alpar@1: /* glpmat.c */
alpar@1: 
alpar@1: /***********************************************************************
alpar@1: *  This code is part of GLPK (GNU Linear Programming Kit).
alpar@1: *
alpar@1: *  Copyright (C) 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008,
alpar@1: *  2009, 2010 Andrew Makhorin, Department for Applied Informatics,
alpar@1: *  Moscow Aviation Institute, Moscow, Russia. All rights reserved.
alpar@1: *  E-mail: <mao@gnu.org>.
alpar@1: *
alpar@1: *  GLPK is free software: you can redistribute it and/or modify it
alpar@1: *  under the terms of the GNU General Public License as published by
alpar@1: *  the Free Software Foundation, either version 3 of the License, or
alpar@1: *  (at your option) any later version.
alpar@1: *
alpar@1: *  GLPK is distributed in the hope that it will be useful, but WITHOUT
alpar@1: *  ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
alpar@1: *  or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public
alpar@1: *  License for more details.
alpar@1: *
alpar@1: *  You should have received a copy of the GNU General Public License
alpar@1: *  along with GLPK. If not, see <http://www.gnu.org/licenses/>.
alpar@1: ***********************************************************************/
alpar@1: 
alpar@1: #include "glpenv.h"
alpar@1: #include "glpmat.h"
alpar@1: #include "glpqmd.h"
alpar@1: #include "amd/amd.h"
alpar@1: #include "colamd/colamd.h"
alpar@1: 
alpar@1: /*----------------------------------------------------------------------
alpar@1: -- check_fvs - check sparse vector in full-vector storage format.
alpar@1: --
alpar@1: -- SYNOPSIS
alpar@1: --
alpar@1: -- #include "glpmat.h"
alpar@1: -- int check_fvs(int n, int nnz, int ind[], double vec[]);
alpar@1: --
alpar@1: -- DESCRIPTION
alpar@1: --
alpar@1: -- The routine check_fvs checks if a given vector of dimension n in
alpar@1: -- full-vector storage format has correct representation.
alpar@1: --
alpar@1: -- RETURNS
alpar@1: --
alpar@1: -- The routine returns one of the following codes:
alpar@1: --
alpar@1: -- 0 - the vector is correct;
alpar@1: -- 1 - the number of elements (n) is negative;
alpar@1: -- 2 - the number of non-zero elements (nnz) is negative;
alpar@1: -- 3 - some element index is out of range;
alpar@1: -- 4 - some element index is duplicate;
alpar@1: -- 5 - some non-zero element is out of pattern. */
alpar@1: 
alpar@1: int check_fvs(int n, int nnz, int ind[], double vec[])
alpar@1: {     int i, t, ret, *flag = NULL;
alpar@1:       /* check the number of elements */
alpar@1:       if (n < 0)
alpar@1:       {  ret = 1;
alpar@1:          goto done;
alpar@1:       }
alpar@1:       /* check the number of non-zero elements */
alpar@1:       if (nnz < 0)
alpar@1:       {  ret = 2;
alpar@1:          goto done;
alpar@1:       }
alpar@1:       /* check vector indices */
alpar@1:       flag = xcalloc(1+n, sizeof(int));
alpar@1:       for (i = 1; i <= n; i++) flag[i] = 0;
alpar@1:       for (t = 1; t <= nnz; t++)
alpar@1:       {  i = ind[t];
alpar@1:          if (!(1 <= i && i <= n))
alpar@1:          {  ret = 3;
alpar@1:             goto done;
alpar@1:          }
alpar@1:          if (flag[i])
alpar@1:          {  ret = 4;
alpar@1:             goto done;
alpar@1:          }
alpar@1:          flag[i] = 1;
alpar@1:       }
alpar@1:       /* check vector elements */
alpar@1:       for (i = 1; i <= n; i++)
alpar@1:       {  if (!flag[i] && vec[i] != 0.0)
alpar@1:          {  ret = 5;
alpar@1:             goto done;
alpar@1:          }
alpar@1:       }
alpar@1:       /* the vector is ok */
alpar@1:       ret = 0;
alpar@1: done: if (flag != NULL) xfree(flag);
alpar@1:       return ret;
alpar@1: }
alpar@1: 
alpar@1: /*----------------------------------------------------------------------
alpar@1: -- check_pattern - check pattern of sparse matrix.
alpar@1: --
alpar@1: -- SYNOPSIS
alpar@1: --
alpar@1: -- #include "glpmat.h"
alpar@1: -- int check_pattern(int m, int n, int A_ptr[], int A_ind[]);
alpar@1: --
alpar@1: -- DESCRIPTION
alpar@1: --
alpar@1: -- The routine check_pattern checks the pattern of a given mxn matrix
alpar@1: -- in storage-by-rows format.
alpar@1: --
alpar@1: -- RETURNS
alpar@1: --
alpar@1: -- The routine returns one of the following codes:
alpar@1: --
alpar@1: -- 0 - the pattern is correct;
alpar@1: -- 1 - the number of rows (m) is negative;
alpar@1: -- 2 - the number of columns (n) is negative;
alpar@1: -- 3 - A_ptr[1] is not 1;
alpar@1: -- 4 - some column index is out of range;
alpar@1: -- 5 - some column indices are duplicate. */
alpar@1: 
alpar@1: int check_pattern(int m, int n, int A_ptr[], int A_ind[])
alpar@1: {     int i, j, ptr, ret, *flag = NULL;
alpar@1:       /* check the number of rows */
alpar@1:       if (m < 0)
alpar@1:       {  ret = 1;
alpar@1:          goto done;
alpar@1:       }
alpar@1:       /* check the number of columns */
alpar@1:       if (n < 0)
alpar@1:       {  ret = 2;
alpar@1:          goto done;
alpar@1:       }
alpar@1:       /* check location A_ptr[1] */
alpar@1:       if (A_ptr[1] != 1)
alpar@1:       {  ret = 3;
alpar@1:          goto done;
alpar@1:       }
alpar@1:       /* check row patterns */
alpar@1:       flag = xcalloc(1+n, sizeof(int));
alpar@1:       for (j = 1; j <= n; j++) flag[j] = 0;
alpar@1:       for (i = 1; i <= m; i++)
alpar@1:       {  /* check pattern of row i */
alpar@1:          for (ptr = A_ptr[i]; ptr < A_ptr[i+1]; ptr++)
alpar@1:          {  j = A_ind[ptr];
alpar@1:             /* check column index */
alpar@1:             if (!(1 <= j && j <= n))
alpar@1:             {  ret = 4;
alpar@1:                goto done;
alpar@1:             }
alpar@1:             /* check for duplication */
alpar@1:             if (flag[j])
alpar@1:             {  ret = 5;
alpar@1:                goto done;
alpar@1:             }
alpar@1:             flag[j] = 1;
alpar@1:          }
alpar@1:          /* clear flags */
alpar@1:          for (ptr = A_ptr[i]; ptr < A_ptr[i+1]; ptr++)
alpar@1:          {  j = A_ind[ptr];
alpar@1:             flag[j] = 0;
alpar@1:          }
alpar@1:       }
alpar@1:       /* the pattern is ok */
alpar@1:       ret = 0;
alpar@1: done: if (flag != NULL) xfree(flag);
alpar@1:       return ret;
alpar@1: }
alpar@1: 
alpar@1: /*----------------------------------------------------------------------
alpar@1: -- transpose - transpose sparse matrix.
alpar@1: --
alpar@1: -- *Synopsis*
alpar@1: --
alpar@1: -- #include "glpmat.h"
alpar@1: -- void transpose(int m, int n, int A_ptr[], int A_ind[],
alpar@1: --    double A_val[], int AT_ptr[], int AT_ind[], double AT_val[]);
alpar@1: --
alpar@1: -- *Description*
alpar@1: --
alpar@1: -- For a given mxn sparse matrix A the routine transpose builds a nxm
alpar@1: -- sparse matrix A' which is a matrix transposed to A.
alpar@1: --
alpar@1: -- The arrays A_ptr, A_ind, and A_val specify a given mxn matrix A to
alpar@1: -- be transposed in storage-by-rows format. The parameter A_val can be
alpar@1: -- NULL, in which case numeric values are not copied. The arrays A_ptr,
alpar@1: -- A_ind, and A_val are not changed on exit.
alpar@1: --
alpar@1: -- On entry the arrays AT_ptr, AT_ind, and AT_val must be allocated,
alpar@1: -- but their content is ignored. On exit the routine stores a resultant
alpar@1: -- nxm matrix A' in these arrays in storage-by-rows format. Note that
alpar@1: -- if the parameter A_val is NULL, the array AT_val is not used.
alpar@1: --
alpar@1: -- The routine transpose has a side effect that elements in rows of the
alpar@1: -- resultant matrix A' follow in ascending their column indices. */
alpar@1: 
alpar@1: void transpose(int m, int n, int A_ptr[], int A_ind[], double A_val[],
alpar@1:       int AT_ptr[], int AT_ind[], double AT_val[])
alpar@1: {     int i, j, t, beg, end, pos, len;
alpar@1:       /* determine row lengths of resultant matrix */
alpar@1:       for (j = 1; j <= n; j++) AT_ptr[j] = 0;
alpar@1:       for (i = 1; i <= m; i++)
alpar@1:       {  beg = A_ptr[i], end = A_ptr[i+1];
alpar@1:          for (t = beg; t < end; t++) AT_ptr[A_ind[t]]++;
alpar@1:       }
alpar@1:       /* set up row pointers of resultant matrix */
alpar@1:       pos = 1;
alpar@1:       for (j = 1; j <= n; j++)
alpar@1:          len = AT_ptr[j], pos += len, AT_ptr[j] = pos;
alpar@1:       AT_ptr[n+1] = pos;
alpar@1:       /* build resultant matrix */
alpar@1:       for (i = m; i >= 1; i--)
alpar@1:       {  beg = A_ptr[i], end = A_ptr[i+1];
alpar@1:          for (t = beg; t < end; t++)
alpar@1:          {  pos = --AT_ptr[A_ind[t]];
alpar@1:             AT_ind[pos] = i;
alpar@1:             if (A_val != NULL) AT_val[pos] = A_val[t];
alpar@1:          }
alpar@1:       }
alpar@1:       return;
alpar@1: }
alpar@1: 
alpar@1: /*----------------------------------------------------------------------
alpar@1: -- adat_symbolic - compute S = P*A*D*A'*P' (symbolic phase).
alpar@1: --
alpar@1: -- *Synopsis*
alpar@1: --
alpar@1: -- #include "glpmat.h"
alpar@1: -- int *adat_symbolic(int m, int n, int P_per[], int A_ptr[],
alpar@1: --    int A_ind[], int S_ptr[]);
alpar@1: --
alpar@1: -- *Description*
alpar@1: --
alpar@1: -- The routine adat_symbolic implements the symbolic phase to compute
alpar@1: -- symmetric matrix S = P*A*D*A'*P', where P is a permutation matrix,
alpar@1: -- A is a given sparse matrix, D is a diagonal matrix, A' is a matrix
alpar@1: -- transposed to A, P' is an inverse of P.
alpar@1: --
alpar@1: -- The parameter m is the number of rows in A and the order of P.
alpar@1: --
alpar@1: -- The parameter n is the number of columns in A and the order of D.
alpar@1: --
alpar@1: -- The array P_per specifies permutation matrix P. It is not changed on
alpar@1: -- exit.
alpar@1: --
alpar@1: -- The arrays A_ptr and A_ind specify the pattern of matrix A. They are
alpar@1: -- not changed on exit.
alpar@1: --
alpar@1: -- On exit the routine stores the pattern of upper triangular part of
alpar@1: -- matrix S without diagonal elements in the arrays S_ptr and S_ind in
alpar@1: -- storage-by-rows format. The array S_ptr should be allocated on entry,
alpar@1: -- however, its content is ignored. The array S_ind is allocated by the
alpar@1: -- routine itself which returns a pointer to it.
alpar@1: --
alpar@1: -- *Returns*
alpar@1: --
alpar@1: -- The routine returns a pointer to the array S_ind. */
alpar@1: 
alpar@1: int *adat_symbolic(int m, int n, int P_per[], int A_ptr[], int A_ind[],
alpar@1:       int S_ptr[])
alpar@1: {     int i, j, t, ii, jj, tt, k, size, len;
alpar@1:       int *S_ind, *AT_ptr, *AT_ind, *ind, *map, *temp;
alpar@1:       /* build the pattern of A', which is a matrix transposed to A, to
alpar@1:          efficiently access A in column-wise manner */
alpar@1:       AT_ptr = xcalloc(1+n+1, sizeof(int));
alpar@1:       AT_ind = xcalloc(A_ptr[m+1], sizeof(int));
alpar@1:       transpose(m, n, A_ptr, A_ind, NULL, AT_ptr, AT_ind, NULL);
alpar@1:       /* allocate the array S_ind */
alpar@1:       size = A_ptr[m+1] - 1;
alpar@1:       if (size < m) size = m;
alpar@1:       S_ind = xcalloc(1+size, sizeof(int));
alpar@1:       /* allocate and initialize working arrays */
alpar@1:       ind = xcalloc(1+m, sizeof(int));
alpar@1:       map = xcalloc(1+m, sizeof(int));
alpar@1:       for (jj = 1; jj <= m; jj++) map[jj] = 0;
alpar@1:       /* compute pattern of S; note that symbolically S = B*B', where
alpar@1:          B = P*A, B' is matrix transposed to B */
alpar@1:       S_ptr[1] = 1;
alpar@1:       for (ii = 1; ii <= m; ii++)
alpar@1:       {  /* compute pattern of ii-th row of S */
alpar@1:          len = 0;
alpar@1:          i = P_per[ii]; /* i-th row of A = ii-th row of B */
alpar@1:          for (t = A_ptr[i]; t < A_ptr[i+1]; t++)
alpar@1:          {  k = A_ind[t];
alpar@1:             /* walk through k-th column of A */
alpar@1:             for (tt = AT_ptr[k]; tt < AT_ptr[k+1]; tt++)
alpar@1:             {  j = AT_ind[tt];
alpar@1:                jj = P_per[m+j]; /* j-th row of A = jj-th row of B */
alpar@1:                /* a[i,k] != 0 and a[j,k] != 0 ergo s[ii,jj] != 0 */
alpar@1:                if (ii < jj && !map[jj]) ind[++len] = jj, map[jj] = 1;
alpar@1:             }
alpar@1:          }
alpar@1:          /* now (ind) is pattern of ii-th row of S */
alpar@1:          S_ptr[ii+1] = S_ptr[ii] + len;
alpar@1:          /* at least (S_ptr[ii+1] - 1) locations should be available in
alpar@1:             the array S_ind */
alpar@1:          if (S_ptr[ii+1] - 1 > size)
alpar@1:          {  temp = S_ind;
alpar@1:             size += size;
alpar@1:             S_ind = xcalloc(1+size, sizeof(int));
alpar@1:             memcpy(&S_ind[1], &temp[1], (S_ptr[ii] - 1) * sizeof(int));
alpar@1:             xfree(temp);
alpar@1:          }
alpar@1:          xassert(S_ptr[ii+1] - 1 <= size);
alpar@1:          /* (ii-th row of S) := (ind) */
alpar@1:          memcpy(&S_ind[S_ptr[ii]], &ind[1], len * sizeof(int));
alpar@1:          /* clear the row pattern map */
alpar@1:          for (t = 1; t <= len; t++) map[ind[t]] = 0;
alpar@1:       }
alpar@1:       /* free working arrays */
alpar@1:       xfree(AT_ptr);
alpar@1:       xfree(AT_ind);
alpar@1:       xfree(ind);
alpar@1:       xfree(map);
alpar@1:       /* reallocate the array S_ind to free unused locations */
alpar@1:       temp = S_ind;
alpar@1:       size = S_ptr[m+1] - 1;
alpar@1:       S_ind = xcalloc(1+size, sizeof(int));
alpar@1:       memcpy(&S_ind[1], &temp[1], size * sizeof(int));
alpar@1:       xfree(temp);
alpar@1:       return S_ind;
alpar@1: }
alpar@1: 
alpar@1: /*----------------------------------------------------------------------
alpar@1: -- adat_numeric - compute S = P*A*D*A'*P' (numeric phase).
alpar@1: --
alpar@1: -- *Synopsis*
alpar@1: --
alpar@1: -- #include "glpmat.h"
alpar@1: -- void adat_numeric(int m, int n, int P_per[],
alpar@1: --    int A_ptr[], int A_ind[], double A_val[], double D_diag[],
alpar@1: --    int S_ptr[], int S_ind[], double S_val[], double S_diag[]);
alpar@1: --
alpar@1: -- *Description*
alpar@1: --
alpar@1: -- The routine adat_numeric implements the numeric phase to compute
alpar@1: -- symmetric matrix S = P*A*D*A'*P', where P is a permutation matrix,
alpar@1: -- A is a given sparse matrix, D is a diagonal matrix, A' is a matrix
alpar@1: -- transposed to A, P' is an inverse of P.
alpar@1: --
alpar@1: -- The parameter m is the number of rows in A and the order of P.
alpar@1: --
alpar@1: -- The parameter n is the number of columns in A and the order of D.
alpar@1: --
alpar@1: -- The matrix P is specified in the array P_per, which is not changed
alpar@1: -- on exit.
alpar@1: --
alpar@1: -- The matrix A is specified in the arrays A_ptr, A_ind, and A_val in
alpar@1: -- storage-by-rows format. These arrays are not changed on exit.
alpar@1: --
alpar@1: -- Diagonal elements of the matrix D are specified in the array D_diag,
alpar@1: -- where D_diag[0] is not used, D_diag[i] = d[i,i] for i = 1, ..., n.
alpar@1: -- The array D_diag is not changed on exit.
alpar@1: --
alpar@1: -- The pattern of the upper triangular part of the matrix S without
alpar@1: -- diagonal elements (previously computed by the routine adat_symbolic)
alpar@1: -- is specified in the arrays S_ptr and S_ind, which are not changed on
alpar@1: -- exit. Numeric values of non-diagonal elements of S are stored in
alpar@1: -- corresponding locations of the array S_val, and values of diagonal
alpar@1: -- elements of S are stored in locations S_diag[1], ..., S_diag[n]. */
alpar@1: 
alpar@1: void adat_numeric(int m, int n, int P_per[],
alpar@1:       int A_ptr[], int A_ind[], double A_val[], double D_diag[],
alpar@1:       int S_ptr[], int S_ind[], double S_val[], double S_diag[])
alpar@1: {     int i, j, t, ii, jj, tt, beg, end, beg1, end1, k;
alpar@1:       double sum, *work;
alpar@1:       work = xcalloc(1+n, sizeof(double));
alpar@1:       for (j = 1; j <= n; j++) work[j] = 0.0;
alpar@1:       /* compute S = B*D*B', where B = P*A, B' is a matrix transposed
alpar@1:          to B */
alpar@1:       for (ii = 1; ii <= m; ii++)
alpar@1:       {  i = P_per[ii]; /* i-th row of A = ii-th row of B */
alpar@1:          /* (work) := (i-th row of A) */
alpar@1:          beg = A_ptr[i], end = A_ptr[i+1];
alpar@1:          for (t = beg; t < end; t++)
alpar@1:             work[A_ind[t]] = A_val[t];
alpar@1:          /* compute ii-th row of S */
alpar@1:          beg = S_ptr[ii], end = S_ptr[ii+1];
alpar@1:          for (t = beg; t < end; t++)
alpar@1:          {  jj = S_ind[t];
alpar@1:             j = P_per[jj]; /* j-th row of A = jj-th row of B */
alpar@1:             /* s[ii,jj] := sum a[i,k] * d[k,k] * a[j,k] */
alpar@1:             sum = 0.0;
alpar@1:             beg1 = A_ptr[j], end1 = A_ptr[j+1];
alpar@1:             for (tt = beg1; tt < end1; tt++)
alpar@1:             {  k = A_ind[tt];
alpar@1:                sum += work[k] * D_diag[k] * A_val[tt];
alpar@1:             }
alpar@1:             S_val[t] = sum;
alpar@1:          }
alpar@1:          /* s[ii,ii] := sum a[i,k] * d[k,k] * a[i,k] */
alpar@1:          sum = 0.0;
alpar@1:          beg = A_ptr[i], end = A_ptr[i+1];
alpar@1:          for (t = beg; t < end; t++)
alpar@1:          {  k = A_ind[t];
alpar@1:             sum += A_val[t] * D_diag[k] * A_val[t];
alpar@1:             work[k] = 0.0;
alpar@1:          }
alpar@1:          S_diag[ii] = sum;
alpar@1:       }
alpar@1:       xfree(work);
alpar@1:       return;
alpar@1: }
alpar@1: 
alpar@1: /*----------------------------------------------------------------------
alpar@1: -- min_degree - minimum degree ordering.
alpar@1: --
alpar@1: -- *Synopsis*
alpar@1: --
alpar@1: -- #include "glpmat.h"
alpar@1: -- void min_degree(int n, int A_ptr[], int A_ind[], int P_per[]);
alpar@1: --
alpar@1: -- *Description*
alpar@1: --
alpar@1: -- The routine min_degree uses the minimum degree ordering algorithm
alpar@1: -- to find a permutation matrix P for a given sparse symmetric positive
alpar@1: -- matrix A which minimizes the number of non-zeros in upper triangular
alpar@1: -- factor U for Cholesky factorization P*A*P' = U'*U.
alpar@1: --
alpar@1: -- The parameter n is the order of matrices A and P.
alpar@1: --
alpar@1: -- The pattern of the given matrix A is specified on entry in the arrays
alpar@1: -- A_ptr and A_ind in storage-by-rows format. Only the upper triangular
alpar@1: -- part without diagonal elements (which all are assumed to be non-zero)
alpar@1: -- should be specified as if A were upper triangular. The arrays A_ptr
alpar@1: -- and A_ind are not changed on exit.
alpar@1: --
alpar@1: -- The permutation matrix P is stored by the routine in the array P_per
alpar@1: -- on exit.
alpar@1: --
alpar@1: -- *Algorithm*
alpar@1: --
alpar@1: -- The routine min_degree is based on some subroutines from the package
alpar@1: -- SPARSPAK (see comments in the module glpqmd). */
alpar@1: 
alpar@1: void min_degree(int n, int A_ptr[], int A_ind[], int P_per[])
alpar@1: {     int i, j, ne, t, pos, len;
alpar@1:       int *xadj, *adjncy, *deg, *marker, *rchset, *nbrhd, *qsize,
alpar@1:          *qlink, nofsub;
alpar@1:       /* determine number of non-zeros in complete pattern */
alpar@1:       ne = A_ptr[n+1] - 1;
alpar@1:       ne += ne;
alpar@1:       /* allocate working arrays */
alpar@1:       xadj = xcalloc(1+n+1, sizeof(int));
alpar@1:       adjncy = xcalloc(1+ne, sizeof(int));
alpar@1:       deg = xcalloc(1+n, sizeof(int));
alpar@1:       marker = xcalloc(1+n, sizeof(int));
alpar@1:       rchset = xcalloc(1+n, sizeof(int));
alpar@1:       nbrhd = xcalloc(1+n, sizeof(int));
alpar@1:       qsize = xcalloc(1+n, sizeof(int));
alpar@1:       qlink = xcalloc(1+n, sizeof(int));
alpar@1:       /* determine row lengths in complete pattern */
alpar@1:       for (i = 1; i <= n; i++) xadj[i] = 0;
alpar@1:       for (i = 1; i <= n; i++)
alpar@1:       {  for (t = A_ptr[i]; t < A_ptr[i+1]; t++)
alpar@1:          {  j = A_ind[t];
alpar@1:             xassert(i < j && j <= n);
alpar@1:             xadj[i]++, xadj[j]++;
alpar@1:          }
alpar@1:       }
alpar@1:       /* set up row pointers for complete pattern */
alpar@1:       pos = 1;
alpar@1:       for (i = 1; i <= n; i++)
alpar@1:          len = xadj[i], pos += len, xadj[i] = pos;
alpar@1:       xadj[n+1] = pos;
alpar@1:       xassert(pos - 1 == ne);
alpar@1:       /* construct complete pattern */
alpar@1:       for (i = 1; i <= n; i++)
alpar@1:       {  for (t = A_ptr[i]; t < A_ptr[i+1]; t++)
alpar@1:          {  j = A_ind[t];
alpar@1:             adjncy[--xadj[i]] = j, adjncy[--xadj[j]] = i;
alpar@1:          }
alpar@1:       }
alpar@1:       /* call the main minimimum degree ordering routine */
alpar@1:       genqmd(&n, xadj, adjncy, P_per, P_per + n, deg, marker, rchset,
alpar@1:          nbrhd, qsize, qlink, &nofsub);
alpar@1:       /* make sure that permutation matrix P is correct */
alpar@1:       for (i = 1; i <= n; i++)
alpar@1:       {  j = P_per[i];
alpar@1:          xassert(1 <= j && j <= n);
alpar@1:          xassert(P_per[n+j] == i);
alpar@1:       }
alpar@1:       /* free working arrays */
alpar@1:       xfree(xadj);
alpar@1:       xfree(adjncy);
alpar@1:       xfree(deg);
alpar@1:       xfree(marker);
alpar@1:       xfree(rchset);
alpar@1:       xfree(nbrhd);
alpar@1:       xfree(qsize);
alpar@1:       xfree(qlink);
alpar@1:       return;
alpar@1: }
alpar@1: 
alpar@1: /**********************************************************************/
alpar@1: 
alpar@1: void amd_order1(int n, int A_ptr[], int A_ind[], int P_per[])
alpar@1: {     /* approximate minimum degree ordering (AMD) */
alpar@1:       int k, ret;
alpar@1:       double Control[AMD_CONTROL], Info[AMD_INFO];
alpar@1:       /* get the default parameters */
alpar@1:       amd_defaults(Control);
alpar@1: #if 0
alpar@1:       /* and print them */
alpar@1:       amd_control(Control);
alpar@1: #endif
alpar@1:       /* make all indices 0-based */
alpar@1:       for (k = 1; k < A_ptr[n+1]; k++) A_ind[k]--;
alpar@1:       for (k = 1; k <= n+1; k++) A_ptr[k]--;
alpar@1:       /* call the ordering routine */
alpar@1:       ret = amd_order(n, &A_ptr[1], &A_ind[1], &P_per[1], Control, Info)
alpar@1:          ;
alpar@1: #if 0
alpar@1:       amd_info(Info);
alpar@1: #endif
alpar@1:       xassert(ret == AMD_OK || ret == AMD_OK_BUT_JUMBLED);
alpar@1:       /* retsore 1-based indices */
alpar@1:       for (k = 1; k <= n+1; k++) A_ptr[k]++;
alpar@1:       for (k = 1; k < A_ptr[n+1]; k++) A_ind[k]++;
alpar@1:       /* patch up permutation matrix */
alpar@1:       memset(&P_per[n+1], 0, n * sizeof(int));
alpar@1:       for (k = 1; k <= n; k++)
alpar@1:       {  P_per[k]++;
alpar@1:          xassert(1 <= P_per[k] && P_per[k] <= n);
alpar@1:          xassert(P_per[n+P_per[k]] == 0);
alpar@1:          P_per[n+P_per[k]] = k;
alpar@1:       }
alpar@1:       return;
alpar@1: }
alpar@1: 
alpar@1: /**********************************************************************/
alpar@1: 
alpar@1: static void *allocate(size_t n, size_t size)
alpar@1: {     void *ptr;
alpar@1:       ptr = xcalloc(n, size);
alpar@1:       memset(ptr, 0, n * size);
alpar@1:       return ptr;
alpar@1: }
alpar@1: 
alpar@1: static void release(void *ptr)
alpar@1: {     xfree(ptr);
alpar@1:       return;
alpar@1: }
alpar@1: 
alpar@1: void symamd_ord(int n, int A_ptr[], int A_ind[], int P_per[])
alpar@1: {     /* approximate minimum degree ordering (SYMAMD) */
alpar@1:       int k, ok;
alpar@1:       int stats[COLAMD_STATS];
alpar@1:       /* make all indices 0-based */
alpar@1:       for (k = 1; k < A_ptr[n+1]; k++) A_ind[k]--;
alpar@1:       for (k = 1; k <= n+1; k++) A_ptr[k]--;
alpar@1:       /* call the ordering routine */
alpar@1:       ok = symamd(n, &A_ind[1], &A_ptr[1], &P_per[1], NULL, stats,
alpar@1:          allocate, release);
alpar@1: #if 0
alpar@1:       symamd_report(stats);
alpar@1: #endif
alpar@1:       xassert(ok);
alpar@1:       /* restore 1-based indices */
alpar@1:       for (k = 1; k <= n+1; k++) A_ptr[k]++;
alpar@1:       for (k = 1; k < A_ptr[n+1]; k++) A_ind[k]++;
alpar@1:       /* patch up permutation matrix */
alpar@1:       memset(&P_per[n+1], 0, n * sizeof(int));
alpar@1:       for (k = 1; k <= n; k++)
alpar@1:       {  P_per[k]++;
alpar@1:          xassert(1 <= P_per[k] && P_per[k] <= n);
alpar@1:          xassert(P_per[n+P_per[k]] == 0);
alpar@1:          P_per[n+P_per[k]] = k;
alpar@1:       }
alpar@1:       return;
alpar@1: }
alpar@1: 
alpar@1: /*----------------------------------------------------------------------
alpar@1: -- chol_symbolic - compute Cholesky factorization (symbolic phase).
alpar@1: --
alpar@1: -- *Synopsis*
alpar@1: --
alpar@1: -- #include "glpmat.h"
alpar@1: -- int *chol_symbolic(int n, int A_ptr[], int A_ind[], int U_ptr[]);
alpar@1: --
alpar@1: -- *Description*
alpar@1: --
alpar@1: -- The routine chol_symbolic implements the symbolic phase of Cholesky
alpar@1: -- factorization A = U'*U, where A is a given sparse symmetric positive
alpar@1: -- definite matrix, U is a resultant upper triangular factor, U' is a
alpar@1: -- matrix transposed to U.
alpar@1: --
alpar@1: -- The parameter n is the order of matrices A and U.
alpar@1: --
alpar@1: -- The pattern of the given matrix A is specified on entry in the arrays
alpar@1: -- A_ptr and A_ind in storage-by-rows format. Only the upper triangular
alpar@1: -- part without diagonal elements (which all are assumed to be non-zero)
alpar@1: -- should be specified as if A were upper triangular. The arrays A_ptr
alpar@1: -- and A_ind are not changed on exit.
alpar@1: --
alpar@1: -- The pattern of the matrix U without diagonal elements (which all are
alpar@1: -- assumed to be non-zero) is stored on exit from the routine in the
alpar@1: -- arrays U_ptr and U_ind in storage-by-rows format. The array U_ptr
alpar@1: -- should be allocated on entry, however, its content is ignored. The
alpar@1: -- array U_ind is allocated by the routine which returns a pointer to it
alpar@1: -- on exit.
alpar@1: --
alpar@1: -- *Returns*
alpar@1: --
alpar@1: -- The routine returns a pointer to the array U_ind.
alpar@1: --
alpar@1: -- *Method*
alpar@1: --
alpar@1: -- The routine chol_symbolic computes the pattern of the matrix U in a
alpar@1: -- row-wise manner. No pivoting is used.
alpar@1: --
alpar@1: -- It is known that to compute the pattern of row k of the matrix U we
alpar@1: -- need to merge the pattern of row k of the matrix A and the patterns
alpar@1: -- of each row i of U, where u[i,k] is non-zero (these rows are already
alpar@1: -- computed and placed above row k).
alpar@1: --
alpar@1: -- However, to reduce the number of rows to be merged the routine uses
alpar@1: -- an advanced algorithm proposed in:
alpar@1: --
alpar@1: -- D.J.Rose, R.E.Tarjan, and G.S.Lueker. Algorithmic aspects of vertex
alpar@1: -- elimination on graphs. SIAM J. Comput. 5, 1976, 266-83.
alpar@1: --
alpar@1: -- The authors of the cited paper show that we have the same result if
alpar@1: -- we merge row k of the matrix A and such rows of the matrix U (among
alpar@1: -- rows 1, ..., k-1) whose leftmost non-diagonal non-zero element is
alpar@1: -- placed in k-th column. This feature signficantly reduces the number
alpar@1: -- of rows to be merged, especially on the final steps, where rows of
alpar@1: -- the matrix U become quite dense.
alpar@1: --
alpar@1: -- To determine rows, which should be merged on k-th step, for a fixed
alpar@1: -- time the routine uses linked lists of row numbers of the matrix U.
alpar@1: -- Location head[k] contains the number of a first row, whose leftmost
alpar@1: -- non-diagonal non-zero element is placed in column k, and location
alpar@1: -- next[i] contains the number of a next row with the same property as
alpar@1: -- row i. */
alpar@1: 
alpar@1: int *chol_symbolic(int n, int A_ptr[], int A_ind[], int U_ptr[])
alpar@1: {     int i, j, k, t, len, size, beg, end, min_j, *U_ind, *head, *next,
alpar@1:          *ind, *map, *temp;
alpar@1:       /* initially we assume that on computing the pattern of U fill-in
alpar@1:          will double the number of non-zeros in A */
alpar@1:       size = A_ptr[n+1] - 1;
alpar@1:       if (size < n) size = n;
alpar@1:       size += size;
alpar@1:       U_ind = xcalloc(1+size, sizeof(int));
alpar@1:       /* allocate and initialize working arrays */
alpar@1:       head = xcalloc(1+n, sizeof(int));
alpar@1:       for (i = 1; i <= n; i++) head[i] = 0;
alpar@1:       next = xcalloc(1+n, sizeof(int));
alpar@1:       ind = xcalloc(1+n, sizeof(int));
alpar@1:       map = xcalloc(1+n, sizeof(int));
alpar@1:       for (j = 1; j <= n; j++) map[j] = 0;
alpar@1:       /* compute the pattern of matrix U */
alpar@1:       U_ptr[1] = 1;
alpar@1:       for (k = 1; k <= n; k++)
alpar@1:       {  /* compute the pattern of k-th row of U, which is the union of
alpar@1:             k-th row of A and those rows of U (among 1, ..., k-1) whose
alpar@1:             leftmost non-diagonal non-zero is placed in k-th column */
alpar@1:          /* (ind) := (k-th row of A) */
alpar@1:          len = A_ptr[k+1] - A_ptr[k];
alpar@1:          memcpy(&ind[1], &A_ind[A_ptr[k]], len * sizeof(int));
alpar@1:          for (t = 1; t <= len; t++)
alpar@1:          {  j = ind[t];
alpar@1:             xassert(k < j && j <= n);
alpar@1:             map[j] = 1;
alpar@1:          }
alpar@1:          /* walk through rows of U whose leftmost non-diagonal non-zero
alpar@1:             is placed in k-th column */
alpar@1:          for (i = head[k]; i != 0; i = next[i])
alpar@1:          {  /* (ind) := (ind) union (i-th row of U) */
alpar@1:             beg = U_ptr[i], end = U_ptr[i+1];
alpar@1:             for (t = beg; t < end; t++)
alpar@1:             {  j = U_ind[t];
alpar@1:                if (j > k && !map[j]) ind[++len] = j, map[j] = 1;
alpar@1:             }
alpar@1:          }
alpar@1:          /* now (ind) is the pattern of k-th row of U */
alpar@1:          U_ptr[k+1] = U_ptr[k] + len;
alpar@1:          /* at least (U_ptr[k+1] - 1) locations should be available in
alpar@1:             the array U_ind */
alpar@1:          if (U_ptr[k+1] - 1 > size)
alpar@1:          {  temp = U_ind;
alpar@1:             size += size;
alpar@1:             U_ind = xcalloc(1+size, sizeof(int));
alpar@1:             memcpy(&U_ind[1], &temp[1], (U_ptr[k] - 1) * sizeof(int));
alpar@1:             xfree(temp);
alpar@1:          }
alpar@1:          xassert(U_ptr[k+1] - 1 <= size);
alpar@1:          /* (k-th row of U) := (ind) */
alpar@1:          memcpy(&U_ind[U_ptr[k]], &ind[1], len * sizeof(int));
alpar@1:          /* determine column index of leftmost non-diagonal non-zero in
alpar@1:             k-th row of U and clear the row pattern map */
alpar@1:          min_j = n + 1;
alpar@1:          for (t = 1; t <= len; t++)
alpar@1:          {  j = ind[t], map[j] = 0;
alpar@1:             if (min_j > j) min_j = j;
alpar@1:          }
alpar@1:          /* include k-th row into corresponding linked list */
alpar@1:          if (min_j <= n) next[k] = head[min_j], head[min_j] = k;
alpar@1:       }
alpar@1:       /* free working arrays */
alpar@1:       xfree(head);
alpar@1:       xfree(next);
alpar@1:       xfree(ind);
alpar@1:       xfree(map);
alpar@1:       /* reallocate the array U_ind to free unused locations */
alpar@1:       temp = U_ind;
alpar@1:       size = U_ptr[n+1] - 1;
alpar@1:       U_ind = xcalloc(1+size, sizeof(int));
alpar@1:       memcpy(&U_ind[1], &temp[1], size * sizeof(int));
alpar@1:       xfree(temp);
alpar@1:       return U_ind;
alpar@1: }
alpar@1: 
alpar@1: /*----------------------------------------------------------------------
alpar@1: -- chol_numeric - compute Cholesky factorization (numeric phase).
alpar@1: --
alpar@1: -- *Synopsis*
alpar@1: --
alpar@1: -- #include "glpmat.h"
alpar@1: -- int chol_numeric(int n,
alpar@1: --    int A_ptr[], int A_ind[], double A_val[], double A_diag[],
alpar@1: --    int U_ptr[], int U_ind[], double U_val[], double U_diag[]);
alpar@1: --
alpar@1: -- *Description*
alpar@1: --
alpar@1: -- The routine chol_symbolic implements the numeric phase of Cholesky
alpar@1: -- factorization A = U'*U, where A is a given sparse symmetric positive
alpar@1: -- definite matrix, U is a resultant upper triangular factor, U' is a
alpar@1: -- matrix transposed to U.
alpar@1: --
alpar@1: -- The parameter n is the order of matrices A and U.
alpar@1: --
alpar@1: -- Upper triangular part of the matrix A without diagonal elements is
alpar@1: -- specified in the arrays A_ptr, A_ind, and A_val in storage-by-rows
alpar@1: -- format. Diagonal elements of A are specified in the array A_diag,
alpar@1: -- where A_diag[0] is not used, A_diag[i] = a[i,i] for i = 1, ..., n.
alpar@1: -- The arrays A_ptr, A_ind, A_val, and A_diag are not changed on exit.
alpar@1: --
alpar@1: -- The pattern of the matrix U without diagonal elements (previously
alpar@1: -- computed with the routine chol_symbolic) is specified in the arrays
alpar@1: -- U_ptr and U_ind, which are not changed on exit. Numeric values of
alpar@1: -- non-diagonal elements of U are stored in corresponding locations of
alpar@1: -- the array U_val, and values of diagonal elements of U are stored in
alpar@1: -- locations U_diag[1], ..., U_diag[n].
alpar@1: --
alpar@1: -- *Returns*
alpar@1: --
alpar@1: -- The routine returns the number of non-positive diagonal elements of
alpar@1: -- the matrix U which have been replaced by a huge positive number (see
alpar@1: -- the method description below). Zero return code means the matrix A
alpar@1: -- has been successfully factorized.
alpar@1: --
alpar@1: -- *Method*
alpar@1: --
alpar@1: -- The routine chol_numeric computes the matrix U in a row-wise manner
alpar@1: -- using standard gaussian elimination technique. No pivoting is used.
alpar@1: --
alpar@1: -- Initially the routine sets U = A, and before k-th elimination step
alpar@1: -- the matrix U is the following:
alpar@1: --
alpar@1: --       1       k         n
alpar@1: --    1  x x x x x x x x x x
alpar@1: --       . x x x x x x x x x
alpar@1: --       . . x x x x x x x x
alpar@1: --       . . . x x x x x x x
alpar@1: --    k  . . . . * * * * * *
alpar@1: --       . . . . * * * * * *
alpar@1: --       . . . . * * * * * *
alpar@1: --       . . . . * * * * * *
alpar@1: --       . . . . * * * * * *
alpar@1: --    n  . . . . * * * * * *
alpar@1: --
alpar@1: -- where 'x' are elements of already computed rows, '*' are elements of
alpar@1: -- the active submatrix. (Note that the lower triangular part of the
alpar@1: -- active submatrix being symmetric is not stored and diagonal elements
alpar@1: -- are stored separately in the array U_diag.)
alpar@1: --
alpar@1: -- The matrix A is assumed to be positive definite. However, if it is
alpar@1: -- close to semi-definite, on some elimination step a pivot u[k,k] may
alpar@1: -- happen to be non-positive due to round-off errors. In this case the
alpar@1: -- routine uses a technique proposed in:
alpar@1: --
alpar@1: -- S.J.Wright. The Cholesky factorization in interior-point and barrier
alpar@1: -- methods. Preprint MCS-P600-0596, Mathematics and Computer Science
alpar@1: -- Division, Argonne National Laboratory, Argonne, Ill., May 1996.
alpar@1: --
alpar@1: -- The routine just replaces non-positive u[k,k] by a huge positive
alpar@1: -- number. This involves non-diagonal elements in k-th row of U to be
alpar@1: -- close to zero that, in turn, involves k-th component of a solution
alpar@1: -- vector to be close to zero. Note, however, that this technique works
alpar@1: -- only if the system A*x = b is consistent. */
alpar@1: 
alpar@1: int chol_numeric(int n,
alpar@1:       int A_ptr[], int A_ind[], double A_val[], double A_diag[],
alpar@1:       int U_ptr[], int U_ind[], double U_val[], double U_diag[])
alpar@1: {     int i, j, k, t, t1, beg, end, beg1, end1, count = 0;
alpar@1:       double ukk, uki, *work;
alpar@1:       work = xcalloc(1+n, sizeof(double));
alpar@1:       for (j = 1; j <= n; j++) work[j] = 0.0;
alpar@1:       /* U := (upper triangle of A) */
alpar@1:       /* note that the upper traingle of A is a subset of U */
alpar@1:       for (i = 1; i <= n; i++)
alpar@1:       {  beg = A_ptr[i], end = A_ptr[i+1];
alpar@1:          for (t = beg; t < end; t++)
alpar@1:             j = A_ind[t], work[j] = A_val[t];
alpar@1:          beg = U_ptr[i], end = U_ptr[i+1];
alpar@1:          for (t = beg; t < end; t++)
alpar@1:             j = U_ind[t], U_val[t] = work[j], work[j] = 0.0;
alpar@1:          U_diag[i] = A_diag[i];
alpar@1:       }
alpar@1:       /* main elimination loop */
alpar@1:       for (k = 1; k <= n; k++)
alpar@1:       {  /* transform k-th row of U */
alpar@1:          ukk = U_diag[k];
alpar@1:          if (ukk > 0.0)
alpar@1:             U_diag[k] = ukk = sqrt(ukk);
alpar@1:          else
alpar@1:             U_diag[k] = ukk = DBL_MAX, count++;
alpar@1:          /* (work) := (transformed k-th row) */
alpar@1:          beg = U_ptr[k], end = U_ptr[k+1];
alpar@1:          for (t = beg; t < end; t++)
alpar@1:             work[U_ind[t]] = (U_val[t] /= ukk);
alpar@1:          /* transform other rows of U */
alpar@1:          for (t = beg; t < end; t++)
alpar@1:          {  i = U_ind[t];
alpar@1:             xassert(i > k);
alpar@1:             /* (i-th row) := (i-th row) - u[k,i] * (k-th row) */
alpar@1:             uki = work[i];
alpar@1:             beg1 = U_ptr[i], end1 = U_ptr[i+1];
alpar@1:             for (t1 = beg1; t1 < end1; t1++)
alpar@1:                U_val[t1] -= uki * work[U_ind[t1]];
alpar@1:             U_diag[i] -= uki * uki;
alpar@1:          }
alpar@1:          /* (work) := 0 */
alpar@1:          for (t = beg; t < end; t++)
alpar@1:             work[U_ind[t]] = 0.0;
alpar@1:       }
alpar@1:       xfree(work);
alpar@1:       return count;
alpar@1: }
alpar@1: 
alpar@1: /*----------------------------------------------------------------------
alpar@1: -- u_solve - solve upper triangular system U*x = b.
alpar@1: --
alpar@1: -- *Synopsis*
alpar@1: --
alpar@1: -- #include "glpmat.h"
alpar@1: -- void u_solve(int n, int U_ptr[], int U_ind[], double U_val[],
alpar@1: --    double U_diag[], double x[]);
alpar@1: --
alpar@1: -- *Description*
alpar@1: --
alpar@1: -- The routine u_solve solves an linear system U*x = b, where U is an
alpar@1: -- upper triangular matrix.
alpar@1: --
alpar@1: -- The parameter n is the order of matrix U.
alpar@1: --
alpar@1: -- The matrix U without diagonal elements is specified in the arrays
alpar@1: -- U_ptr, U_ind, and U_val in storage-by-rows format. Diagonal elements
alpar@1: -- of U are specified in the array U_diag, where U_diag[0] is not used,
alpar@1: -- U_diag[i] = u[i,i] for i = 1, ..., n. All these four arrays are not
alpar@1: -- changed on exit.
alpar@1: --
alpar@1: -- The right-hand side vector b is specified on entry in the array x,
alpar@1: -- where x[0] is not used, and x[i] = b[i] for i = 1, ..., n. On exit
alpar@1: -- the routine stores computed components of the vector of unknowns x
alpar@1: -- in the array x in the same manner. */
alpar@1: 
alpar@1: void u_solve(int n, int U_ptr[], int U_ind[], double U_val[],
alpar@1:       double U_diag[], double x[])
alpar@1: {     int i, t, beg, end;
alpar@1:       double temp;
alpar@1:       for (i = n; i >= 1; i--)
alpar@1:       {  temp = x[i];
alpar@1:          beg = U_ptr[i], end = U_ptr[i+1];
alpar@1:          for (t = beg; t < end; t++)
alpar@1:             temp -= U_val[t] * x[U_ind[t]];
alpar@1:          xassert(U_diag[i] != 0.0);
alpar@1:          x[i] = temp / U_diag[i];
alpar@1:       }
alpar@1:       return;
alpar@1: }
alpar@1: 
alpar@1: /*----------------------------------------------------------------------
alpar@1: -- ut_solve - solve lower triangular system U'*x = b.
alpar@1: --
alpar@1: -- *Synopsis*
alpar@1: --
alpar@1: -- #include "glpmat.h"
alpar@1: -- void ut_solve(int n, int U_ptr[], int U_ind[], double U_val[],
alpar@1: --    double U_diag[], double x[]);
alpar@1: --
alpar@1: -- *Description*
alpar@1: --
alpar@1: -- The routine ut_solve solves an linear system U'*x = b, where U is a
alpar@1: -- matrix transposed to an upper triangular matrix.
alpar@1: --
alpar@1: -- The parameter n is the order of matrix U.
alpar@1: --
alpar@1: -- The matrix U without diagonal elements is specified in the arrays
alpar@1: -- U_ptr, U_ind, and U_val in storage-by-rows format. Diagonal elements
alpar@1: -- of U are specified in the array U_diag, where U_diag[0] is not used,
alpar@1: -- U_diag[i] = u[i,i] for i = 1, ..., n. All these four arrays are not
alpar@1: -- changed on exit.
alpar@1: --
alpar@1: -- The right-hand side vector b is specified on entry in the array x,
alpar@1: -- where x[0] is not used, and x[i] = b[i] for i = 1, ..., n. On exit
alpar@1: -- the routine stores computed components of the vector of unknowns x
alpar@1: -- in the array x in the same manner. */
alpar@1: 
alpar@1: void ut_solve(int n, int U_ptr[], int U_ind[], double U_val[],
alpar@1:       double U_diag[], double x[])
alpar@1: {     int i, t, beg, end;
alpar@1:       double temp;
alpar@1:       for (i = 1; i <= n; i++)
alpar@1:       {  xassert(U_diag[i] != 0.0);
alpar@1:          temp = (x[i] /= U_diag[i]);
alpar@1:          if (temp == 0.0) continue;
alpar@1:          beg = U_ptr[i], end = U_ptr[i+1];
alpar@1:          for (t = beg; t < end; t++)
alpar@1:             x[U_ind[t]] -= U_val[t] * temp;
alpar@1:       }
alpar@1:       return;
alpar@1: }
alpar@1: 
alpar@1: /* eof */