/* 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 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 */