eqnsys.cpp

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/*
 * eqnsys.cpp - equation system solver class implementation
 *
 * Copyright (C) 2004, 2005, 2006, 2007, 2008 Stefan Jahn <stefan@lkcc.org>
 *
 * This 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 2, or (at your option)
 * any later version.
 *
 * This software 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 this package; see the file COPYING.  If not, write to
 * the Free Software Foundation, Inc., 51 Franklin Street - Fifth Floor,
 * Boston, MA 02110-1301, USA.
 *
 * $Id$
 *
 */

// the types required for qucs library files are defined
// in qucs_typedefs.h, created by configure from
// qucs_typedefs.h.in
#include "qucs_typedefs.h"

#include <assert.h>
#include <time.h>
#include <cmath>
#include <float.h>

#include "compat.h"
#include "logging.h"
#include "precision.h"
#include "complex.h"
#include "tmatrix.h"
#include "eqnsys.h"
#include "exception.h"
#include "exceptionstack.h"

#define Swap(type,a,b) { type t; t = a; a = b; b = t; }

namespace qucs {

template <class nr_type_t>
eqnsys<nr_type_t>::eqnsys () {
  A = V = NULL;
  B = X = NULL;
  S = E = NULL;
  T = R = NULL;
  nPvt = NULL;
  cMap = rMap = NULL;
  update = 1;
  pivoting = PIVOT_PARTIAL;
  N = 0;
}

template <class nr_type_t>
eqnsys<nr_type_t>::~eqnsys () {
  if (R != NULL) delete R;
  if (T != NULL) delete T;
  if (B != NULL) delete B;
  if (S != NULL) delete S;
  if (E != NULL) delete E;
  if (V != NULL) delete V;
  if (rMap != NULL) delete[] rMap;
  if (cMap != NULL) delete[] cMap;
  if (nPvt != NULL) delete[] nPvt;
}

template <class nr_type_t>
eqnsys<nr_type_t>::eqnsys (eqnsys & e) {
  A = e.A;
  V = NULL;
  S = E = NULL;
  T = R = NULL;
  B = e.B ? new tvector<nr_type_t> (*(e.B)) : NULL;
  cMap = rMap = NULL;
  nPvt = NULL;
  update = 1;
  X = e.X;
  N = 0;
}

template <class nr_type_t>
void eqnsys<nr_type_t>::passEquationSys (tmatrix<nr_type_t> * nA,
                                         tvector<nr_type_t> * refX,
                                         tvector<nr_type_t> * nB) {
  if (nA != NULL) {
    A = nA;
    update = 1;
    if (N != A->getCols ()) {
      N = A->getCols ();
      if (cMap) delete[] cMap; cMap = new int[N];
      if (rMap) delete[] rMap; rMap = new int[N];
      if (nPvt) delete[] nPvt; nPvt = new nr_double_t[N];
    }
  }
  else {
    update = 0;
  }
  if (B != NULL) delete B;
  B = new tvector<nr_type_t> (*nB);
  X = refX;
}

template <class nr_type_t>
void eqnsys<nr_type_t>::solve (void) {
#if DEBUG && 0
  time_t t = time (NULL);
#endif
  switch (algo) {
  case ALGO_INVERSE:
    solve_inverse ();
    break;
  case ALGO_GAUSS:
    solve_gauss ();
    break;
  case ALGO_GAUSS_JORDAN:
    solve_gauss_jordan ();
    break;
  case ALGO_LU_DECOMPOSITION_CROUT:
    solve_lu_crout ();
    break;
  case ALGO_LU_DECOMPOSITION_DOOLITTLE:
    solve_lu_doolittle ();
    break;
  case ALGO_LU_FACTORIZATION_CROUT:
    factorize_lu_crout ();
    break;
  case ALGO_LU_FACTORIZATION_DOOLITTLE:
    factorize_lu_doolittle ();
    break;
  case ALGO_LU_SUBSTITUTION_CROUT:
    substitute_lu_crout ();
    break;
  case ALGO_LU_SUBSTITUTION_DOOLITTLE:
    substitute_lu_doolittle ();
    break;
  case ALGO_JACOBI: case ALGO_GAUSS_SEIDEL:
    solve_iterative ();
    break;
  case ALGO_SOR:
    solve_sor ();
    break;
  case ALGO_QR_DECOMPOSITION:
    solve_qr ();
    break;
  case ALGO_QR_DECOMPOSITION_LS:
    solve_qr_ls ();
    break;
  case ALGO_SV_DECOMPOSITION:
    solve_svd ();
    break;
  case ALGO_QR_DECOMPOSITION_2:
    solve_qrh ();
    break;
  }
#if DEBUG && 0
  logprint (LOG_STATUS, "NOTIFY: %dx%d eqnsys solved in %ld seconds\n",
            A->getRows (), A->getCols (), time (NULL) - t);
#endif
}

template <class nr_type_t>
void eqnsys<nr_type_t>::solve_inverse (void) {
  *X = inverse (*A) * *B;
}

#define A_ (*A)
#define X_ (*X)
#define B_ (*B)

template <class nr_type_t>
void eqnsys<nr_type_t>::solve_gauss (void) {
  nr_double_t MaxPivot;
  nr_type_t f;
  int i, c, r, pivot;

  // triangulate the matrix
  for (i = 0; i < N; i++) {
    // find maximum column value for pivoting
    for (MaxPivot = 0, pivot = r = i; r < N; r++) {
      if (abs (A_(r, i)) > MaxPivot) {
        MaxPivot = abs (A_(r, i));
        pivot = r;
      }
    }
    // exchange rows if necessary
    assert (MaxPivot != 0);
    if (i != pivot) {
      A->exchangeRows (i, pivot);
      B->exchangeRows (i, pivot);
    }
    // compute new rows and columns
    for (r = i + 1; r < N; r++) {
      f = A_(r, i) / A_(i, i);
      for (c = i + 1; c < N; c++) A_(r, c) -= f * A_(i, c);
      B_(r) -= f * B_(i);
    }
  }

  // backward substitution
  for (i = N - 1; i >= 0; i--) {
    f = B_(i);
    for (c = i + 1; c < N; c++) f -= A_(i, c) * X_(c);
    X_(i) = f / A_(i, i);
  }
}

template <class nr_type_t>
void eqnsys<nr_type_t>::solve_gauss_jordan (void) {
  nr_double_t MaxPivot;
  nr_type_t f;
  int i, c, r, pivot;

  // create the eye matrix
  for (i = 0; i < N; i++) {
    // find maximum column value for pivoting
    for (MaxPivot = 0, pivot = r = i; r < N; r++) {
      if (abs (A_(r, i)) > MaxPivot) {
        MaxPivot = abs (A_(r, i));
        pivot = r;
      }
    }
    // exchange rows if necessary
    assert (MaxPivot != 0);
    if (i != pivot) {
      A->exchangeRows (i, pivot);
      B->exchangeRows (i, pivot);
    }

    // compute current row
    f = A_(i, i);
    for (c = i + 1; c < N; c++) A_(i, c) /= f;
    B_(i) /= f;

    // compute new rows and columns
    for (r = 0; r < N; r++) {
      if (r != i) {
        f = A_(r, i);
        for (c = i + 1; c < N; c++) A_(r, c) -= f * A_(i, c);
        B_(r) -= f * B_(i);
      }
    }
  }

  // right hand side is now the solution
  *X = *B;
}

#define LU_FAILURE 0
#define VIRTUAL_RES(txt,i) {                                      \
  qucs::exception * e = new qucs::exception (EXCEPTION_SINGULAR); \
  e->setText (txt);                                               \
  e->setData (rMap[i]);                                           \
  A_(i, i) = NR_TINY; /* virtual resistance to ground */          \
  throw_exception (e); }

template <class nr_type_t>
void eqnsys<nr_type_t>::solve_lu_crout (void) {

  // skip decomposition if requested
  if (update) {
    // perform LU composition
    factorize_lu_crout ();
  }

  // finally solve the equation system
  substitute_lu_crout ();
}

template <class nr_type_t>
void eqnsys<nr_type_t>::solve_lu_doolittle (void) {

  // skip decomposition if requested
  if (update) {
    // perform LU composition
    factorize_lu_doolittle ();
  }

  // finally solve the equation system
  substitute_lu_doolittle ();
}

template <class nr_type_t>
void eqnsys<nr_type_t>::factorize_lu_crout (void) {
  nr_double_t d, MaxPivot;
  nr_type_t f;
  int k, c, r, pivot;

  // initialize pivot exchange table
  for (r = 0; r < N; r++) {
    for (MaxPivot = 0, c = 0; c < N; c++)
      if ((d = abs (A_(r, c))) > MaxPivot)
        MaxPivot = d;
    if (MaxPivot <= 0) MaxPivot = NR_TINY;
    nPvt[r] = 1 / MaxPivot;
    rMap[r] = r;
  }

  // decompose the matrix into L (lower) and U (upper) matrix
  for (c = 0; c < N; c++) {
    // upper matrix entries
    for (r = 0; r < c; r++) {
      f = A_(r, c);
      for (k = 0; k < r; k++) f -= A_(r, k) * A_(k, c);
      A_(r, c) = f / A_(r, r);
    }
    // lower matrix entries
    for (MaxPivot = 0, pivot = r; r < N; r++) {
      f = A_(r, c);
      for (k = 0; k < c; k++) f -= A_(r, k) * A_(k, c);
      A_(r, c) = f;
      // larger pivot ?
      if ((d = nPvt[r] * abs (f)) > MaxPivot) {
        MaxPivot = d;
        pivot = r;
      }
    }

    // check pivot element and throw appropriate exception
    if (MaxPivot <= 0) {
#if LU_FAILURE
      qucs::exception * e = new qucs::exception (EXCEPTION_PIVOT);
      e->setText ("no pivot != 0 found during Crout LU decomposition");
      e->setData (c);
      throw_exception (e);
      goto fail;
#else /* insert virtual resistance */
      VIRTUAL_RES ("no pivot != 0 found during Crout LU decomposition", c);
#endif
    }

    // swap matrix rows if necessary and remember that step in the
    // exchange table
    if (c != pivot) {
      A->exchangeRows (c, pivot);
      Swap (int, rMap[c], rMap[pivot]);
      Swap (nr_double_t, nPvt[c], nPvt[pivot]);
    }
  }
#if LU_FAILURE
 fail:
#endif
}

template <class nr_type_t>
void eqnsys<nr_type_t>::factorize_lu_doolittle (void) {
  nr_double_t d, MaxPivot;
  nr_type_t f;
  int k, c, r, pivot;

  // initialize pivot exchange table
  for (r = 0; r < N; r++) {
    for (MaxPivot = 0, c = 0; c < N; c++)
      if ((d = abs (A_(r, c))) > MaxPivot)
        MaxPivot = d;
    if (MaxPivot <= 0) MaxPivot = NR_TINY;
    nPvt[r] = 1 / MaxPivot;
    rMap[r] = r;
  }

  // decompose the matrix into L (lower) and U (upper) matrix
  for (c = 0; c < N; c++) {
    // upper matrix entries
    for (r = 0; r < c; r++) {
      f = A_(r, c);
      for (k = 0; k < r; k++) f -= A_(r, k) * A_(k, c);
      A_(r, c) = f;
    }
    // lower matrix entries
    for (MaxPivot = 0, pivot = r; r < N; r++) {
      f = A_(r, c);
      for (k = 0; k < c; k++) f -= A_(r, k) * A_(k, c);
      A_(r, c) = f;
      // larger pivot ?
      if ((d = nPvt[r] * abs (f)) > MaxPivot) {
        MaxPivot = d;
        pivot = r;
      }
    }

    // check pivot element and throw appropriate exception
    if (MaxPivot <= 0) {
#if LU_FAILURE
      qucs::exception * e = new qucs::exception (EXCEPTION_PIVOT);
      e->setText ("no pivot != 0 found during Doolittle LU decomposition");
      e->setData (c);
      throw_exception (e);
      goto fail;
#else /* insert virtual resistance */
      VIRTUAL_RES ("no pivot != 0 found during Doolittle LU decomposition", c);
#endif
    }

    // swap matrix rows if necessary and remember that step in the
    // exchange table
    if (c != pivot) {
      A->exchangeRows (c, pivot);
      Swap (int, rMap[c], rMap[pivot]);
      Swap (nr_double_t, nPvt[c], nPvt[pivot]);
    }

    // finally divide by the pivot element
    if (c < N - 1) {
      f = 1.0 / A_(c, c);
      for (r = c + 1; r < N; r++) A_(r, c) *= f;
    }
  }
#if LU_FAILURE
 fail:
#endif
}

template <class nr_type_t>
void eqnsys<nr_type_t>::substitute_lu_crout (void) {
  nr_type_t f;
  int i, c;

  // forward substitution in order to solve LY = B
  for (i = 0; i < N; i++) {
    f = B_(rMap[i]);
    for (c = 0; c < i; c++) f -= A_(i, c) * X_(c);
    X_(i) = f / A_(i, i);
  }

  // backward substitution in order to solve UX = Y
  for (i = N - 1; i >= 0; i--) {
    f = X_(i);
    for (c = i + 1; c < N; c++) f -= A_(i, c) * X_(c);
    // remember that the Uii diagonal are ones only in Crout's definition
    X_(i) = f;
  }
}

template <class nr_type_t>
void eqnsys<nr_type_t>::substitute_lu_doolittle (void) {
  nr_type_t f;
  int i, c;

  // forward substitution in order to solve LY = B
  for (i = 0; i < N; i++) {
    f = B_(rMap[i]);
    for (c = 0; c < i; c++) f -= A_(i, c) * X_(c);
    // remember that the Lii diagonal are ones only in Doolittle's definition
    X_(i) = f;
  }

  // backward substitution in order to solve UX = Y
  for (i = N - 1; i >= 0; i--) {
    f = X_(i);
    for (c = i + 1; c < N; c++) f -= A_(i, c) * X_(c);
    X_(i) = f / A_(i, i);
  }
}

template <class nr_type_t>
void eqnsys<nr_type_t>::solve_iterative (void) {
  nr_type_t f;
  int error, conv, i, c, r;
  int MaxIter = N; // -> less than N^3 operations
  nr_double_t reltol = 1e-4;
  nr_double_t abstol = NR_TINY;
  nr_double_t diff, crit;

  // ensure that all diagonal values are non-zero
  ensure_diagonal ();

  // try to raise diagonal dominance
  preconditioner ();

  // decide here about possible convergence
  if ((crit = convergence_criteria ()) >= 1) {
#if DEBUG && 0
    logprint (LOG_STATUS, "NOTIFY: convergence criteria: %g >= 1 (%dx%d)\n",
              crit, N, N);
#endif
    //solve_lu ();
    //return;
  }

  // normalize the equation system to have ones on its diagonal
  for (r = 0; r < N; r++) {
    f = A_(r, r);
    assert (f != 0); // singular matrix
    for (c = 0; c < N; c++) A_(r, c) /= f;
    B_(r) /= f;
  }

  // the current X vector is a good initial guess for the iteration
  tvector<nr_type_t> * Xprev = new tvector<nr_type_t> (*X);

  // start iterating here
  i = 0; error = 0;
  do {
    // compute new solution vector
    for (r = 0; r < N; r++) {
      for (f = 0, c = 0; c < N; c++) {
        if (algo == ALGO_GAUSS_SEIDEL) {
          // Gauss-Seidel
          if (c < r)      f += A_(r, c) * X_(c);
          else if (c > r) f += A_(r, c) * Xprev->get (c);
        }
        else {
          // Jacobi
          if (c != r) f += A_(r, c) * Xprev->get (c);
        }
      }
      X_(r) = B_(r) - f;
    }
    // check for convergence
    for (conv = 1, r = 0; r < N; r++) {
      diff = abs (X_(r) - Xprev->get (r));
      if (diff >= abstol + reltol * abs (X_(r))) {
        conv = 0;
        break;
      }
      if (!std::isfinite (diff)) { error++; break; }
    }
    // save last values
    *Xprev = *X;
  }
  while (++i < MaxIter && !conv);

  delete Xprev;

  if (!conv || error) {
    logprint (LOG_ERROR,
              "WARNING: no convergence after %d %s iterations\n",
              i, algo == ALGO_JACOBI ? "jacobi" : "gauss-seidel");
    solve_lu_crout ();
  }
#if DEBUG && 0
  else {
    logprint (LOG_STATUS,
              "NOTIFY: %s convergence after %d iterations\n",
              algo == ALGO_JACOBI ? "jacobi" : "gauss-seidel", i);
  }
#endif
}

template <class nr_type_t>
void eqnsys<nr_type_t>::solve_sor (void) {
  nr_type_t f;
  int error, conv, i, c, r;
  int MaxIter = N; // -> less than N^3 operations
  nr_double_t reltol = 1e-4;
  nr_double_t abstol = NR_TINY;
  nr_double_t diff, crit, l = 1, d, s;

  // ensure that all diagonal values are non-zero
  ensure_diagonal ();

  // try to raise diagonal dominance
  preconditioner ();

  // decide here about possible convergence
  if ((crit = convergence_criteria ()) >= 1) {
#if DEBUG && 0
    logprint (LOG_STATUS, "NOTIFY: convergence criteria: %g >= 1 (%dx%d)\n",
              crit, N, N);
#endif
    //solve_lu ();
    //return;
  }

  // normalize the equation system to have ones on its diagonal
  for (r = 0; r < N; r++) {
    f = A_(r, r);
    assert (f != 0); // singular matrix
    for (c = 0; c < N; c++) A_(r, c) /= f;
    B_(r) /= f;
  }

  // the current X vector is a good initial guess for the iteration
  tvector<nr_type_t> * Xprev = new tvector<nr_type_t> (*X);

  // start iterating here
  i = 0; error = 0;
  do {
    // compute new solution vector
    for (r = 0; r < N; r++) {
      for (f = 0, c = 0; c < N; c++) {
        if (c < r)      f += A_(r, c) * X_(c);
        else if (c > r) f += A_(r, c) * Xprev->get (c);
      }
      X_(r) = (1 - l) * Xprev->get (r) + l * (B_(r) - f);
    }
    // check for convergence
    for (s = 0, d = 0, conv = 1, r = 0; r < N; r++) {
      diff = abs (X_(r) - Xprev->get (r));
      if (diff >= abstol + reltol * abs (X_(r))) {
        conv = 0;
        break;
      }
      d += diff; s += abs (X_(r));
      if (!std::isfinite (diff)) { error++; break; }
    }
    if (!error) {
      // adjust relaxation based on average errors
      if ((s == 0 && d == 0) || d >= abstol * N + reltol * s) {
        // values <= 1 -> non-convergence to convergence
        if (l >= 0.6) l -= 0.1;
        if (l >= 1.0) l = 1.0;
      }
      else {
        // values >= 1 -> faster convergence
        if (l < 1.5) l += 0.01;
        if (l < 1.0) l = 1.0;
      }
    }
    // save last values
    *Xprev = *X;
  }
  while (++i < MaxIter && !conv);

  delete Xprev;

  if (!conv || error) {
    logprint (LOG_ERROR,
              "WARNING: no convergence after %d sor iterations (l = %g)\n",
              i, l);
    solve_lu_crout ();
  }
#if DEBUG && 0
  else {
    logprint (LOG_STATUS,
              "NOTIFY: sor convergence after %d iterations\n", i);
  }
#endif
}

template <class nr_type_t>
nr_double_t eqnsys<nr_type_t>::convergence_criteria (void) {
  nr_double_t f = 0;
  for (int r = 0; r < A->getCols (); r++) {
    for (int c = 0; c < A->getCols (); c++) {
      if (r != c) f += norm (A_(r, c) / A_(r, r));
    }
  }
  return sqrt (f);
}

template <class nr_type_t>
void eqnsys<nr_type_t>::ensure_diagonal (void) {
  ensure_diagonal_MNA ();
}

template <class nr_type_t>
void eqnsys<nr_type_t>::ensure_diagonal_MNA (void) {
  int restart, exchanged, begin = 0, pairs;
  int pivot1, pivot2, i;
  do {
    restart = exchanged = 0;
    /* search for zero diagonals with lone pairs */
    for (i = begin; i < N; i++) {
      if (A_(i, i) == 0) {
        pairs = countPairs (i, pivot1, pivot2);
        if (pairs == 1) { /* lone pair found, substitute rows */
          A->exchangeRows (i, pivot1);
          B->exchangeRows (i, pivot1);
          exchanged = 1;
        }
        else if ((pairs > 1) && !restart) {
          restart = 1;
          begin = i;
        }
      }
    }

    /* all lone pairs are gone, look for zero diagonals with multiple pairs */
    if (restart) {
      for (i = begin; !exchanged && i < N; i++) {
        if (A_(i, i) == 0) {
          pairs = countPairs (i, pivot1, pivot2);
          A->exchangeRows (i, pivot1);
          B->exchangeRows (i, pivot1);
          exchanged = 1;
        }
      }
    }
  }
  while (restart);
}

template <class nr_type_t>
int eqnsys<nr_type_t>::countPairs (int i, int& r1, int& r2) {
  int pairs = 0;
  for (int r = 0; r < N; r++) {
    if (fabs (real (A_(r, i))) == 1.0) {
      r1 = r;
      pairs++;
      for (r++; r < N; r++) {
        if (fabs (real (A_(r, i))) == 1.0) {
          r2 = r;
          if (++pairs >= 2) return pairs;
        }
      }
    }
  }
  return pairs;
}

template <class nr_type_t>
void eqnsys<nr_type_t>::preconditioner (void) {
  int pivot, r;
  nr_double_t MaxPivot;
  for (int i = 0; i < N; i++) {
    // find maximum column value for pivoting
    for (MaxPivot = 0, pivot = i, r = 0; r < N; r++) {
      if (abs (A_(r, i)) > MaxPivot &&
          abs (A_(i, r)) >= abs (A_(r, r))) {
        MaxPivot = abs (A_(r, i));
        pivot = r;
      }
    }
    // swap matrix rows if possible
    if (i != pivot) {
      A->exchangeRows (i, pivot);
      B->exchangeRows (i, pivot);
    }
  }
}

#define R_ (*R)
#define T_ (*T)

template <class nr_type_t>
void eqnsys<nr_type_t>::solve_qrh (void) {
  factorize_qrh ();
  substitute_qrh ();
}

template <class nr_type_t>
void eqnsys<nr_type_t>::solve_qr (void) {
  factorize_qr_householder ();
  substitute_qr_householder ();
}

template <class nr_type_t>
void eqnsys<nr_type_t>::solve_qr_ls (void) {
  A->transpose ();
  factorize_qr_householder ();
  substitute_qr_householder_ls ();
}

static void
euclidian_update (nr_double_t a, nr_double_t& n, nr_double_t& scale) {
  nr_double_t x, ax;
  if ((x = a) != 0) {
    ax = fabs (x);
    if (scale < ax) {
      x = scale / ax;
      n = 1 + n * x * x;
      scale = ax;
    }
    else {
      x = ax / scale;
      n += x * x;
    }
  }
}

template <class nr_type_t>
nr_double_t eqnsys<nr_type_t>::euclidian_c (int c, int r) {
  nr_double_t scale = 0, n = 1;
  for (int i = r; i < N; i++) {
    euclidian_update (real (A_(i, c)), n, scale);
    euclidian_update (imag (A_(i, c)), n, scale);
  }
  return scale * sqrt (n);
}

template <class nr_type_t>
nr_double_t eqnsys<nr_type_t>::euclidian_r (int r, int c) {
  nr_double_t scale = 0, n = 1;
  for (int i = c; i < N; i++) {
    euclidian_update (real (A_(r, i)), n, scale);
    euclidian_update (imag (A_(r, i)), n, scale);
  }
  return scale * sqrt (n);
}

template <typename nr_type_t>
inline nr_type_t cond_conj (nr_type_t t) {
  return std::conj(t);
}

template <>
inline double cond_conj (double t) {
  return t;
}

template <class nr_type_t>
void eqnsys<nr_type_t>::factorize_qrh (void) {
  int c, r, k, pivot;
  nr_type_t f, t;
  nr_double_t s, MaxPivot;

  if (R) delete R; R = new tvector<nr_type_t> (N);

  for (c = 0; c < N; c++) {
    // compute column norms and save in work array
    nPvt[c] = euclidian_c (c);
    cMap[c] = c; // initialize permutation vector
  }

  for (c = 0; c < N; c++) {

    // put column with largest norm into pivot position
    MaxPivot = nPvt[c]; pivot = c;
    for (r = c + 1; r < N; r++) {
      if ((s = nPvt[r]) > MaxPivot) {
        pivot = r;
        MaxPivot = s;
      }
    }
    if (pivot != c) {
      A->exchangeCols (pivot, c);
      Swap (int, cMap[pivot], cMap[c]);
      Swap (nr_double_t, nPvt[pivot], nPvt[c]);
    }

    // compute householder vector
    if (c < N) {
      nr_type_t a, b;
      s = euclidian_c (c, c + 1);
      a = A_(c, c);
      b = -sign (a) * xhypot (a, s); // Wj
      t = xhypot (s, a - b);         // || Vi - Wi ||
      R_(c) = b;
      // householder vector entries Ui
      A_(c, c) = (a - b) / t;
      for (r = c + 1; r < N; r++) A_(r, c) /= t;
    }
    else {
      R_(c) = A_(c, c);
    }

    // apply householder transformation to remaining columns
    for (r = c + 1; r < N; r++) {
      for (f = 0, k = c; k < N; k++) f += cond_conj (A_(k, c)) * A_(k, r);
      for (k = c; k < N; k++) A_(k, r) -= 2.0 * f * A_(k, c);
    }

    // update norms of remaining columns too
    for (r = c + 1; r < N; r++) {
      nPvt[r] = euclidian_c (r, c + 1);
    }
  }
}

template <class nr_type_t>
void eqnsys<nr_type_t>::factorize_qr_householder (void) {
  int c, r, pivot;
  nr_double_t s, MaxPivot;

  if (T) delete T; T = new tvector<nr_type_t> (N);

  for (c = 0; c < N; c++) {
    // compute column norms and save in work array
    nPvt[c] = euclidian_c (c);
    cMap[c] = c; // initialize permutation vector
  }

  for (c = 0; c < N; c++) {

    // put column with largest norm into pivot position
    MaxPivot = nPvt[c]; pivot = c;
    for (r = c + 1; r < N; r++)
      if ((s = nPvt[r]) > MaxPivot) {
        pivot = r;
        MaxPivot = s;
      }
    if (pivot != c) {
      A->exchangeCols (pivot, c);
      Swap (int, cMap[pivot], cMap[c]);
      Swap (nr_double_t, nPvt[pivot], nPvt[c]);
    }

    // compute and apply householder vector
    T_(c) = householder_left (c);

    // update norms of remaining columns too
    for (r = c + 1; r < N; r++) {
      if ((s = nPvt[r]) > 0) {
        nr_double_t y = 0;
        nr_double_t t = norm (A_(c, r) / s);
        if (t < 1)
          y = s * sqrt (1 - t);
        if (fabs (y / s) < NR_TINY)
          nPvt[r] = euclidian_c (r, c + 1);
        else
          nPvt[r] = y;
      }
    }
  }
}

template <class nr_type_t>
void eqnsys<nr_type_t>::substitute_qrh (void) {
  int c, r;
  nr_type_t f;

  // form the new right hand side Q'B
  for (c = 0; c < N - 1; c++) {
    // scalar product u_k^T * B
    for (f = 0, r = c; r < N; r++) f += cond_conj (A_(r, c)) * B_(r);
    // z - 2 * f * u_k
    for (r = c; r < N; r++) B_(r) -= 2.0 * f * A_(r, c);
  }

  // backward substitution in order to solve RX = Q'B
  for (r = N - 1; r >= 0; r--) {
    f = B_(r);
    for (c = r + 1; c < N; c++) f -= A_(r, c) * X_(cMap[c]);
    if (abs (R_(r)) > NR_EPSI)
      X_(cMap[r]) = f / R_(r);
    else
      X_(cMap[r]) = 0;
  }
}

template <class nr_type_t>
void eqnsys<nr_type_t>::substitute_qr_householder (void) {
  int c, r;
  nr_type_t f;

  // form the new right hand side Q'B
  for (c = 0; c < N; c++) {
    if (T_(c) != 0) {
      // scalar product u' * B
      for (f = B_(c), r = c + 1; r < N; r++) f += cond_conj (A_(r, c)) * B_(r);
      // z - T * f * u
      f *= cond_conj (T_(c)); B_(c) -= f;
      for (r = c + 1; r < N; r++) B_(r) -= f * A_(r, c);
    }
  }

  // backward substitution in order to solve RX = Q'B
  for (r = N - 1; r >= 0; r--) {
    for (f = B_(r), c = r + 1; c < N; c++) f -= A_(r, c) * X_(cMap[c]);
    if (abs (A_(r, r)) > NR_EPSI)
      X_(cMap[r]) = f / A_(r, r);
    else
      X_(cMap[r]) = 0;
  }
}

template <class nr_type_t>
void eqnsys<nr_type_t>::substitute_qr_householder_ls (void) {
  int c, r;
  nr_type_t f;

  // forward substitution in order to solve R'X = B
  for (r = 0; r < N; r++) {
    for (f = B_(r), c = 0; c < r; c++) f -= A_(c, r) * B_(c);
    if (abs (A_(r, r)) > NR_EPSI)
      B_(r) = f / A_(r, r);
    else
      B_(r) = 0;
  }

  // compute the least square solution QX
  for (c = N - 1; c >= 0; c--) {
    if (T_(c) != 0) {
      // scalar product u' * B
      for (f = B_(c), r = c + 1; r < N; r++) f += cond_conj (A_(r, c)) * B_(r);
      // z - T * f * u_k
      f *= T_(c); B_(c) -= f;
      for (r = c + 1; r < N; r++) B_(r) -= f * A_(r, c);
    }
  }

  // permute solution vector
  for (r = 0; r < N; r++) X_(cMap[r]) = B_(r);
}

#define sign_(a) (real (a) < 0 ? -1 : 1)

template <class nr_type_t>
nr_type_t eqnsys<nr_type_t>::householder_create_left (int c) {
  nr_type_t a, b, t;
  nr_double_t s, g;
  s = euclidian_c (c, c + 1);
  if (s == 0 && imag (A_(c, c)) == 0) {
    // no reflection necessary
    t = 0;
  }
  else {
    // calculate householder reflection
    a = A_(c, c);
    g = sign_(a) * xhypot (a, s);
    b = a + g;
    t = b / g;
    // store householder vector
    for (int r = c + 1; r < N; r++) A_(r, c) /= b;
    A_(c, c) = -g;
  }
  return t;
}

template <class nr_type_t>
nr_type_t eqnsys<nr_type_t>::householder_left (int c) {
  // compute householder vector
  nr_type_t t = householder_create_left (c);
  // apply householder transformation to remaining columns if necessary
  if (t != 0) {
    householder_apply_left (c, t);
  }
  return t;
}

template <class nr_type_t>
nr_type_t eqnsys<nr_type_t>::householder_right (int r) {
  // compute householder vector
  nr_type_t t = householder_create_right (r);
  // apply householder transformation to remaining rows if necessary
  if (t != 0) {
    householder_apply_right (r, t);
  }
  return t;
}

template <class nr_type_t>
nr_type_t eqnsys<nr_type_t>::householder_create_right (int r) {
  nr_type_t a, b, t;
  nr_double_t s, g;
  s = euclidian_r (r, r + 2);
  if (s == 0 && imag (A_(r, r + 1)) == 0) {
    // no reflection necessary
    t = 0;
  }
  else {
    // calculate householder reflection
    a = A_(r, r + 1);
    g = sign_(a) * xhypot (a, s);
    b = a + g;
    t = b / g;
    // store householder vector
    for (int c = r + 2; c < N; c++) A_(r, c) /= b;
    A_(r, r + 1) = -g;
  }
  return t;
}

template <class nr_type_t>
void eqnsys<nr_type_t>::householder_apply_left (int c, nr_type_t t) {
  nr_type_t f;
  int k, r;
  // apply the householder vector to each right-hand column
  for (r = c + 1; r < N; r++) {
    // calculate f = u' * A (a scalar product)
    f = A_(c, r);
    for (k = c + 1; k < N; k++) f += cond_conj (A_(k, c)) * A_(k, r);
    // calculate A -= T * f * u
    f *= cond_conj (t); A_(c, r) -= f;
    for (k = c + 1; k < N; k++) A_(k, r) -= f * A_(k, c);
  }
}

template <class nr_type_t>
void eqnsys<nr_type_t>::householder_apply_right (int r, nr_type_t t) {
  nr_type_t f;
  int k, c;
  // apply the householder vector to each downward row
  for (c = r + 1; c < N; c++) {
    // calculate f = u' * A (a scalar product)
    f = A_(c, r + 1);
    for (k = r + 2; k < N; k++) f += cond_conj (A_(r, k)) * A_(c, k);
    // calculate A -= T * f * u
    f *= cond_conj (t); A_(c, r + 1) -= f;
    for (k = r + 2; k < N; k++) A_(c, k) -= f * A_(r, k);
  }
}

// Some helper defines for SVD.
#define V_ (*V)
#define S_ (*S)
#define E_ (*E)
#define U_ (*A)

template <class nr_type_t>
void eqnsys<nr_type_t>::householder_apply_right_extern (int r, nr_type_t t) {
  nr_type_t f;
  int k, c;
  // apply the householder vector to each downward row
  for (c = r + 1; c < N; c++) {
    // calculate f = u' * A (a scalar product)
    f = V_(c, r + 1);
    for (k = r + 2; k < N; k++) f += cond_conj (A_(r, k)) * V_(c, k);
    // calculate A -= T * f * u
    f *= cond_conj (t); V_(c, r + 1) -= f;
    for (k = r + 2; k < N; k++) V_(c, k) -= f * A_(r, k);
  }
}

template <class nr_type_t>
void eqnsys<nr_type_t>::solve_svd (void) {
  factorize_svd ();
  chop_svd ();
  substitute_svd ();
}

template <class nr_type_t>
void eqnsys<nr_type_t>::chop_svd (void) {
  int c;
  nr_double_t Max, Min;
  Max = 0.0;
  for (c = 0; c < N; c++) if (fabs (S_(c)) > Max) Max = fabs (S_(c));
  Min = Max * NR_EPSI;
  for (c = 0; c < N; c++) if (fabs (S_(c)) < Min) S_(c) = 0.0;
}

template <class nr_type_t>
void eqnsys<nr_type_t>::substitute_svd (void) {
  int c, r;
  nr_type_t f;
  // calculate U'B
  for (c = 0; c < N; c++) {
    f = 0.0;
    // non-zero result only if S is non-zero
    if (S_(c) != 0.0) {
      for (r = 0; r < N; r++) f += cond_conj (U_(r, c)) * B_(r);
      // this is the divide by S
      f /= S_(c);
    }
    R_(c) = f;
  }
  // matrix multiply by V to get the final solution
  for (r = 0; r < N; r++) {
    for (f = 0.0, c = 0; c < N; c++) f += cond_conj (V_(c, r)) * R_(c);
    X_(r) = f;
  }
}

template <class nr_type_t>
void eqnsys<nr_type_t>::factorize_svd (void) {
  int i, j, l;
  nr_type_t t;

  // allocate space for vectors and matrices
  if (R) delete R; R = new tvector<nr_type_t> (N);
  if (T) delete T; T = new tvector<nr_type_t> (N);
  if (V) delete V; V = new tmatrix<nr_type_t> (N);
  if (S) delete S; S = new tvector<nr_double_t> (N);
  if (E) delete E; E = new tvector<nr_double_t> (N);

  // bidiagonalization through householder transformations
  for (i = 0; i < N; i++) {
    T_(i) = householder_left (i);
    if (i < N - 1) R_(i) = householder_right (i);
  }

  // copy over the real valued bidiagonal values
  for (i = 0; i < N; i++) S_(i) = real (A_(i, i));
  for (E_(0) = 0, i = 1; i < N; i++) E_(i) = real (A_(i - 1, i));

  // backward accumulation of right-hand householder transformations
  // yields the V' matrix
  for (l = N, i = N - 1; i >= 0; l = i--) {
    if (i < N - 1) {
      if ((t = R_(i)) != 0.0) {
        householder_apply_right_extern (i, cond_conj (t));
      }
      else for (j = l; j < N; j++) // cleanup this row
        V_(i, j) = V_(j, i) = 0.0;
    }
    V_(i, i) = 1.0;
  }

  // backward accumulation of left-hand householder transformations
  // yields the U matrix in place of the A matrix
  for (l = N, i = N - 1; i >= 0; l = i--) {
    for (j = l; j < N; j++) // cleanup upper row
      A_(i, j) = 0.0;
    if ((t = T_(i)) != 0.0) {
      householder_apply_left (i, cond_conj (t));
      for (j = l; j < N; j++) A_(j, i) *= -t;
    }
    else for (j = l; j < N; j++) // cleanup this column
      A_(j, i) = 0.0;
    A_(i, i) = 1.0 - t;
  }

  // S and E contain diagonal and super-diagonal, A contains U, V'
  // calculated; now diagonalization can begin
  diagonalize_svd ();
}

#ifndef MAX
# define MAX(x,y) (((x) > (y)) ? (x) : (y))
#endif

static nr_double_t
givens (nr_double_t a, nr_double_t b, nr_double_t& c, nr_double_t& s) {
  nr_double_t z = xhypot (a, b);
  c = a / z;
  s = b / z;
  return z;
}

template <class nr_type_t>
void eqnsys<nr_type_t>::givens_apply_u (int c1, int c2,
                                        nr_double_t c, nr_double_t s) {
  for (int i = 0; i < N; i++) {
    nr_type_t y = U_(i, c1);
    nr_type_t z = U_(i, c2);
    U_(i, c1) = y * c + z * s;
    U_(i, c2) = z * c - y * s;
  }
}

template <class nr_type_t>
void eqnsys<nr_type_t>::givens_apply_v (int r1, int r2,
                                        nr_double_t c, nr_double_t s) {
  for (int i = 0; i < N; i++) {
    nr_type_t y = V_(r1, i);
    nr_type_t z = V_(r2, i);
    V_(r1, i) = y * c + z * s;
    V_(r2, i) = z * c - y * s;
  }
}

template <class nr_type_t>
void eqnsys<nr_type_t>::diagonalize_svd (void) {
  bool split;
  int i, l, j, its, k, n, MaxIters = 30;
  nr_double_t an, f, g, h, d, c, s, b, a;

  // find largest bidiagonal value
  for (an = 0, i = 0; i < N; i++)
    an = MAX (an, fabs (S_(i)) + fabs (E_(i)));

  // diagonalize the bidiagonal matrix (stored as super-diagonal
  // vector E and diagonal vector S)
  for (k = N - 1; k >= 0; k--) {
    // loop over singular values
    for (its = 0; its <= MaxIters; its++) {
      split = true;
      // check for a zero entry along the super-diagonal E, if there
      // is one, it is possible to QR iterate on two separate matrices
      // above and below it
      for (n = 0, l = k; l >= 1; l--) {
        // note that E_(0) is always zero
        n = l - 1;
        if (fabs (E_(l)) + an == an) { split = false; break; }
        if (fabs (S_(n)) + an == an) break;
      }
      // if there is a zero on the diagonal S, it is possible to zero
      // out the corresponding super-diagonal E entry to its right
      if (split) {
        // cancellation of E_(l), if l > 0
        c = 0.0;
        s = 1.0;
        for (i = l; i <= k; i++) {
          f = -s * E_(i);
          E_(i) *= c;
          if (fabs (f) + an == an) break;
          g = S_(i);
          S_(i) = givens (f, g, c, s);
          // apply inverse rotation to U
          givens_apply_u (n, i, c, s);
        }
      }

      d = S_(k);
      // convergence
      if (l == k) {
        // singular value is made non-negative
        if (d < 0.0) {
          S_(k) = -d;
          for (j = 0; j < N; j++) V_(k, j) = -V_(k, j);
        }
        break;
      }
      if (its == MaxIters) {
        logprint (LOG_ERROR, "WARNING: no convergence in %d SVD iterations\n",
                  MaxIters);
      }

      // shift from bottom 2-by-2 minor
      a = S_(l);
      n = k - 1;
      b = S_(n);
      g = E_(n);
      h = E_(k);

      // compute QR shift value (as close as possible to the largest
      // eigenvalue of the 2-by-2 minor matrix)
      f  = (b - d) * (b + d) + (g - h) * (g + h);
      f /= 2.0 * h * b;
      f += sign_(f) * xhypot (f, 1.0);
      f  = ((a - d) * (a + d) + h * (b / f - h)) / a;
      // f => (B_{ll}^2 - u) / B_{ll}
      // u => eigenvalue of T = B' * B nearer T_{22} (Wilkinson shift)

      // next QR transformation
      c = s = 1.0;
      for (j = l; j <= n; j++) {
        i = j + 1;
        g = E_(i);
        b = S_(i);
        h = s * g; // h => right-hand non-zero to annihilate
        g *= c;
        E_(j) = givens (f, h, c, s);
        // perform the rotation
        f = a * c + g * s;
        g = g * c - a * s;
        h = b * s;
        b *= c;
        // here: +-   -+
        //       | f g | = B * V'_j (also first V'_1)
        //       | h b |
        //       +-   -+

        // accumulate the rotation in V'
        givens_apply_v (j, i, c, s);
        d = S_(j) = xhypot (f, h);
        // rotation can be arbitrary if d = 0
        if (d != 0.0) {
          // d => non-zero result on diagonal
          d = 1.0 / d;
          // rotation coefficients to annihilate the lower non-zero
          c = f * d;
          s = h * d;
        }
        f = c * g + s * b;
        a = c * b - s * g;
        // here: +-   -+
        //       | d f | => U_j * B
        //       | 0 a |
        //       +-   -+

        // accumulate rotation into U
        givens_apply_u (j, i, c, s);
      }
      E_(l) = 0;
      E_(k) = f;
      S_(k) = a;
    }
  }
}

} // namespace qucs

#undef V_