lemon-project-template-glpk: deps/glpk/src/glpmat.c comparison

lemon-project-template-glpk

comparison deps/glpk/src/glpmat.c @ 9:33de93886c88

Import GLPK 4.47

author	Alpar Juttner <alpar@cs.elte.hu>
date	Sun, 06 Nov 2011 20:59:10 +0100
parents
children

comparison

equal deleted inserted replaced

--1:000000000000
+:e0504fc901e0
+/* glpmat.c */
+/***********************************************************************
+*  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: <mao@gnu.org>.
+*
+*  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 <http://www.gnu.org/licenses/>.
+***********************************************************************/
+#include "glpenv.h"
+#include "glpmat.h"
+#include "glpqmd.h"
+#include "amd/amd.h"
+#include "colamd/colamd.h"
+/*----------------------------------------------------------------------
+-- check_fvs - check sparse vector in full-vector storage format.
+--
+-- SYNOPSIS
+--
+-- #include "glpmat.h"
+-- int check_fvs(int n, int nnz, int ind[], double vec[]);
+--
+-- DESCRIPTION
+--
+-- The routine check_fvs checks if a given vector of dimension n in
+-- full-vector storage format has correct representation.
+--
+-- RETURNS
+--
+-- The routine returns one of the following codes:
+--
+-- 0 - the vector is correct;
+-- 1 - the number of elements (n) is negative;
+-- 2 - the number of non-zero elements (nnz) is negative;
+-- 3 - some element index is out of range;
+-- 4 - some element index is duplicate;
+-- 5 - some non-zero element is out of pattern. */
+int check_fvs(int n, int nnz, int ind[], double vec[])
+{     int i, t, ret, *flag = NULL;
+/* check the number of elements */
+if (n < 0)
+{  ret = 1;
+goto done;
+}
+/* check the number of non-zero elements */
+if (nnz < 0)
+{  ret = 2;
+goto done;
+}
+/* check vector indices */
+flag = xcalloc(1+n, sizeof(int));
+for (i = 1; i <= n; i++) flag[i] = 0;
+for (t = 1; t <= nnz; t++)
+{  i = ind[t];
+if (!(1 <= i && i <= n))
+{  ret = 3;
+goto done;
+}
+if (flag[i])
+{  ret = 4;
+goto done;
+}
+flag[i] = 1;
+}
+/* check vector elements */
+for (i = 1; i <= n; i++)
+{  if (!flag[i] && vec[i] != 0.0)
+{  ret = 5;
+goto done;
+}
+}
+/* the vector is ok */
+ret = 0;
+done: if (flag != NULL) xfree(flag);
+return ret;
+}
+/*----------------------------------------------------------------------
+-- check_pattern - check pattern of sparse matrix.
+--
+-- SYNOPSIS
+--
+-- #include "glpmat.h"
+-- int check_pattern(int m, int n, int A_ptr[], int A_ind[]);
+--
+-- DESCRIPTION
+--
+-- The routine check_pattern checks the pattern of a given mxn matrix
+-- in storage-by-rows format.
+--
+-- RETURNS
+--
+-- The routine returns one of the following codes:
+--
+-- 0 - the pattern is correct;
+-- 1 - the number of rows (m) is negative;
+-- 2 - the number of columns (n) is negative;
+-- 3 - A_ptr[1] is not 1;
+-- 4 - some column index is out of range;
+-- 5 - some column indices are duplicate. */
+int check_pattern(int m, int n, int A_ptr[], int A_ind[])
+{     int i, j, ptr, ret, *flag = NULL;
+/* check the number of rows */
+if (m < 0)
+{  ret = 1;
+goto done;
+}
+/* check the number of columns */
+if (n < 0)
+{  ret = 2;
+goto done;
+}
+/* check location A_ptr[1] */
+if (A_ptr[1] != 1)
+{  ret = 3;
+goto done;
+}
+/* check row patterns */
+flag = xcalloc(1+n, sizeof(int));
+for (j = 1; j <= n; j++) flag[j] = 0;
+for (i = 1; i <= m; i++)
+{  /* check pattern of row i */
+for (ptr = A_ptr[i]; ptr < A_ptr[i+1]; ptr++)
+{  j = A_ind[ptr];
+/* check column index */
+if (!(1 <= j && j <= n))
+{  ret = 4;
+goto done;
+}
+/* check for duplication */
+if (flag[j])
+{  ret = 5;
+goto done;
+}
+flag[j] = 1;
+}
+/* clear flags */
+for (ptr = A_ptr[i]; ptr < A_ptr[i+1]; ptr++)
+{  j = A_ind[ptr];
+flag[j] = 0;
+}
+}
+/* the pattern is ok */
+ret = 0;
+done: if (flag != NULL) xfree(flag);
+return ret;
+}
+/*----------------------------------------------------------------------
+-- transpose - transpose sparse matrix.
+--
+-- *Synopsis*
+--
+-- #include "glpmat.h"
+-- void transpose(int m, int n, int A_ptr[], int A_ind[],
+--    double A_val[], int AT_ptr[], int AT_ind[], double AT_val[]);
+--
+-- *Description*
+--
+-- For a given mxn sparse matrix A the routine transpose builds a nxm
+-- sparse matrix A' which is a matrix transposed to A.
+--
+-- The arrays A_ptr, A_ind, and A_val specify a given mxn matrix A to
+-- be transposed in storage-by-rows format. The parameter A_val can be
+-- NULL, in which case numeric values are not copied. The arrays A_ptr,
+-- A_ind, and A_val are not changed on exit.
+--
+-- On entry the arrays AT_ptr, AT_ind, and AT_val must be allocated,
+-- but their content is ignored. On exit the routine stores a resultant
+-- nxm matrix A' in these arrays in storage-by-rows format. Note that
+-- if the parameter A_val is NULL, the array AT_val is not used.
+--
+-- The routine transpose has a side effect that elements in rows of the
+-- resultant matrix A' follow in ascending their column indices. */
+void transpose(int m, int n, int A_ptr[], int A_ind[], double A_val[],
+int AT_ptr[], int AT_ind[], double AT_val[])
+{     int i, j, t, beg, end, pos, len;
+/* determine row lengths of resultant matrix */
+for (j = 1; j <= n; j++) AT_ptr[j] = 0;
+for (i = 1; i <= m; i++)
+{  beg = A_ptr[i], end = A_ptr[i+1];
+for (t = beg; t < end; t++) AT_ptr[A_ind[t]]++;
+}
+/* set up row pointers of resultant matrix */
+pos = 1;
+for (j = 1; j <= n; j++)
+len = AT_ptr[j], pos += len, AT_ptr[j] = pos;
+AT_ptr[n+1] = pos;
+/* build resultant matrix */
+for (i = m; i >= 1; i--)
+{  beg = A_ptr[i], end = A_ptr[i+1];
+for (t = beg; t < end; t++)
+{  pos = --AT_ptr[A_ind[t]];
+AT_ind[pos] = i;
+if (A_val != NULL) AT_val[pos] = A_val[t];
+}
+}
+return;
+}
+/*----------------------------------------------------------------------
+-- adat_symbolic - compute S = P*A*D*A'*P' (symbolic phase).
+--
+-- *Synopsis*
+--
+-- #include "glpmat.h"
+-- int *adat_symbolic(int m, int n, int P_per[], int A_ptr[],
+--    int A_ind[], int S_ptr[]);
+--
+-- *Description*
+--
+-- The routine adat_symbolic implements the symbolic phase to compute
+-- symmetric matrix S = P*A*D*A'*P', where P is a permutation matrix,
+-- A is a given sparse matrix, D is a diagonal matrix, A' is a matrix
+-- transposed to A, P' is an inverse of P.
+--
+-- The parameter m is the number of rows in A and the order of P.
+--
+-- The parameter n is the number of columns in A and the order of D.
+--
+-- The array P_per specifies permutation matrix P. It is not changed on
+-- exit.
+--
+-- The arrays A_ptr and A_ind specify the pattern of matrix A. They are
+-- not changed on exit.
+--
+-- On exit the routine stores the pattern of upper triangular part of
+-- matrix S without diagonal elements in the arrays S_ptr and S_ind in
+-- storage-by-rows format. The array S_ptr should be allocated on entry,
+-- however, its content is ignored. The array S_ind is allocated by the
+-- routine itself which returns a pointer to it.
+--
+-- *Returns*
+--
+-- The routine returns a pointer to the array S_ind. */
+int *adat_symbolic(int m, int n, int P_per[], int A_ptr[], int A_ind[],
+int S_ptr[])
+{     int i, j, t, ii, jj, tt, k, size, len;
+int *S_ind, *AT_ptr, *AT_ind, *ind, *map, *temp;
+/* build the pattern of A', which is a matrix transposed to A, to
+efficiently access A in column-wise manner */
+AT_ptr = xcalloc(1+n+1, sizeof(int));
+AT_ind = xcalloc(A_ptr[m+1], sizeof(int));
+transpose(m, n, A_ptr, A_ind, NULL, AT_ptr, AT_ind, NULL);
+/* allocate the array S_ind */
+size = A_ptr[m+1] - 1;
+if (size < m) size = m;
+S_ind = xcalloc(1+size, sizeof(int));
+/* allocate and initialize working arrays */
+ind = xcalloc(1+m, sizeof(int));
+map = xcalloc(1+m, sizeof(int));
+for (jj = 1; jj <= m; jj++) map[jj] = 0;
+/* compute pattern of S; note that symbolically S = B*B', where
+B = P*A, B' is matrix transposed to B */
+S_ptr[1] = 1;
+for (ii = 1; ii <= m; ii++)
+{  /* compute pattern of ii-th row of S */
+len = 0;
+i = P_per[ii]; /* i-th row of A = ii-th row of B */
+for (t = A_ptr[i]; t < A_ptr[i+1]; t++)
+{  k = A_ind[t];
+/* walk through k-th column of A */
+for (tt = AT_ptr[k]; tt < AT_ptr[k+1]; tt++)
+{  j = AT_ind[tt];
+jj = P_per[m+j]; /* j-th row of A = jj-th row of B */
+/* a[i,k] != 0 and a[j,k] != 0 ergo s[ii,jj] != 0 */
+if (ii < jj && !map[jj]) ind[++len] = jj, map[jj] = 1;
+}
+}
+/* now (ind) is pattern of ii-th row of S */
+S_ptr[ii+1] = S_ptr[ii] + len;
+/* at least (S_ptr[ii+1] - 1) locations should be available in
+the array S_ind */
+if (S_ptr[ii+1] - 1 > size)
+{  temp = S_ind;
+size += size;
+S_ind = xcalloc(1+size, sizeof(int));
+memcpy(&S_ind[1], &temp[1], (S_ptr[ii] - 1) * sizeof(int));
+xfree(temp);
+}
+xassert(S_ptr[ii+1] - 1 <= size);
+/* (ii-th row of S) := (ind) */
+memcpy(&S_ind[S_ptr[ii]], &ind[1], len * sizeof(int));
+/* clear the row pattern map */
+for (t = 1; t <= len; t++) map[ind[t]] = 0;
+}
+/* free working arrays */
+xfree(AT_ptr);
+xfree(AT_ind);
+xfree(ind);
+xfree(map);
+/* reallocate the array S_ind to free unused locations */
+temp = S_ind;
+size = S_ptr[m+1] - 1;
+S_ind = xcalloc(1+size, sizeof(int));
+memcpy(&S_ind[1], &temp[1], size * sizeof(int));
+xfree(temp);
+return S_ind;
+}
+/*----------------------------------------------------------------------
+-- adat_numeric - compute S = P*A*D*A'*P' (numeric phase).
+--
+-- *Synopsis*
+--
+-- #include "glpmat.h"
+-- void adat_numeric(int m, int n, int P_per[],
+--    int A_ptr[], int A_ind[], double A_val[], double D_diag[],
+--    int S_ptr[], int S_ind[], double S_val[], double S_diag[]);
+--
+-- *Description*
+--
+-- The routine adat_numeric implements the numeric phase to compute
+-- symmetric matrix S = P*A*D*A'*P', where P is a permutation matrix,
+-- A is a given sparse matrix, D is a diagonal matrix, A' is a matrix
+-- transposed to A, P' is an inverse of P.
+--
+-- The parameter m is the number of rows in A and the order of P.
+--
+-- The parameter n is the number of columns in A and the order of D.
+--
+-- The matrix P is specified in the array P_per, which is not changed
+-- on exit.
+--
+-- The matrix A is specified in the arrays A_ptr, A_ind, and A_val in
+-- storage-by-rows format. These arrays are not changed on exit.
+--
+-- Diagonal elements of the matrix D are specified in the array D_diag,
+-- where D_diag[0] is not used, D_diag[i] = d[i,i] for i = 1, ..., n.
+-- The array D_diag is not changed on exit.
+--
+-- The pattern of the upper triangular part of the matrix S without
+-- diagonal elements (previously computed by the routine adat_symbolic)
+-- is specified in the arrays S_ptr and S_ind, which are not changed on
+-- exit. Numeric values of non-diagonal elements of S are stored in
+-- corresponding locations of the array S_val, and values of diagonal
+-- elements of S are stored in locations S_diag[1], ..., S_diag[n]. */
+void adat_numeric(int m, int n, int P_per[],
+int A_ptr[], int A_ind[], double A_val[], double D_diag[],
+int S_ptr[], int S_ind[], double S_val[], double S_diag[])
+{     int i, j, t, ii, jj, tt, beg, end, beg1, end1, k;
+double sum, *work;
+work = xcalloc(1+n, sizeof(double));
+for (j = 1; j <= n; j++) work[j] = 0.0;
+/* compute S = B*D*B', where B = P*A, B' is a matrix transposed
+to B */
+for (ii = 1; ii <= m; ii++)
+{  i = P_per[ii]; /* i-th row of A = ii-th row of B */
+/* (work) := (i-th row of A) */
+beg = A_ptr[i], end = A_ptr[i+1];
+for (t = beg; t < end; t++)
+work[A_ind[t]] = A_val[t];
+/* compute ii-th row of S */
+beg = S_ptr[ii], end = S_ptr[ii+1];
+for (t = beg; t < end; t++)
+{  jj = S_ind[t];
+j = P_per[jj]; /* j-th row of A = jj-th row of B */
+/* s[ii,jj] := sum a[i,k] * d[k,k] * a[j,k] */
+sum = 0.0;
+beg1 = A_ptr[j], end1 = A_ptr[j+1];
+for (tt = beg1; tt < end1; tt++)
+{  k = A_ind[tt];
+sum += work[k] * D_diag[k] * A_val[tt];
+}
+S_val[t] = sum;
+}
+/* s[ii,ii] := sum a[i,k] * d[k,k] * a[i,k] */
+sum = 0.0;
+beg = A_ptr[i], end = A_ptr[i+1];
+for (t = beg; t < end; t++)
+{  k = A_ind[t];
+sum += A_val[t] * D_diag[k] * A_val[t];
+work[k] = 0.0;
+}
+S_diag[ii] = sum;
+}
+xfree(work);
+return;
+}
+/*----------------------------------------------------------------------
+-- min_degree - minimum degree ordering.
+--
+-- *Synopsis*
+--
+-- #include "glpmat.h"
+-- void min_degree(int n, int A_ptr[], int A_ind[], int P_per[]);
+--
+-- *Description*
+--
+-- The routine min_degree uses the minimum degree ordering algorithm
+-- to find a permutation matrix P for a given sparse symmetric positive
+-- matrix A which minimizes the number of non-zeros in upper triangular
+-- factor U for Cholesky factorization P*A*P' = U'*U.
+--
+-- The parameter n is the order of matrices A and P.
+--
+-- The pattern of the given matrix A is specified on entry in the arrays
+-- A_ptr and A_ind in storage-by-rows format. Only the upper triangular
+-- part without diagonal elements (which all are assumed to be non-zero)
+-- should be specified as if A were upper triangular. The arrays A_ptr
+-- and A_ind are not changed on exit.
+--
+-- The permutation matrix P is stored by the routine in the array P_per
+-- on exit.
+--
+-- *Algorithm*
+--
+-- The routine min_degree is based on some subroutines from the package
+-- SPARSPAK (see comments in the module glpqmd). */
+void min_degree(int n, int A_ptr[], int A_ind[], int P_per[])
+{     int i, j, ne, t, pos, len;
+int *xadj, *adjncy, *deg, *marker, *rchset, *nbrhd, *qsize,
+*qlink, nofsub;
+/* determine number of non-zeros in complete pattern */
+ne = A_ptr[n+1] - 1;
+ne += ne;
+/* allocate working arrays */
+xadj = xcalloc(1+n+1, sizeof(int));
+adjncy = xcalloc(1+ne, sizeof(int));
+deg = xcalloc(1+n, sizeof(int));
+marker = xcalloc(1+n, sizeof(int));
+rchset = xcalloc(1+n, sizeof(int));
+nbrhd = xcalloc(1+n, sizeof(int));
+qsize = xcalloc(1+n, sizeof(int));
+qlink = xcalloc(1+n, sizeof(int));
+/* determine row lengths in complete pattern */
+for (i = 1; i <= n; i++) xadj[i] = 0;
+for (i = 1; i <= n; i++)
+{  for (t = A_ptr[i]; t < A_ptr[i+1]; t++)
+{  j = A_ind[t];
+xassert(i < j && j <= n);
+xadj[i]++, xadj[j]++;
+}
+}
+/* set up row pointers for complete pattern */
+pos = 1;
+for (i = 1; i <= n; i++)
+len = xadj[i], pos += len, xadj[i] = pos;
+xadj[n+1] = pos;
+xassert(pos - 1 == ne);
+/* construct complete pattern */
+for (i = 1; i <= n; i++)
+{  for (t = A_ptr[i]; t < A_ptr[i+1]; t++)
+{  j = A_ind[t];
+adjncy[--xadj[i]] = j, adjncy[--xadj[j]] = i;
+}
+}
+/* call the main minimimum degree ordering routine */
+genqmd(&n, xadj, adjncy, P_per, P_per + n, deg, marker, rchset,
+nbrhd, qsize, qlink, &nofsub);
+/* make sure that permutation matrix P is correct */
+for (i = 1; i <= n; i++)
+{  j = P_per[i];
+xassert(1 <= j && j <= n);
+xassert(P_per[n+j] == i);
+}
+/* free working arrays */
+xfree(xadj);
+xfree(adjncy);
+xfree(deg);
+xfree(marker);
+xfree(rchset);
+xfree(nbrhd);
+xfree(qsize);
+xfree(qlink);
+return;
+}
+/**********************************************************************/
+void amd_order1(int n, int A_ptr[], int A_ind[], int P_per[])
+{     /* approximate minimum degree ordering (AMD) */
+int k, ret;
+double Control[AMD_CONTROL], Info[AMD_INFO];
+/* get the default parameters */
+amd_defaults(Control);
+#if 0
+/* and print them */
+amd_control(Control);
+#endif
+/* make all indices 0-based */
+for (k = 1; k < A_ptr[n+1]; k++) A_ind[k]--;
+for (k = 1; k <= n+1; k++) A_ptr[k]--;
+/* call the ordering routine */
+ret = amd_order(n, &A_ptr[1], &A_ind[1], &P_per[1], Control, Info)
+;
+#if 0
+amd_info(Info);
+#endif
+xassert(ret == AMD_OK || ret == AMD_OK_BUT_JUMBLED);
+/* retsore 1-based indices */
+for (k = 1; k <= n+1; k++) A_ptr[k]++;
+for (k = 1; k < A_ptr[n+1]; k++) A_ind[k]++;
+/* patch up permutation matrix */
+memset(&P_per[n+1], 0, n * sizeof(int));
+for (k = 1; k <= n; k++)
+{  P_per[k]++;
+xassert(1 <= P_per[k] && P_per[k] <= n);
+xassert(P_per[n+P_per[k]] == 0);
+P_per[n+P_per[k]] = k;
+}
+return;
+}
+/**********************************************************************/
+static void *allocate(size_t n, size_t size)
+{     void *ptr;
+ptr = xcalloc(n, size);
+memset(ptr, 0, n * size);
+return ptr;
+}
+static void release(void *ptr)
+{     xfree(ptr);
+return;
+}
+void symamd_ord(int n, int A_ptr[], int A_ind[], int P_per[])
+{     /* approximate minimum degree ordering (SYMAMD) */
+int k, ok;
+int stats[COLAMD_STATS];
+/* make all indices 0-based */
+for (k = 1; k < A_ptr[n+1]; k++) A_ind[k]--;
+for (k = 1; k <= n+1; k++) A_ptr[k]--;
+/* call the ordering routine */
+ok = symamd(n, &A_ind[1], &A_ptr[1], &P_per[1], NULL, stats,
+allocate, release);
+#if 0
+symamd_report(stats);
+#endif
+xassert(ok);
+/* restore 1-based indices */
+for (k = 1; k <= n+1; k++) A_ptr[k]++;
+for (k = 1; k < A_ptr[n+1]; k++) A_ind[k]++;
+/* patch up permutation matrix */
+memset(&P_per[n+1], 0, n * sizeof(int));
+for (k = 1; k <= n; k++)
+{  P_per[k]++;
+xassert(1 <= P_per[k] && P_per[k] <= n);
+xassert(P_per[n+P_per[k]] == 0);
+P_per[n+P_per[k]] = k;
+}
+return;
+}
+/*----------------------------------------------------------------------
+-- chol_symbolic - compute Cholesky factorization (symbolic phase).
+--
+-- *Synopsis*
+--
+-- #include "glpmat.h"
+-- int *chol_symbolic(int n, int A_ptr[], int A_ind[], int U_ptr[]);
+--
+-- *Description*
+--
+-- The routine chol_symbolic implements the symbolic phase of Cholesky
+-- factorization A = U'*U, where A is a given sparse symmetric positive
+-- definite matrix, U is a resultant upper triangular factor, U' is a
+-- matrix transposed to U.
+--
+-- The parameter n is the order of matrices A and U.
+--
+-- The pattern of the given matrix A is specified on entry in the arrays
+-- A_ptr and A_ind in storage-by-rows format. Only the upper triangular
+-- part without diagonal elements (which all are assumed to be non-zero)
+-- should be specified as if A were upper triangular. The arrays A_ptr
+-- and A_ind are not changed on exit.
+--
+-- The pattern of the matrix U without diagonal elements (which all are
+-- assumed to be non-zero) is stored on exit from the routine in the
+-- arrays U_ptr and U_ind in storage-by-rows format. The array U_ptr
+-- should be allocated on entry, however, its content is ignored. The
+-- array U_ind is allocated by the routine which returns a pointer to it
+-- on exit.
+--
+-- *Returns*
+--
+-- The routine returns a pointer to the array U_ind.
+--
+-- *Method*
+--
+-- The routine chol_symbolic computes the pattern of the matrix U in a
+-- row-wise manner. No pivoting is used.
+--
+-- It is known that to compute the pattern of row k of the matrix U we
+-- need to merge the pattern of row k of the matrix A and the patterns
+-- of each row i of U, where u[i,k] is non-zero (these rows are already
+-- computed and placed above row k).
+--
+-- However, to reduce the number of rows to be merged the routine uses
+-- an advanced algorithm proposed in:
+--
+-- D.J.Rose, R.E.Tarjan, and G.S.Lueker. Algorithmic aspects of vertex
+-- elimination on graphs. SIAM J. Comput. 5, 1976, 266-83.
+--
+-- The authors of the cited paper show that we have the same result if
+-- we merge row k of the matrix A and such rows of the matrix U (among
+-- rows 1, ..., k-1) whose leftmost non-diagonal non-zero element is
+-- placed in k-th column. This feature signficantly reduces the number
+-- of rows to be merged, especially on the final steps, where rows of
+-- the matrix U become quite dense.
+--
+-- To determine rows, which should be merged on k-th step, for a fixed
+-- time the routine uses linked lists of row numbers of the matrix U.
+-- Location head[k] contains the number of a first row, whose leftmost
+-- non-diagonal non-zero element is placed in column k, and location
+-- next[i] contains the number of a next row with the same property as
+-- row i. */
+int *chol_symbolic(int n, int A_ptr[], int A_ind[], int U_ptr[])
+{     int i, j, k, t, len, size, beg, end, min_j, *U_ind, *head, *next,
+*ind, *map, *temp;
+/* initially we assume that on computing the pattern of U fill-in
+will double the number of non-zeros in A */
+size = A_ptr[n+1] - 1;
+if (size < n) size = n;
+size += size;
+U_ind = xcalloc(1+size, sizeof(int));
+/* allocate and initialize working arrays */
+head = xcalloc(1+n, sizeof(int));
+for (i = 1; i <= n; i++) head[i] = 0;
+next = xcalloc(1+n, sizeof(int));
+ind = xcalloc(1+n, sizeof(int));
+map = xcalloc(1+n, sizeof(int));
+for (j = 1; j <= n; j++) map[j] = 0;
+/* compute the pattern of matrix U */
+U_ptr[1] = 1;
+for (k = 1; k <= n; k++)
+{  /* compute the pattern of k-th row of U, which is the union of
+k-th row of A and those rows of U (among 1, ..., k-1) whose
+leftmost non-diagonal non-zero is placed in k-th column */
+/* (ind) := (k-th row of A) */
+len = A_ptr[k+1] - A_ptr[k];
+memcpy(&ind[1], &A_ind[A_ptr[k]], len * sizeof(int));
+for (t = 1; t <= len; t++)
+{  j = ind[t];
+xassert(k < j && j <= n);
+map[j] = 1;
+}
+/* walk through rows of U whose leftmost non-diagonal non-zero
+is placed in k-th column */
+for (i = head[k]; i != 0; i = next[i])
+{  /* (ind) := (ind) union (i-th row of U) */
+beg = U_ptr[i], end = U_ptr[i+1];
+for (t = beg; t < end; t++)
+{  j = U_ind[t];
+if (j > k && !map[j]) ind[++len] = j, map[j] = 1;
+}
+}
+/* now (ind) is the pattern of k-th row of U */
+U_ptr[k+1] = U_ptr[k] + len;
+/* at least (U_ptr[k+1] - 1) locations should be available in
+the array U_ind */
+if (U_ptr[k+1] - 1 > size)
+{  temp = U_ind;
+size += size;
+U_ind = xcalloc(1+size, sizeof(int));
+memcpy(&U_ind[1], &temp[1], (U_ptr[k] - 1) * sizeof(int));
+xfree(temp);
+}
+xassert(U_ptr[k+1] - 1 <= size);
+/* (k-th row of U) := (ind) */
+memcpy(&U_ind[U_ptr[k]], &ind[1], len * sizeof(int));
+/* determine column index of leftmost non-diagonal non-zero in
+k-th row of U and clear the row pattern map */
+min_j = n + 1;
+for (t = 1; t <= len; t++)
+{  j = ind[t], map[j] = 0;
+if (min_j > j) min_j = j;
+}
+/* include k-th row into corresponding linked list */
+if (min_j <= n) next[k] = head[min_j], head[min_j] = k;
+}
+/* free working arrays */
+xfree(head);
+xfree(next);
+xfree(ind);
+xfree(map);
+/* reallocate the array U_ind to free unused locations */
+temp = U_ind;
+size = U_ptr[n+1] - 1;
+U_ind = xcalloc(1+size, sizeof(int));
+memcpy(&U_ind[1], &temp[1], size * sizeof(int));
+xfree(temp);
+return U_ind;
+}
+/*----------------------------------------------------------------------
+-- chol_numeric - compute Cholesky factorization (numeric phase).
+--
+-- *Synopsis*
+--
+-- #include "glpmat.h"
+-- int chol_numeric(int n,
+--    int A_ptr[], int A_ind[], double A_val[], double A_diag[],
+--    int U_ptr[], int U_ind[], double U_val[], double U_diag[]);
+--
+-- *Description*
+--
+-- The routine chol_symbolic implements the numeric phase of Cholesky
+-- factorization A = U'*U, where A is a given sparse symmetric positive
+-- definite matrix, U is a resultant upper triangular factor, U' is a
+-- matrix transposed to U.
+--
+-- The parameter n is the order of matrices A and U.
+--
+-- Upper triangular part of the matrix A without diagonal elements is
+-- specified in the arrays A_ptr, A_ind, and A_val in storage-by-rows
+-- format. Diagonal elements of A are specified in the array A_diag,
+-- where A_diag[0] is not used, A_diag[i] = a[i,i] for i = 1, ..., n.
+-- The arrays A_ptr, A_ind, A_val, and A_diag are not changed on exit.
+--
+-- The pattern of the matrix U without diagonal elements (previously
+-- computed with the routine chol_symbolic) is specified in the arrays
+-- U_ptr and U_ind, which are not changed on exit. Numeric values of
+-- non-diagonal elements of U are stored in corresponding locations of
+-- the array U_val, and values of diagonal elements of U are stored in
+-- locations U_diag[1], ..., U_diag[n].
+--
+-- *Returns*
+--
+-- The routine returns the number of non-positive diagonal elements of
+-- the matrix U which have been replaced by a huge positive number (see
+-- the method description below). Zero return code means the matrix A
+-- has been successfully factorized.
+--
+-- *Method*
+--
+-- The routine chol_numeric computes the matrix U in a row-wise manner
+-- using standard gaussian elimination technique. No pivoting is used.
+--
+-- Initially the routine sets U = A, and before k-th elimination step
+-- the matrix U is the following:
+--
+--       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 'x' are elements of already computed rows, '*' are elements of
+-- the active submatrix. (Note that the lower triangular part of the
+-- active submatrix being symmetric is not stored and diagonal elements
+-- are stored separately in the array U_diag.)
+--
+-- The matrix A is assumed to be positive definite. However, if it is
+-- close to semi-definite, on some elimination step a pivot u[k,k] may
+-- happen to be non-positive due to round-off errors. In this case the
+-- routine uses a technique proposed in:
+--
+-- S.J.Wright. The Cholesky factorization in interior-point and barrier
+-- methods. Preprint MCS-P600-0596, Mathematics and Computer Science
+-- Division, Argonne National Laboratory, Argonne, Ill., May 1996.
+--
+-- The routine just replaces non-positive u[k,k] by a huge positive
+-- number. This involves non-diagonal elements in k-th row of U to be
+-- close to zero that, in turn, involves k-th component of a solution
+-- vector to be close to zero. Note, however, that this technique works
+-- only if the system A*x = b is consistent. */
+int chol_numeric(int n,
+int A_ptr[], int A_ind[], double A_val[], double A_diag[],
+int U_ptr[], int U_ind[], double U_val[], double U_diag[])
+{     int i, j, k, t, t1, beg, end, beg1, end1, count = 0;
+double ukk, uki, *work;
+work = xcalloc(1+n, sizeof(double));
+for (j = 1; j <= n; j++) work[j] = 0.0;
+/* U := (upper triangle of A) */
+/* note that the upper traingle of A is a subset of U */
+for (i = 1; i <= n; i++)
+{  beg = A_ptr[i], end = A_ptr[i+1];
+for (t = beg; t < end; t++)
+j = A_ind[t], work[j] = A_val[t];
+beg = U_ptr[i], end = U_ptr[i+1];
+for (t = beg; t < end; t++)
+j = U_ind[t], U_val[t] = work[j], work[j] = 0.0;
+U_diag[i] = A_diag[i];
+}
+/* main elimination loop */
+for (k = 1; k <= n; k++)
+{  /* transform k-th row of U */
+ukk = U_diag[k];
+if (ukk > 0.0)
+U_diag[k] = ukk = sqrt(ukk);
+else
+U_diag[k] = ukk = DBL_MAX, count++;
+/* (work) := (transformed k-th row) */
+beg = U_ptr[k], end = U_ptr[k+1];
+for (t = beg; t < end; t++)
+work[U_ind[t]] = (U_val[t] /= ukk);
+/* transform other rows of U */
+for (t = beg; t < end; t++)
+{  i = U_ind[t];
+xassert(i > k);
+/* (i-th row) := (i-th row) - u[k,i] * (k-th row) */
+uki = work[i];
+beg1 = U_ptr[i], end1 = U_ptr[i+1];
+for (t1 = beg1; t1 < end1; t1++)
+U_val[t1] -= uki * work[U_ind[t1]];
+U_diag[i] -= uki * uki;
+}
+/* (work) := 0 */
+for (t = beg; t < end; t++)
+work[U_ind[t]] = 0.0;
+}
+xfree(work);
+return count;
+}
+/*----------------------------------------------------------------------
+-- u_solve - solve upper triangular system U*x = b.
+--
+-- *Synopsis*
+--
+-- #include "glpmat.h"
+-- void u_solve(int n, int U_ptr[], int U_ind[], double U_val[],
+--    double U_diag[], double x[]);
+--
+-- *Description*
+--
+-- The routine u_solve solves an linear system U*x = b, where U is an
+-- upper triangular matrix.
+--
+-- The parameter n is the order of matrix U.
+--
+-- The matrix U without diagonal elements is specified in the arrays
+-- U_ptr, U_ind, and U_val in storage-by-rows format. Diagonal elements
+-- of U are specified in the array U_diag, where U_diag[0] is not used,
+-- U_diag[i] = u[i,i] for i = 1, ..., n. All these four arrays are not
+-- changed on exit.
+--
+-- The right-hand side vector b is specified on entry in the array x,
+-- where x[0] is not used, and x[i] = b[i] for i = 1, ..., n. On exit
+-- the routine stores computed components of the vector of unknowns x
+-- in the array x in the same manner. */
+void u_solve(int n, int U_ptr[], int U_ind[], double U_val[],
+double U_diag[], double x[])
+{     int i, t, beg, end;
+double temp;
+for (i = n; i >= 1; i--)
+{  temp = x[i];
+beg = U_ptr[i], end = U_ptr[i+1];
+for (t = beg; t < end; t++)
+temp -= U_val[t] * x[U_ind[t]];
+xassert(U_diag[i] != 0.0);
+x[i] = temp / U_diag[i];
+}
+return;
+}
+/*----------------------------------------------------------------------
+-- ut_solve - solve lower triangular system U'*x = b.
+--
+-- *Synopsis*
+--
+-- #include "glpmat.h"
+-- void ut_solve(int n, int U_ptr[], int U_ind[], double U_val[],
+--    double U_diag[], double x[]);
+--
+-- *Description*
+--
+-- The routine ut_solve solves an linear system U'*x = b, where U is a
+-- matrix transposed to an upper triangular matrix.
+--
+-- The parameter n is the order of matrix U.
+--
+-- The matrix U without diagonal elements is specified in the arrays
+-- U_ptr, U_ind, and U_val in storage-by-rows format. Diagonal elements
+-- of U are specified in the array U_diag, where U_diag[0] is not used,
+-- U_diag[i] = u[i,i] for i = 1, ..., n. All these four arrays are not
+-- changed on exit.
+--
+-- The right-hand side vector b is specified on entry in the array x,
+-- where x[0] is not used, and x[i] = b[i] for i = 1, ..., n. On exit
+-- the routine stores computed components of the vector of unknowns x
+-- in the array x in the same manner. */
+void ut_solve(int n, int U_ptr[], int U_ind[], double U_val[],
+double U_diag[], double x[])
+{     int i, t, beg, end;
+double temp;
+for (i = 1; i <= n; i++)
+{  xassert(U_diag[i] != 0.0);
+temp = (x[i] /= U_diag[i]);
+if (temp == 0.0) continue;
+beg = U_ptr[i], end = U_ptr[i+1];
+for (t = beg; t < end; t++)
+x[U_ind[t]] -= U_val[t] * temp;
+}
+return;
+}
+/* eof */