/* 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 */