/* 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, 2011 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 */