr++) + x[sv_ind[ptr]] -= sv_val[ptr] * xk; + } + } + } + else + { /* solve the system F'*x = b */ + for (i = n; i >= 1; i--) + { k = pp_row[i]; + xk = x[k]; + if (xk != 0.0) + { beg = fr_ptr[k]; + end = beg + fr_len[k] - 1; + for (ptr = beg; ptr <= end; ptr++) + x[sv_ind[ptr]] -= sv_val[ptr] * xk; + } + } + } + return; +} + +/*********************************************************************** +* NAME +* +* luf_v_solve - solve system V*x = b or V'*x = b +* +* SYNOPSIS +* +* #include "glpluf.h" +* void luf_v_solve(LUF *luf, int tr, double x[]); +* +* DESCRIPTION +* +* The routine luf_v_solve solves either the system V*x = b (if the +* flag tr is zero) or the system V'*x = b (if the flag tr is non-zero), +* where the matrix V is a component of LU-factorization specified by +* the parameter luf, V' is a matrix transposed to V. +* +* On entry the array x should contain elements of the right-hand side +* vector b in locations x[1], ..., x[n], where n is the order of the +* matrix V. On exit this array will contain elements of the solution +* vector x in the same locations. */ + +void luf_v_solve(LUF *luf, int tr, double x[]) +{ int n = luf->n; + int *vr_ptr = luf->vr_ptr; + int *vr_len = luf->vr_len; + double *vr_piv = luf->vr_piv; + int *vc_ptr = luf->vc_ptr; + int *vc_len = luf->vc_len; + int *pp_row = luf->pp_row; + int *qq_col = luf->qq_col; + int *sv_ind = luf->sv_ind; + double *sv_val = luf->sv_val; + double *b = luf->work; + int i, j, k, beg, end, ptr; + double temp; + if (!luf->valid) + xfault("luf_v_solve: LU-factorization is not valid\n"); + for (k = 1; k <= n; k++) b[k] = x[k], x[k] = 0.0; + if (!tr) + { /* solve the system V*x = b */ + for (k = n; k >= 1; k--) + { i = pp_row[k], j = qq_col[k]; + temp = b[i]; + if (temp != 0.0) + { x[j] = (temp /= vr_piv[i]); + beg = vc_ptr[j]; + end = beg + vc_len[j] - 1; + for (ptr = beg; ptr <= end; ptr++) + b[sv_ind[ptr]] -= sv_val[ptr] * temp; + } + } + } + else + { /* solve the system V'*x = b */ + for (k = 1; k <= n; k++) + { i = pp_row[k], j = qq_col[k]; + temp = b[j]; + if (temp != 0.0) + { x[i] = (temp /= vr_piv[i]); + beg = vr_ptr[i]; + end = beg + vr_len[i] - 1; + for (ptr = beg; ptr <= end; ptr++) + b[sv_ind[ptr]] -= sv_val[ptr] * temp; + } + } + } + return; +} + +/*********************************************************************** +* NAME +* +* luf_a_solve - solve system A*x = b or A'*x = b +* +* SYNOPSIS +* +* #include "glpluf.h" +* void luf_a_solve(LUF *luf, int tr, double x[]); +* +* DESCRIPTION +* +* The routine luf_a_solve solves either the system A*x = b (if the +* flag tr is zero) or the system A'*x = b (if the flag tr is non-zero), +* where the parameter luf specifies LU-factorization of the matrix A, +* A' is a matrix transposed to A. +* +* On entry the array x should contain elements of the right-hand side +* vector b in locations x[1], ..., x[n], where n is the order of the +* matrix A. On exit this array will contain elements of the solution +* vector x in the same locations. */ + +void luf_a_solve(LUF *luf, int tr, double x[]) +{ if (!luf->valid) + xfault("luf_a_solve: LU-factorization is not valid\n"); + if (!tr) + { /* A = F*V, therefore inv(A) = inv(V)*inv(F) */ + luf_f_solve(luf, 0, x); + luf_v_solve(luf, 0, x); + } + else + { /* A' = V'*F', therefore inv(A') = inv(F')*inv(V') */ + luf_v_solve(luf, 1, x); + luf_f_solve(luf, 1, x); + } + return; +} + +/*********************************************************************** +* NAME +* +* luf_delete_it - delete LU-factorization +* +* SYNOPSIS +* +* #include "glpluf.h" +* void luf_delete_it(LUF *luf); +* +* DESCRIPTION +* +* The routine luf_delete deletes LU-factorization specified by the +* parameter luf and frees all the memory allocated to this program +* object. */ + +void luf_delete_it(LUF *luf) +{ if (luf->fr_ptr != NULL) xfree(luf->fr_ptr); + if (luf->fr_len != NULL) xfree(luf->fr_len); + if (luf->fc_ptr != NULL) xfree(luf->fc_ptr); + if (luf->fc_len != NULL) xfree(luf->fc_len); + if (luf->vr_ptr != NULL) xfree(luf->vr_ptr); + if (luf->vr_len != NULL) xfree(luf->vr_len); + if (luf->vr_cap != NULL) xfree(luf->vr_cap); + if (luf->vr_piv != NULL) xfree(luf->vr_piv); + if (luf->vc_ptr != NULL) xfree(luf->vc_ptr); + if (luf->vc_len != NULL) xfree(luf->vc_len); + if (luf->vc_cap != NULL) xfree(luf->vc_cap); + if (luf->pp_row != NULL) xfree(luf->pp_row); + if (luf->pp_col != NULL) xfree(luf->pp_col); + if (luf->qq_row != NULL) xfree(luf->qq_row); + if (luf->qq_col != NULL) xfree(luf->qq_col); + if (luf->sv_ind != NULL) xfree(luf->sv_ind); + if (luf->sv_val != NULL) xfree(luf->sv_val); + if (luf->sv_prev != NULL) xfree(luf->sv_prev); + if (luf->sv_next != NULL) xfree(luf->sv_next); + if (luf->vr_max != NULL) xfree(luf->vr_max); + if (luf->rs_head != NULL) xfree(luf->rs_head); + if (luf->rs_prev != NULL) xfree(luf->rs_prev); + if (luf->rs_next != NULL) xfree(luf->rs_next); + if (luf->cs_head != NULL) xfree(luf->cs_head); + if (luf->cs_prev != NULL) xfree(luf->cs_prev); + if (luf->cs_next != NULL) xfree(luf->cs_next); + if (luf->flag != NULL) xfree(luf->flag); + if (luf->work != NULL) xfree(luf->work); + xfree(luf); + return; +} + +/* eof */