/* glpscf.c (Schur complement factorization) */

 /***********************************************************************

 *  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 "glpscf.h"

 #define xfault xerror

 #define _GLPSCF_DEBUG 0

 #define eps 1e-10

 /***********************************************************************

 *  NAME

 *

 *  scf_create_it - create Schur complement factorization

 *

 *  SYNOPSIS

 *

 *  #include "glpscf.h"

 *  SCF *scf_create_it(int n_max);

 *

 *  DESCRIPTION

 *

 *  The routine scf_create_it creates the factorization of matrix C,

 *  which initially has no rows and columns.

 *

 *  The parameter n_max specifies the maximal order of matrix C to be

 *  factorized, 1 <= n_max <= 32767.

 *

 *  RETURNS

 *

 *  The routine scf_create_it returns a pointer to the structure SCF,

 *  which defines the factorization. */

 SCF *scf_create_it(int n_max)

 {     SCF *scf;

 #if _GLPSCF_DEBUG

       xprintf("scf_create_it: warning: debug mode enabled\n");

 #endif

       if (!(1 <= n_max && n_max <= 32767))

          xfault("scf_create_it: n_max = %d; invalid parameter\n",

             n_max);

       scf = xmalloc(sizeof(SCF));

       scf->n_max = n_max;

       scf->n = 0;

       scf->f = xcalloc(1 + n_max * n_max, sizeof(double));

       scf->u = xcalloc(1 + n_max * (n_max + 1) / 2, sizeof(double));

       scf->p = xcalloc(1 + n_max, sizeof(int));

       scf->t_opt = SCF_TBG;

       scf->rank = 0;

 #if _GLPSCF_DEBUG

       scf->c = xcalloc(1 + n_max * n_max, sizeof(double));

 #else

       scf->c = NULL;

 #endif

       scf->w = xcalloc(1 + n_max, sizeof(double));

       return scf;

 }

 /***********************************************************************

 *  The routine f_loc determines location of matrix element F[i,j] in

 *  the one-dimensional array f. */

 static int f_loc(SCF *scf, int i, int j)

 {     int n_max = scf->n_max;

       int n = scf->n;

       xassert(1 <= i && i <= n);

       xassert(1 <= j && j <= n);

       return (i - 1) * n_max + j;

 }

 /***********************************************************************

 *  The routine u_loc determines location of matrix element U[i,j] in

 *  the one-dimensional array u. */

 static int u_loc(SCF *scf, int i, int j)

 {     int n_max = scf->n_max;

       int n = scf->n;

       xassert(1 <= i && i <= n);

       xassert(i <= j && j <= n);

       return (i - 1) * n_max + j - i * (i - 1) / 2;

 }

 /***********************************************************************

 *  The routine bg_transform applies Bartels-Golub version of gaussian

 *  elimination to restore triangular structure of matrix U.

 *

 *  On entry matrix U has the following structure:

 *

 *        1       k         n

 *     1  * * * * * * * * * *

 *        . * * * * * * * * *

 *        . . * * * * * * * *

 *        . . . * * * * * * *

 *     k  . . . . * * * * * *

 *        . . . . . * * * * *

 *        . . . . . . * * * *

 *        . . . . . . . * * *

 *        . . . . . . . . * *

 *     n  . . . . # # # # # #

 *

 *  where '#' is a row spike to be eliminated.

 *

 *  Elements of n-th row are passed separately in locations un[k], ...,

 *  un[n]. On exit the content of the array un is destroyed.

 *

 *  REFERENCES

 *

 *  R.H.Bartels, G.H.Golub, "The Simplex Method of Linear Programming

 *  Using LU-decomposition", Comm. ACM, 12, pp. 266-68, 1969. */

 static void bg_transform(SCF *scf, int k, double un[])

 {     int n = scf->n;

       double *f = scf->f;

       double *u = scf->u;

       int j, k1, kj, kk, n1, nj;

       double t;

       xassert(1 <= k && k <= n);

       /* main elimination loop */

       for (k = k; k < n; k++)

       {  /* determine location of U[k,k] */

          kk = u_loc(scf, k, k);

          /* determine location of F[k,1] */

          k1 = f_loc(scf, k, 1);

          /* determine location of F[n,1] */

          n1 = f_loc(scf, n, 1);

          /* if |U[k,k]| < |U[n,k]|, interchange k-th and n-th rows to

             provide |U[k,k]| >= |U[n,k]| */

          if (fabs(u[kk]) < fabs(un[k]))

          {  /* interchange k-th and n-th rows of matrix U */

             for (j = k, kj = kk; j <= n; j++, kj++)

                t = u[kj], u[kj] = un[j], un[j] = t;

             /* interchange k-th and n-th rows of matrix F to keep the

                main equality F * C = U * P */

             for (j = 1, kj = k1, nj = n1; j <= n; j++, kj++, nj++)

                t = f[kj], f[kj] = f[nj], f[nj] = t;

          }

          /* now |U[k,k]| >= |U[n,k]| */

          /* if U[k,k] is too small in the magnitude, replace U[k,k] and

             U[n,k] by exact zero */

          if (fabs(u[kk]) < eps) u[kk] = un[k] = 0.0;

          /* if U[n,k] is already zero, elimination is not needed */

          if (un[k] == 0.0) continue;

          /* compute gaussian multiplier t = U[n,k] / U[k,k] */

          t = un[k] / u[kk];

          /* apply gaussian elimination to nullify U[n,k] */

          /* (n-th row of U) := (n-th row of U) - t * (k-th row of U) */

          for (j = k+1, kj = kk+1; j <= n; j++, kj++)

             un[j] -= t * u[kj];

          /* (n-th row of F) := (n-th row of F) - t * (k-th row of F)

             to keep the main equality F * C = U * P */

          for (j = 1, kj = k1, nj = n1; j <= n; j++, kj++, nj++)

             f[nj] -= t * f[kj];

       }

       /* if U[n,n] is too small in the magnitude, replace it by exact

          zero */

       if (fabs(un[n]) < eps) un[n] = 0.0;

       /* store U[n,n] in a proper location */

       u[u_loc(scf, n, n)] = un[n];

       return;

 }

 /***********************************************************************

 *  The routine givens computes the parameters of Givens plane rotation

 *  c = cos(teta) and s = sin(teta) such that:

 *

 *     ( c -s ) ( a )   ( r )

 *     (      ) (   ) = (   ) ,

 *     ( s  c ) ( b )   ( 0 )

 *

 *  where a and b are given scalars.

 *

 *  REFERENCES

 *

 *  G.H.Golub, C.F.Van Loan, "Matrix Computations", 2nd ed. */

 static void givens(double a, double b, double *c, double *s)

 {     double t;

       if (b == 0.0)

          (*c) = 1.0, (*s) = 0.0;

       else if (fabs(a) <= fabs(b))

          t = - a / b, (*s) = 1.0 / sqrt(1.0 + t * t), (*c) = (*s) * t;

       else

          t = - b / a, (*c) = 1.0 / sqrt(1.0 + t * t), (*s) = (*c) * t;

       return;

 }

 /*----------------------------------------------------------------------

 *  The routine gr_transform applies Givens plane rotations to restore

 *  triangular structure of matrix U.

 *

 *  On entry matrix U has the following structure:

 *

 *        1       k         n

 *     1  * * * * * * * * * *

 *        . * * * * * * * * *

 *        . . * * * * * * * *

 *        . . . * * * * * * *

 *     k  . . . . * * * * * *

 *        . . . . . * * * * *

 *        . . . . . . * * * *

 *        . . . . . . . * * *

 *        . . . . . . . . * *

 *     n  . . . . # # # # # #

 *

 *  where '#' is a row spike to be eliminated.

 *

 *  Elements of n-th row are passed separately in locations un[k], ...,

 *  un[n]. On exit the content of the array un is destroyed.

 *

 *  REFERENCES

 *

 *  R.H.Bartels, G.H.Golub, "The Simplex Method of Linear Programming

 *  Using LU-decomposition", Comm. ACM, 12, pp. 266-68, 1969. */

 static void gr_transform(SCF *scf, int k, double un[])

 {     int n = scf->n;

       double *f = scf->f;

       double *u = scf->u;

       int j, k1, kj, kk, n1, nj;

       double c, s;

       xassert(1 <= k && k <= n);

       /* main elimination loop */

       for (k = k; k < n; k++)

       {  /* determine location of U[k,k] */

          kk = u_loc(scf, k, k);

          /* determine location of F[k,1] */

          k1 = f_loc(scf, k, 1);

          /* determine location of F[n,1] */

          n1 = f_loc(scf, n, 1);

          /* if both U[k,k] and U[n,k] are too small in the magnitude,

             replace them by exact zero */

          if (fabs(u[kk]) < eps && fabs(un[k]) < eps)

             u[kk] = un[k] = 0.0;

          /* if U[n,k] is already zero, elimination is not needed */

          if (un[k] == 0.0) continue;

          /* compute the parameters of Givens plane rotation */

          givens(u[kk], un[k], &c, &s);

          /* apply Givens rotation to k-th and n-th rows of matrix U */

          for (j = k, kj = kk; j <= n; j++, kj++)

          {  double ukj = u[kj], unj = un[j];

             u[kj] = c * ukj - s * unj;

             un[j] = s * ukj + c * unj;

          }

          /* apply Givens rotation to k-th and n-th rows of matrix F

             to keep the main equality F * C = U * P */

          for (j = 1, kj = k1, nj = n1; j <= n; j++, kj++, nj++)

          {  double fkj = f[kj], fnj = f[nj];

             f[kj] = c * fkj - s * fnj;

             f[nj] = s * fkj + c * fnj;

          }

       }

       /* if U[n,n] is too small in the magnitude, replace it by exact

          zero */

       if (fabs(un[n]) < eps) un[n] = 0.0;

       /* store U[n,n] in a proper location */

       u[u_loc(scf, n, n)] = un[n];

       return;

 }

 /***********************************************************************

 *  The routine transform restores triangular structure of matrix U.

 *  It is a driver to the routines bg_transform and gr_transform (see

 *  comments to these routines above). */

 static void transform(SCF *scf, int k, double un[])

 {     switch (scf->t_opt)

       {  case SCF_TBG:

             bg_transform(scf, k, un);

             break;

          case SCF_TGR:

             gr_transform(scf, k, un);

             break;

          default:

             xassert(scf != scf);

       }

       return;

 }

 /***********************************************************************

 *  The routine estimate_rank estimates the rank of matrix C.

 *

 *  Since all transformations applied to matrix F are non-singular,

 *  and F is assumed to be well conditioned, from the main equaility

 *  F * C = U * P it follows that rank(C) = rank(U), where rank(U) is

 *  estimated as the number of non-zero diagonal elements of U. */

 static int estimate_rank(SCF *scf)

 {     int n_max = scf->n_max;

       int n = scf->n;

       double *u = scf->u;

       int i, ii, inc, rank = 0;

       for (i = 1, ii = u_loc(scf, i, i), inc = n_max; i <= n;

          i++, ii += inc, inc--)

          if (u[ii] != 0.0) rank++;

       return rank;

 }

 #if _GLPSCF_DEBUG

 /***********************************************************************

 *  The routine check_error computes the maximal relative error between

 *  left- and right-hand sides of the main equality F * C = U * P. (This

 *  routine is intended only for debugging.) */

 static void check_error(SCF *scf, const char *func)

 {     int n = scf->n;

       double *f = scf->f;

       double *u = scf->u;

       int *p = scf->p;

       double *c = scf->c;

       int i, j, k;

       double d, dmax = 0.0, s, t;

       xassert(c != NULL);

       for (i = 1; i <= n; i++)

       {  for (j = 1; j <= n; j++)

          {  /* compute element (i,j) of product F * C */

             s = 0.0;

             for (k = 1; k <= n; k++)

                s += f[f_loc(scf, i, k)] * c[f_loc(scf, k, j)];

             /* compute element (i,j) of product U * P */

             k = p[j];

             t = (i <= k ? u[u_loc(scf, i, k)] : 0.0);

             /* compute the maximal relative error */

             d = fabs(s - t) / (1.0 + fabs(t));

             if (dmax < d) dmax = d;

          }

       }

       if (dmax > 1e-8)

          xprintf("%s: dmax = %g; relative error too large\n", func,

             dmax);

       return;

 }

 #endif

 /***********************************************************************

 *  NAME

 *

 *  scf_update_exp - update factorization on expanding C

 *

 *  SYNOPSIS

 *

 *  #include "glpscf.h"

 *  int scf_update_exp(SCF *scf, const double x[], const double y[],

 *     double z);

 *

 *  DESCRIPTION

 *

 *  The routine scf_update_exp updates the factorization of matrix C on

 *  expanding it by adding a new row and column as follows:

 *

 *             ( C  x )

 *     new C = (      )

 *             ( y' z )

 *

 *  where x[1,...,n] is a new column, y[1,...,n] is a new row, and z is

 *  a new diagonal element.

 *

 *  If on entry the factorization is empty, the parameters x and y can

 *  be specified as NULL.

 *

 *  RETURNS

 *

 *  0  The factorization has been successfully updated.

 *

 *  SCF_ESING

 *     The factorization has been successfully updated, however, new

 *     matrix C is singular within working precision. Note that the new

 *     factorization remains valid.

 *

 *  SCF_ELIMIT

 *     There is not enough room to expand the factorization, because

 *     n = n_max. The factorization remains unchanged.

 *

 *  ALGORITHM

 *

 *  We can see that:

 *

 *     ( F  0 ) ( C  x )   ( FC  Fx )   ( UP  Fx )

 *     (      ) (      ) = (        ) = (        ) =

 *     ( 0  1 ) ( y' z )   ( y'   z )   ( y'   z )

 *

 *        ( U   Fx ) ( P  0 )

 *     =  (        ) (      ),

 *        ( y'P' z ) ( 0  1 )

 *

 *  therefore to keep the main equality F * C = U * P we can take:

 *

 *             ( F  0 )           ( U   Fx )           ( P  0 )

 *     new F = (      ),  new U = (        ),  new P = (      ),

 *             ( 0  1 )           ( y'P' z )           ( 0  1 )

 *

 *  and eliminate the row spike y'P' in the last row of new U to restore

 *  its upper triangular structure. */

 int scf_update_exp(SCF *scf, const double x[], const double y[],

       double z)

 {     int n_max = scf->n_max;

       int n = scf->n;

       double *f = scf->f;

       double *u = scf->u;

       int *p = scf->p;

 #if _GLPSCF_DEBUG

       double *c = scf->c;

 #endif

       double *un = scf->w;

       int i, ij, in, j, k, nj, ret = 0;

       double t;

       /* check if the factorization can be expanded */

       if (n == n_max)

       {  /* there is not enough room */

          ret = SCF_ELIMIT;

          goto done;

       }

       /* increase the order of the factorization */

       scf->n = ++n;

       /* fill new zero column of matrix F */

       for (i = 1, in = f_loc(scf, i, n); i < n; i++, in += n_max)

          f[in] = 0.0;

       /* fill new zero row of matrix F */

       for (j = 1, nj = f_loc(scf, n, j); j < n; j++, nj++)

          f[nj] = 0.0;

       /* fill new unity diagonal element of matrix F */

       f[f_loc(scf, n, n)] = 1.0;

       /* compute new column of matrix U, which is (old F) * x */

       for (i = 1; i < n; i++)

       {  /* u[i,n] := (i-th row of old F) * x */

          t = 0.0;

          for (j = 1, ij = f_loc(scf, i, 1); j < n; j++, ij++)

             t += f[ij] * x[j];

          u[u_loc(scf, i, n)] = t;

       }

       /* compute new (spiked) row of matrix U, which is (old P) * y */

       for (j = 1; j < n; j++) un[j] = y[p[j]];

       /* store new diagonal element of matrix U, which is z */

       un[n] = z;

       /* expand matrix P */

       p[n] = n;

 #if _GLPSCF_DEBUG

       /* expand matrix C */

       /* fill its new column, which is x */

       for (i = 1, in = f_loc(scf, i, n); i < n; i++, in += n_max)

          c[in] = x[i];

       /* fill its new row, which is y */

       for (j = 1, nj = f_loc(scf, n, j); j < n; j++, nj++)

          c[nj] = y[j];

       /* fill its new diagonal element, which is z */

       c[f_loc(scf, n, n)] = z;

 #endif

       /* restore upper triangular structure of matrix U */

       for (k = 1; k < n; k++)

          if (un[k] != 0.0) break;

       transform(scf, k, un);

       /* estimate the rank of matrices C and U */

       scf->rank = estimate_rank(scf);

       if (scf->rank != n) ret = SCF_ESING;

 #if _GLPSCF_DEBUG

       /* check that the factorization is accurate enough */

       check_error(scf, "scf_update_exp");

 #endif

 done: return ret;

 }

 /***********************************************************************

 *  The routine solve solves the system C * x = b.

 *

 *  From the main equation F * C = U * P it follows that:

 *

 *     C * x = b  =>  F * C * x = F * b  =>  U * P * x = F * b  =>

 *

 *     P * x = inv(U) * F * b  =>  x = P' * inv(U) * F * b.

 *

 *  On entry the array x contains right-hand side vector b. On exit this

 *  array contains solution vector x. */

 static void solve(SCF *scf, double x[])

 {     int n = scf->n;

       double *f = scf->f;

       double *u = scf->u;

       int *p = scf->p;

       double *y = scf->w;

       int i, j, ij;

       double t;

       /* y := F * b */

       for (i = 1; i <= n; i++)

       {  /* y[i] = (i-th row of F) * b */

          t = 0.0;

          for (j = 1, ij = f_loc(scf, i, 1); j <= n; j++, ij++)

             t += f[ij] * x[j];

          y[i] = t;

       }

       /* y := inv(U) * y */

       for (i = n; i >= 1; i--)

       {  t = y[i];

          for (j = n, ij = u_loc(scf, i, n); j > i; j--, ij--)

             t -= u[ij] * y[j];

          y[i] = t / u[ij];

       }

       /* x := P' * y */

       for (i = 1; i <= n; i++) x[p[i]] = y[i];

       return;

 }

 /***********************************************************************

 *  The routine tsolve solves the transposed system C' * x = b.

 *

 *  From the main equation F * C = U * P it follows that:

 *

 *     C' * F' = P' * U',

 *

 *  therefore:

 *

 *     C' * x = b  =>  C' * F' * inv(F') * x = b  =>

 *

 *     P' * U' * inv(F') * x = b  =>  U' * inv(F') * x = P * b  =>

 *

 *     inv(F') * x = inv(U') * P * b  =>  x = F' * inv(U') * P * b.

 *

 *  On entry the array x contains right-hand side vector b. On exit this

 *  array contains solution vector x. */

 static void tsolve(SCF *scf, double x[])

 {     int n = scf->n;

       double *f = scf->f;

       double *u = scf->u;

       int *p = scf->p;

       double *y = scf->w;

       int i, j, ij;

       double t;

       /* y := P * b */

       for (i = 1; i <= n; i++) y[i] = x[p[i]];

       /* y := inv(U') * y */

       for (i = 1; i <= n; i++)

       {  /* compute y[i] */

          ij = u_loc(scf, i, i);

          t = (y[i] /= u[ij]);

          /* substitute y[i] in other equations */

          for (j = i+1, ij++; j <= n; j++, ij++)

             y[j] -= u[ij] * t;

       }

       /* x := F' * y (computed as linear combination of rows of F) */

       for (j = 1; j <= n; j++) x[j] = 0.0;

       for (i = 1; i <= n; i++)

       {  t = y[i]; /* coefficient of linear combination */

          for (j = 1, ij = f_loc(scf, i, 1); j <= n; j++, ij++)

             x[j] += f[ij] * t;

       }

       return;

 }

 /***********************************************************************

 *  NAME

 *

 *  scf_solve_it - solve either system C * x = b or C' * x = b

 *

 *  SYNOPSIS

 *

 *  #include "glpscf.h"

 *  void scf_solve_it(SCF *scf, int tr, double x[]);

 *

 *  DESCRIPTION

 *

 *  The routine scf_solve_it solves either the system C * x = b (if tr

 *  is zero) or the system C' * x = b, where C' is a matrix transposed

 *  to C (if tr is non-zero). C is assumed to be non-singular.

 *

 *  On entry the array x should contain the right-hand side vector b in

 *  locations x[1], ..., x[n], where n is the order of matrix C. On exit

 *  the array x contains the solution vector x in the same locations. */

 void scf_solve_it(SCF *scf, int tr, double x[])

 {     if (scf->rank < scf->n)

          xfault("scf_solve_it: singular matrix\n");

       if (!tr)

          solve(scf, x);

       else

          tsolve(scf, x);

       return;

 }

 void scf_reset_it(SCF *scf)

 {     /* reset factorization for empty matrix C */

       scf->n = scf->rank = 0;

       return;

 }

 /***********************************************************************

 *  NAME

 *

 *  scf_delete_it - delete Schur complement factorization

 *

 *  SYNOPSIS

 *

 *  #include "glpscf.h"

 *  void scf_delete_it(SCF *scf);

 *

 *  DESCRIPTION

 *

 *  The routine scf_delete_it deletes the specified factorization and

 *  frees all the memory allocated to this object. */

 void scf_delete_it(SCF *scf)

 {     xfree(scf->f);

       xfree(scf->u);

       xfree(scf->p);

 #if _GLPSCF_DEBUG

       xfree(scf->c);

 #endif

       xfree(scf->w);

       xfree(scf);

       return;

 }

 /* eof */