diff -r d59bea55db9b -r c445c931472f src/glplux.c --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/src/glplux.c Mon Dec 06 13:09:21 2010 +0100 @@ -0,0 +1,1028 @@ +/* glplux.c */ + +/*********************************************************************** +* 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 "glplux.h" +#define xfault xerror +#define dmp_create_poolx(size) dmp_create_pool() + +/*---------------------------------------------------------------------- +// lux_create - create LU-factorization. +// +// SYNOPSIS +// +// #include "glplux.h" +// LUX *lux_create(int n); +// +// DESCRIPTION +// +// The routine lux_create creates LU-factorization data structure for +// a matrix of the order n. Initially the factorization corresponds to +// the unity matrix (F = V = P = Q = I, so A = I). +// +// RETURNS +// +// The routine returns a pointer to the created LU-factorization data +// structure, which represents the unity matrix of the order n. */ + +LUX *lux_create(int n) +{ LUX *lux; + int k; + if (n < 1) + xfault("lux_create: n = %d; invalid parameter\n", n); + lux = xmalloc(sizeof(LUX)); + lux->n = n; + lux->pool = dmp_create_poolx(sizeof(LUXELM)); + lux->F_row = xcalloc(1+n, sizeof(LUXELM *)); + lux->F_col = xcalloc(1+n, sizeof(LUXELM *)); + lux->V_piv = xcalloc(1+n, sizeof(mpq_t)); + lux->V_row = xcalloc(1+n, sizeof(LUXELM *)); + lux->V_col = xcalloc(1+n, sizeof(LUXELM *)); + lux->P_row = xcalloc(1+n, sizeof(int)); + lux->P_col = xcalloc(1+n, sizeof(int)); + lux->Q_row = xcalloc(1+n, sizeof(int)); + lux->Q_col = xcalloc(1+n, sizeof(int)); + for (k = 1; k <= n; k++) + { lux->F_row[k] = lux->F_col[k] = NULL; + mpq_init(lux->V_piv[k]); + mpq_set_si(lux->V_piv[k], 1, 1); + lux->V_row[k] = lux->V_col[k] = NULL; + lux->P_row[k] = lux->P_col[k] = k; + lux->Q_row[k] = lux->Q_col[k] = k; + } + lux->rank = n; + return lux; +} + +/*---------------------------------------------------------------------- +// 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. */ + +static void initialize(LUX *lux, int (*col)(void *info, int j, + int ind[], mpq_t val[]), void *info, LUXWKA *wka) +{ int n = lux->n; + DMP *pool = lux->pool; + LUXELM **F_row = lux->F_row; + LUXELM **F_col = lux->F_col; + mpq_t *V_piv = lux->V_piv; + LUXELM **V_row = lux->V_row; + LUXELM **V_col = lux->V_col; + int *P_row = lux->P_row; + int *P_col = lux->P_col; + int *Q_row = lux->Q_row; + int *Q_col = lux->Q_col; + int *R_len = wka->R_len; + int *R_head = wka->R_head; + int *R_prev = wka->R_prev; + int *R_next = wka->R_next; + int *C_len = wka->C_len; + int *C_head = wka->C_head; + int *C_prev = wka->C_prev; + int *C_next = wka->C_next; + LUXELM *fij, *vij; + int i, j, k, len, *ind; + mpq_t *val; + /* F := I */ + for (i = 1; i <= n; i++) + { while (F_row[i] != NULL) + { fij = F_row[i], F_row[i] = fij->r_next; + mpq_clear(fij->val); + dmp_free_atom(pool, fij, sizeof(LUXELM)); + } + } + for (j = 1; j <= n; j++) F_col[j] = NULL; + /* V := 0 */ + for (k = 1; k <= n; k++) mpq_set_si(V_piv[k], 0, 1); + for (i = 1; i <= n; i++) + { while (V_row[i] != NULL) + { vij = V_row[i], V_row[i] = vij->r_next; + mpq_clear(vij->val); + dmp_free_atom(pool, vij, sizeof(LUXELM)); + } + } + for (j = 1; j <= n; j++) V_col[j] = NULL; + /* V := A */ + ind = xcalloc(1+n, sizeof(int)); + val = xcalloc(1+n, sizeof(mpq_t)); + for (k = 1; k <= n; k++) mpq_init(val[k]); + for (j = 1; j <= n; j++) + { /* obtain j-th column of matrix A */ + len = col(info, j, ind, val); + if (!(0 <= len && len <= n)) + xfault("lux_decomp: j = %d: len = %d; invalid column length" + "\n", j, len); + /* copy elements of j-th column to matrix V */ + for (k = 1; k <= len; k++) + { /* get row index of a[i,j] */ + i = ind[k]; + if (!(1 <= i && i <= n)) + xfault("lux_decomp: j = %d: i = %d; row index out of ran" + "ge\n", j, i); + /* check for duplicate indices */ + if (V_row[i] != NULL && V_row[i]->j == j) + xfault("lux_decomp: j = %d: i = %d; duplicate row indice" + "s not allowed\n", j, i); + /* check for zero value */ + if (mpq_sgn(val[k]) == 0) + xfault("lux_decomp: j = %d: i = %d; zero elements not al" + "lowed\n", j, i); + /* add new element v[i,j] = a[i,j] to V */ + vij = dmp_get_atom(pool, sizeof(LUXELM)); + vij->i = i, vij->j = j; + mpq_init(vij->val); + mpq_set(vij->val, val[k]); + vij->r_prev = NULL; + vij->r_next = V_row[i]; + vij->c_prev = NULL; + vij->c_next = V_col[j]; + if (vij->r_next != NULL) vij->r_next->r_prev = vij; + if (vij->c_next != NULL) vij->c_next->c_prev = vij; + V_row[i] = V_col[j] = vij; + } + } + xfree(ind); + for (k = 1; k <= n; k++) mpq_clear(val[k]); + xfree(val); + /* P := Q := I */ + for (k = 1; k <= n; k++) + P_row[k] = P_col[k] = Q_row[k] = Q_col[k] = k; + /* the rank of A and V is not determined yet */ + lux->rank = -1; + /* initially the entire matrix V is active */ + /* determine its row lengths */ + for (i = 1; i <= n; i++) + { len = 0; + for (vij = V_row[i]; vij != NULL; vij = vij->r_next) len++; + R_len[i] = len; + } + /* build linked lists of active rows */ + for (len = 0; len <= n; len++) R_head[len] = 0; + for (i = 1; i <= n; i++) + { len = R_len[i]; + R_prev[i] = 0; + R_next[i] = R_head[len]; + if (R_next[i] != 0) R_prev[R_next[i]] = i; + R_head[len] = i; + } + /* determine its column lengths */ + for (j = 1; j <= n; j++) + { len = 0; + for (vij = V_col[j]; vij != NULL; vij = vij->c_next) len++; + C_len[j] = len; + } + /* build linked lists of active columns */ + for (len = 0; len <= n; len++) C_head[len] = 0; + for (j = 1; j <= n; j++) + { len = C_len[j]; + C_prev[j] = 0; + C_next[j] = C_head[len]; + if (C_next[j] != 0) C_prev[C_next[j]] = j; + C_head[len] = j; + } + return; +} + +/*---------------------------------------------------------------------- +// find_pivot - choose a pivot element. +// +// This routine chooses a pivot element v[p,q] in the active submatrix +// of 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 does not 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). +// +// Due to exact arithmetic any non-zero element of the active submatrix +// can be chosen as a pivot. However, to keep sparsity of the matrix V +// the routine uses Markowitz strategy, trying to choose such element +// v[p,q], which has smallest Markowitz cost (nr[p]-1) * (nc[q]-1), +// where nr[p] and nc[q] are the number of non-zero elements, resp., in +// p-th row and in 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. +// +// On exit the routine returns a pointer to a pivot v[p,q] chosen, or +// NULL, if the active submatrix is empty. */ + +static LUXELM *find_pivot(LUX *lux, LUXWKA *wka) +{ int n = lux->n; + LUXELM **V_row = lux->V_row; + LUXELM **V_col = lux->V_col; + int *R_len = wka->R_len; + int *R_head = wka->R_head; + int *R_next = wka->R_next; + int *C_len = wka->C_len; + int *C_head = wka->C_head; + int *C_next = wka->C_next; + LUXELM *piv, *some, *vij; + int i, j, len, min_len, ncand, piv_lim = 5; + double best, cost; + /* nothing is chosen so far */ + piv = NULL, 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 a pivot */ + j = C_head[1]; + if (j != 0) + { xassert(C_len[j] == 1); + piv = V_col[j]; + xassert(piv != NULL && piv->c_next == NULL); + goto done; + } + /* if in the active submatrix there is a row that has the only + non-zero (row singleton), choose it as a pivot */ + i = R_head[1]; + if (i != 0) + { xassert(R_len[i] == 1); + piv = V_row[i]; + xassert(piv != NULL && piv->r_next == NULL); + 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 having len non-zeros */ + for (j = C_head[len]; j != 0; j = C_next[j]) + { /* j-th column has len non-zeros */ + /* find an element in the row of minimal length */ + some = NULL, min_len = INT_MAX; + for (vij = V_col[j]; vij != NULL; vij = vij->c_next) + { if (min_len > R_len[vij->i]) + some = vij, min_len = R_len[vij->i]; + /* if Markowitz cost of this 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) + { piv = some; + goto done; + } + } + /* j-th column has been scanned */ + /* the minimal element found is a next pivot candidate */ + xassert(some != NULL); + ncand++; + /* compute its Markowitz cost */ + cost = (double)(min_len - 1) * (double)(len - 1); + /* choose between the current candidate and this element */ + if (cost < best) piv = some, best = cost; + /* if piv_lim candidates have been considered, there is a + doubt that a much better candidate exists; therefore it + is the time to terminate the search */ + if (ncand == piv_lim) goto done; + } + /* now consider active rows having len non-zeros */ + for (i = R_head[len]; i != 0; i = R_next[i]) + { /* i-th row has len non-zeros */ + /* find an element in the column of minimal length */ + some = NULL, min_len = INT_MAX; + for (vij = V_row[i]; vij != NULL; vij = vij->r_next) + { if (min_len > C_len[vij->j]) + some = vij, min_len = C_len[vij->j]; + /* if Markowitz cost of this 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) + { piv = some; + goto done; + } + } + /* i-th row has been scanned */ + /* the minimal element found is a next pivot candidate */ + xassert(some != NULL); + ncand++; + /* compute its Markowitz cost */ + cost = (double)(len - 1) * (double)(min_len - 1); + /* choose between the current candidate and this element */ + if (cost < best) piv = some, best = cost; + /* if piv_lim candidates have been considered, there is a + doubt that a much better candidate exists; therefore it + is the time to terminate the search */ + if (ncand == piv_lim) goto done; + } + } +done: /* bring the pivot v[p,q] to the factorizing routine */ + return piv; +} + +/*---------------------------------------------------------------------- +// 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 parameter piv specifies the pivot element v[p,q] = 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. */ + +static void eliminate(LUX *lux, LUXWKA *wka, LUXELM *piv, int flag[], + mpq_t work[]) +{ DMP *pool = lux->pool; + LUXELM **F_row = lux->F_row; + LUXELM **F_col = lux->F_col; + mpq_t *V_piv = lux->V_piv; + LUXELM **V_row = lux->V_row; + LUXELM **V_col = lux->V_col; + int *R_len = wka->R_len; + int *R_head = wka->R_head; + int *R_prev = wka->R_prev; + int *R_next = wka->R_next; + int *C_len = wka->C_len; + int *C_head = wka->C_head; + int *C_prev = wka->C_prev; + int *C_next = wka->C_next; + LUXELM *fip, *vij, *vpj, *viq, *next; + mpq_t temp; + int i, j, p, q; + mpq_init(temp); + /* determine row and column indices of the pivot v[p,q] */ + xassert(piv != NULL); + p = piv->i, q = piv->j; + /* remove p-th (pivot) row from the active set; it will never + return there */ + if (R_prev[p] == 0) + R_head[R_len[p]] = R_next[p]; + else + R_next[R_prev[p]] = R_next[p]; + if (R_next[p] == 0) + ; + else + R_prev[R_next[p]] = R_prev[p]; + /* remove q-th (pivot) column from the active set; it will never + return there */ + if (C_prev[q] == 0) + C_head[C_len[q]] = C_next[q]; + else + C_next[C_prev[q]] = C_next[q]; + if (C_next[q] == 0) + ; + else + C_prev[C_next[q]] = C_prev[q]; + /* store the pivot value in a separate array */ + mpq_set(V_piv[p], piv->val); + /* remove the pivot from p-th row */ + if (piv->r_prev == NULL) + V_row[p] = piv->r_next; + else + piv->r_prev->r_next = piv->r_next; + if (piv->r_next == NULL) + ; + else + piv->r_next->r_prev = piv->r_prev; + R_len[p]--; + /* remove the pivot from q-th column */ + if (piv->c_prev == NULL) + V_col[q] = piv->c_next; + else + piv->c_prev->c_next = piv->c_next; + if (piv->c_next == NULL) + ; + else + piv->c_next->c_prev = piv->c_prev; + C_len[q]--; + /* free the space occupied by the pivot */ + mpq_clear(piv->val); + dmp_free_atom(pool, piv, sizeof(LUXELM)); + /* walk through p-th (pivot) row, which already does not contain + the pivot v[p,q], and do the following... */ + for (vpj = V_row[p]; vpj != NULL; vpj = vpj->r_next) + { /* get column index of v[p,j] */ + j = vpj->j; + /* store v[p,j] in the working array */ + flag[j] = 1; + mpq_set(work[j], vpj->val); + /* remove j-th column from the active set; it will return there + later with a new length */ + if (C_prev[j] == 0) + C_head[C_len[j]] = C_next[j]; + else + C_next[C_prev[j]] = C_next[j]; + if (C_next[j] == 0) + ; + else + C_prev[C_next[j]] = C_prev[j]; + /* v[p,j] leaves the active submatrix, so remove it from j-th + column; however, v[p,j] is kept in p-th row */ + if (vpj->c_prev == NULL) + V_col[j] = vpj->c_next; + else + vpj->c_prev->c_next = vpj->c_next; + if (vpj->c_next == NULL) + ; + else + vpj->c_next->c_prev = vpj->c_prev; + C_len[j]--; + } + /* now walk through q-th (pivot) column, which already does not + contain the pivot v[p,q], and perform gaussian elimination */ + while (V_col[q] != NULL) + { /* element v[i,q] has to be eliminated */ + viq = V_col[q]; + /* get row index of v[i,q] */ + i = viq->i; + /* remove i-th row from the active set; later it will return + there with a new length */ + if (R_prev[i] == 0) + R_head[R_len[i]] = R_next[i]; + else + R_next[R_prev[i]] = R_next[i]; + if (R_next[i] == 0) + ; + else + R_prev[R_next[i]] = R_prev[i]; + /* compute gaussian multiplier f[i,p] = v[i,q] / v[p,q] and + store it in the matrix F */ + fip = dmp_get_atom(pool, sizeof(LUXELM)); + fip->i = i, fip->j = p; + mpq_init(fip->val); + mpq_div(fip->val, viq->val, V_piv[p]); + fip->r_prev = NULL; + fip->r_next = F_row[i]; + fip->c_prev = NULL; + fip->c_next = F_col[p]; + if (fip->r_next != NULL) fip->r_next->r_prev = fip; + if (fip->c_next != NULL) fip->c_next->c_prev = fip; + F_row[i] = F_col[p] = fip; + /* v[i,q] has to be eliminated, so remove it from i-th row */ + if (viq->r_prev == NULL) + V_row[i] = viq->r_next; + else + viq->r_prev->r_next = viq->r_next; + if (viq->r_next == NULL) + ; + else + viq->r_next->r_prev = viq->r_prev; + R_len[i]--; + /* and also from q-th column */ + V_col[q] = viq->c_next; + C_len[q]--; + /* free the space occupied by v[i,q] */ + mpq_clear(viq->val); + dmp_free_atom(pool, viq, sizeof(LUXELM)); + /* perform gaussian transformation: + (i-th row) := (i-th row) - f[i,p] * (p-th row) + note that now p-th row, which is in the working array, + does not contain the pivot v[p,q], and i-th row does not + contain the element v[i,q] to be eliminated */ + /* walk through i-th row and transform existing non-zero + elements */ + for (vij = V_row[i]; vij != NULL; vij = next) + { next = vij->r_next; + /* get column index of v[i,j] */ + j = vij->j; + /* v[i,j] := v[i,j] - f[i,p] * v[p,j] */ + if (flag[j]) + { /* v[p,j] != 0 */ + flag[j] = 0; + mpq_mul(temp, fip->val, work[j]); + mpq_sub(vij->val, vij->val, temp); + if (mpq_sgn(vij->val) == 0) + { /* new v[i,j] is zero, so remove it from the active + submatrix */ + /* remove v[i,j] from i-th row */ + if (vij->r_prev == NULL) + V_row[i] = vij->r_next; + else + vij->r_prev->r_next = vij->r_next; + if (vij->r_next == NULL) + ; + else + vij->r_next->r_prev = vij->r_prev; + R_len[i]--; + /* remove v[i,j] from j-th column */ + if (vij->c_prev == NULL) + V_col[j] = vij->c_next; + else + vij->c_prev->c_next = vij->c_next; + if (vij->c_next == NULL) + ; + else + vij->c_next->c_prev = vij->c_prev; + C_len[j]--; + /* free the space occupied by v[i,j] */ + mpq_clear(vij->val); + dmp_free_atom(pool, vij, sizeof(LUXELM)); + } + } + } + /* now flag is the pattern of the set v[p,*] \ v[i,*] */ + /* walk through p-th (pivot) row and create new elements in + i-th row, which appear due to fill-in */ + for (vpj = V_row[p]; vpj != NULL; vpj = vpj->r_next) + { j = vpj->j; + if (flag[j]) + { /* create new non-zero v[i,j] = 0 - f[i,p] * v[p,j] and + add it to i-th row and j-th column */ + vij = dmp_get_atom(pool, sizeof(LUXELM)); + vij->i = i, vij->j = j; + mpq_init(vij->val); + mpq_mul(vij->val, fip->val, work[j]); + mpq_neg(vij->val, vij->val); + vij->r_prev = NULL; + vij->r_next = V_row[i]; + vij->c_prev = NULL; + vij->c_next = V_col[j]; + if (vij->r_next != NULL) vij->r_next->r_prev = vij; + if (vij->c_next != NULL) vij->c_next->c_prev = vij; + V_row[i] = V_col[j] = vij; + R_len[i]++, C_len[j]++; + } + else + { /* there is no fill-in, because v[i,j] already exists in + i-th row; restore the flag, which was reset before */ + flag[j] = 1; + } + } + /* now i-th row has been completely transformed and can return + to the active set with a new length */ + R_prev[i] = 0; + R_next[i] = R_head[R_len[i]]; + if (R_next[i] != 0) R_prev[R_next[i]] = i; + R_head[R_len[i]] = i; + } + /* at this point q-th (pivot) column must be empty */ + xassert(C_len[q] == 0); + /* walk through p-th (pivot) row again and do the following... */ + for (vpj = V_row[p]; vpj != NULL; vpj = vpj->r_next) + { /* get column index of v[p,j] */ + j = vpj->j; + /* erase v[p,j] from the working array */ + flag[j] = 0; + mpq_set_si(work[j], 0, 1); + /* now j-th column has been completely transformed, so it can + return to the active list with a new length */ + C_prev[j] = 0; + C_next[j] = C_head[C_len[j]]; + if (C_next[j] != 0) C_prev[C_next[j]] = j; + C_head[C_len[j]] = j; + } + mpq_clear(temp); + /* return to the factorizing routine */ + return; +} + +/*---------------------------------------------------------------------- +// lux_decomp - compute LU-factorization. +// +// SYNOPSIS +// +// #include "glplux.h" +// int lux_decomp(LUX *lux, int (*col)(void *info, int j, int ind[], +// mpq_t val[]), void *info); +// +// DESCRIPTION +// +// The routine lux_decomp computes LU-factorization of a given square +// matrix A. +// +// The parameter lux specifies LU-factorization data structure built by +// means of the routine lux_create. +// +// The formal routine col specifies the original matrix A. In order to +// obtain j-th column of the matrix A the routine lux_decomp calls the +// routine col with the parameter j (1 <= j <= n, where n is the order +// of A). In response the routine col should store row indices and +// numerical values of non-zero elements of j-th column of A to the +// locations ind[1], ..., ind[len] and val[1], ..., val[len], resp., +// where len is the number of non-zeros in j-th column, which should be +// returned on exit. Neiter zero nor duplicate elements are allowed. +// +// The parameter info is a transit pointer passed to the formal routine +// col; it can be used for various purposes. +// +// RETURNS +// +// The routine lux_decomp returns the singularity flag. Zero flag means +// that the original matrix A is non-singular while non-zero flag means +// that A is (exactly!) singular. +// +// Note that LU-factorization is valid in both cases, however, in case +// of singularity some rows of the matrix V (including pivot elements) +// will be empty. +// +// REPAIRING SINGULAR MATRIX +// +// If the routine lux_decomp returns non-zero flag, 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 the routine lux_decomp is used for reinverting the basis +// matrix within the simplex method procedure. +// +// On exit linearly dependent columns of the matrix U have the numbers +// rank+1, rank+2, ..., n, where rank is the exact rank of the matrix A +// stored by the routine to the member lux->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 the numbers +// Q_col[rank+1], Q_col[rank+2], ..., Q_col[n], where Q_col is an array +// representing the permutation matrix Q in column-like format. 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 P_row[j], where P_row is an array +// representing the permutation matrix P in row-like format. Thus, each +// j-th linearly dependent column of the matrix U should be replaced by +// a column of the unity matrix with the number P_row[j]. +// +// The code that repairs the matrix A may look like follows: +// +// for (j = rank+1; j <= n; j++) +// { replace column Q_col[j] of the matrix A by column P_row[j] of +// the unity matrix; +// } +// +// where rank, P_row, and Q_col are members of the structure LUX. */ + +int lux_decomp(LUX *lux, int (*col)(void *info, int j, int ind[], + mpq_t val[]), void *info) +{ int n = lux->n; + LUXELM **V_row = lux->V_row; + LUXELM **V_col = lux->V_col; + int *P_row = lux->P_row; + int *P_col = lux->P_col; + int *Q_row = lux->Q_row; + int *Q_col = lux->Q_col; + LUXELM *piv, *vij; + LUXWKA *wka; + int i, j, k, p, q, t, *flag; + mpq_t *work; + /* allocate working area */ + wka = xmalloc(sizeof(LUXWKA)); + wka->R_len = xcalloc(1+n, sizeof(int)); + wka->R_head = xcalloc(1+n, sizeof(int)); + wka->R_prev = xcalloc(1+n, sizeof(int)); + wka->R_next = xcalloc(1+n, sizeof(int)); + wka->C_len = xcalloc(1+n, sizeof(int)); + wka->C_head = xcalloc(1+n, sizeof(int)); + wka->C_prev = xcalloc(1+n, sizeof(int)); + wka->C_next = xcalloc(1+n, sizeof(int)); + /* initialize LU-factorization data structures */ + initialize(lux, col, info, wka); + /* allocate working arrays */ + flag = xcalloc(1+n, sizeof(int)); + work = xcalloc(1+n, sizeof(mpq_t)); + for (k = 1; k <= n; k++) + { flag[k] = 0; + mpq_init(work[k]); + } + /* main elimination loop */ + for (k = 1; k <= n; k++) + { /* choose a pivot element v[p,q] */ + piv = find_pivot(lux, wka); + if (piv == NULL) + { /* no pivot can be chosen, because the active submatrix is + empty */ + break; + } + /* determine row and column indices of the pivot element */ + p = piv->i, q = piv->j; + /* 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 = P_col[p], j = Q_row[q]; + xassert(k <= i && i <= n && k <= j && j <= n); + /* permute k-th and i-th rows of the matrix U */ + t = P_row[k]; + P_row[i] = t, P_col[t] = i; + P_row[k] = p, P_col[p] = k; + /* permute k-th and j-th columns of the matrix U */ + t = Q_col[k]; + Q_col[j] = t, Q_row[t] = j; + Q_col[k] = q, Q_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] */ + eliminate(lux, wka, piv, flag, work); + } + /* determine the rank of A (and V) */ + lux->rank = k - 1; + /* free working arrays */ + xfree(flag); + for (k = 1; k <= n; k++) mpq_clear(work[k]); + xfree(work); + /* build column lists of the matrix V using its row lists */ + for (j = 1; j <= n; j++) + xassert(V_col[j] == NULL); + for (i = 1; i <= n; i++) + { for (vij = V_row[i]; vij != NULL; vij = vij->r_next) + { j = vij->j; + vij->c_prev = NULL; + vij->c_next = V_col[j]; + if (vij->c_next != NULL) vij->c_next->c_prev = vij; + V_col[j] = vij; + } + } + /* free working area */ + xfree(wka->R_len); + xfree(wka->R_head); + xfree(wka->R_prev); + xfree(wka->R_next); + xfree(wka->C_len); + xfree(wka->C_head); + xfree(wka->C_prev); + xfree(wka->C_next); + xfree(wka); + /* return to the calling program */ + return (lux->rank < n); +} + +/*---------------------------------------------------------------------- +// lux_f_solve - solve system F*x = b or F'*x = b. +// +// SYNOPSIS +// +// #include "glplux.h" +// void lux_f_solve(LUX *lux, int tr, mpq_t x[]); +// +// DESCRIPTION +// +// The routine lux_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 lux, 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 lux_f_solve(LUX *lux, int tr, mpq_t x[]) +{ int n = lux->n; + LUXELM **F_row = lux->F_row; + LUXELM **F_col = lux->F_col; + int *P_row = lux->P_row; + LUXELM *fik, *fkj; + int i, j, k; + mpq_t temp; + mpq_init(temp); + if (!tr) + { /* solve the system F*x = b */ + for (j = 1; j <= n; j++) + { k = P_row[j]; + if (mpq_sgn(x[k]) != 0) + { for (fik = F_col[k]; fik != NULL; fik = fik->c_next) + { mpq_mul(temp, fik->val, x[k]); + mpq_sub(x[fik->i], x[fik->i], temp); + } + } + } + } + else + { /* solve the system F'*x = b */ + for (i = n; i >= 1; i--) + { k = P_row[i]; + if (mpq_sgn(x[k]) != 0) + { for (fkj = F_row[k]; fkj != NULL; fkj = fkj->r_next) + { mpq_mul(temp, fkj->val, x[k]); + mpq_sub(x[fkj->j], x[fkj->j], temp); + } + } + } + } + mpq_clear(temp); + return; +} + +/*---------------------------------------------------------------------- +// lux_v_solve - solve system V*x = b or V'*x = b. +// +// SYNOPSIS +// +// #include "glplux.h" +// void lux_v_solve(LUX *lux, int tr, double x[]); +// +// DESCRIPTION +// +// The routine lux_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 lux, 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 lux_v_solve(LUX *lux, int tr, mpq_t x[]) +{ int n = lux->n; + mpq_t *V_piv = lux->V_piv; + LUXELM **V_row = lux->V_row; + LUXELM **V_col = lux->V_col; + int *P_row = lux->P_row; + int *Q_col = lux->Q_col; + LUXELM *vij; + int i, j, k; + mpq_t *b, temp; + b = xcalloc(1+n, sizeof(mpq_t)); + for (k = 1; k <= n; k++) + mpq_init(b[k]), mpq_set(b[k], x[k]), mpq_set_si(x[k], 0, 1); + mpq_init(temp); + if (!tr) + { /* solve the system V*x = b */ + for (k = n; k >= 1; k--) + { i = P_row[k], j = Q_col[k]; + if (mpq_sgn(b[i]) != 0) + { mpq_set(x[j], b[i]); + mpq_div(x[j], x[j], V_piv[i]); + for (vij = V_col[j]; vij != NULL; vij = vij->c_next) + { mpq_mul(temp, vij->val, x[j]); + mpq_sub(b[vij->i], b[vij->i], temp); + } + } + } + } + else + { /* solve the system V'*x = b */ + for (k = 1; k <= n; k++) + { i = P_row[k], j = Q_col[k]; + if (mpq_sgn(b[j]) != 0) + { mpq_set(x[i], b[j]); + mpq_div(x[i], x[i], V_piv[i]); + for (vij = V_row[i]; vij != NULL; vij = vij->r_next) + { mpq_mul(temp, vij->val, x[i]); + mpq_sub(b[vij->j], b[vij->j], temp); + } + } + } + } + for (k = 1; k <= n; k++) mpq_clear(b[k]); + mpq_clear(temp); + xfree(b); + return; +} + +/*---------------------------------------------------------------------- +// lux_solve - solve system A*x = b or A'*x = b. +// +// SYNOPSIS +// +// #include "glplux.h" +// void lux_solve(LUX *lux, int tr, mpq_t x[]); +// +// DESCRIPTION +// +// The routine lux_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 lux 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 lux_solve(LUX *lux, int tr, mpq_t x[]) +{ if (lux->rank < lux->n) + xfault("lux_solve: LU-factorization has incomplete rank\n"); + if (!tr) + { /* A = F*V, therefore inv(A) = inv(V)*inv(F) */ + lux_f_solve(lux, 0, x); + lux_v_solve(lux, 0, x); + } + else + { /* A' = V'*F', therefore inv(A') = inv(F')*inv(V') */ + lux_v_solve(lux, 1, x); + lux_f_solve(lux, 1, x); + } + return; +} + +/*---------------------------------------------------------------------- +// lux_delete - delete LU-factorization. +// +// SYNOPSIS +// +// #include "glplux.h" +// void lux_delete(LUX *lux); +// +// DESCRIPTION +// +// The routine lux_delete deletes LU-factorization data structure, +// which the parameter lux points to, freeing all the memory allocated +// to this object. */ + +void lux_delete(LUX *lux) +{ int n = lux->n; + LUXELM *fij, *vij; + int i; + for (i = 1; i <= n; i++) + { for (fij = lux->F_row[i]; fij != NULL; fij = fij->r_next) + mpq_clear(fij->val); + mpq_clear(lux->V_piv[i]); + for (vij = lux->V_row[i]; vij != NULL; vij = vij->r_next) + mpq_clear(vij->val); + } + dmp_delete_pool(lux->pool); + xfree(lux->F_row); + xfree(lux->F_col); + xfree(lux->V_piv); + xfree(lux->V_row); + xfree(lux->V_col); + xfree(lux->P_row); + xfree(lux->P_col); + xfree(lux->Q_row); + xfree(lux->Q_col); + xfree(lux); + return; +} + +/* eof */