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svdstep.c

/* linalg/svdstep.c 
 *
 * Copyright (C) 2007, 2010 Brian Gough
 * 
 * This program 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.
 * 
 * This program 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 program; if not, write to the Free Software
 * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
 */

static void
chop_small_elements (gsl_vector * d, gsl_vector * f)
{
  const size_t N = d->size;
  double d_i = gsl_vector_get (d, 0);

  size_t i;

  for (i = 0; i < N - 1; i++)
    {
      double f_i = gsl_vector_get (f, i);
      double d_ip1 = gsl_vector_get (d, i + 1);

      if (fabs (f_i) < GSL_DBL_EPSILON * (fabs (d_i) + fabs (d_ip1)))
        {
          gsl_vector_set (f, i, 0.0);
        }

      d_i = d_ip1;
    }

}

static double
trailing_eigenvalue (const gsl_vector * d, const gsl_vector * f)
{
  const size_t n = d->size;

  double da = gsl_vector_get (d, n - 2);
  double db = gsl_vector_get (d, n - 1);
  double fa = (n > 2) ? gsl_vector_get (f, n - 3) : 0.0;
  double fb = gsl_vector_get (f, n - 2);

  double mu;

#if GOLUB_VAN_LOAN_8_3_2

  /* Golub and van Loan, Algorithm 8.3.2 
     The full SVD algorithm is described in section 8.6.2 */

  double ta = da * da + fa * fa;
  double tb = db * db + fb * fb;
  double tab = da * fb;

  double dt = (ta - tb) / 2.0;


  if (dt >= 0)
    {
      mu = tb - (tab * tab) / (dt + hypot (dt, tab));
    }
  else 
    {
      mu = tb + (tab * tab) / ((-dt) + hypot (dt, tab));
    }

#else
  {
    /* We can compute mu more accurately than using the formula above
       since we know the roots cannot be negative.  This also avoids
       the possibility of NaNs in the formula above.

       The matrix is [ da^2 + fa^2,  da fb      ;
                       da fb      , db^2 + fb^2 ]
       and mu is the eigenvalue closest to the bottom right element.
    */
    
    double ta = da * da + fa * fa;
    double tb = db * db + fb * fb;
    double tab = da * fb;
    
    double dt = (ta - tb) / 2.0;
    
    double S = ta + tb;
    double da2 = da * da, db2 = db * db;
    double fa2 = fa * fa, fb2 = fb * fb;
    double P = (da2 * db2) + (fa2 * db2) + (fa2 * fb2);
    double D = hypot(dt, tab);
    double r1 = S/2 + D;
    
    if (dt >= 0)
      {
        /* tb < ta, choose smaller root */
        mu = (r1 > 0) ?  P / r1 : 0.0;
      }
    else 
      {
        /* tb > ta, choose larger root */
        mu = r1;
      }
  }
    
#endif

  return mu;
}

static void
create_schur (double d0, double f0, double d1, double * c, double * s)
{
  double apq = 2.0 * d0 * f0;

  if (d0 == 0 || f0 == 0)
    {
      *c = 1.0;
      *s = 0.0;
      return;
    }

  /* Check if we need to rescale to avoid underflow/overflow */
  if (fabs(d0) < GSL_SQRT_DBL_MIN || fabs(d0) > GSL_SQRT_DBL_MAX
      || fabs(f0) < GSL_SQRT_DBL_MIN || fabs(f0) > GSL_SQRT_DBL_MAX
      || fabs(d1) < GSL_SQRT_DBL_MIN || fabs(d1) > GSL_SQRT_DBL_MAX)
    {
      double scale;
      int d0_exp, f0_exp;
      frexp(d0, &d0_exp);
      frexp(f0, &f0_exp);
      /* Bring |d0*f0| into the range GSL_DBL_MIN to GSL_DBL_MAX */
      scale = ldexp(1.0, -(d0_exp + f0_exp)/4);
      d0 *= scale;
      f0 *= scale;
      d1 *= scale;
      apq = 2.0 * d0 * f0;
    }

  if (apq != 0.0)
    {
      double t;
      double tau = (f0*f0 + (d1 + d0)*(d1 - d0)) / apq;
      
      if (tau >= 0.0)
        {
          t = 1.0/(tau + hypot(1.0, tau));
        }
      else
        {
          t = -1.0/(-tau + hypot(1.0, tau));
        }

      *c = 1.0 / hypot(1.0, t);
      *s = t * (*c);
    }
  else
    {
      *c = 1.0;
      *s = 0.0;
    }
}

static void
svd2 (gsl_vector * d, gsl_vector * f, gsl_matrix * U, gsl_matrix * V)
{
  size_t i;
  double c, s, a11, a12, a21, a22;

  const size_t M = U->size1;
  const size_t N = V->size1;

  double d0 = gsl_vector_get (d, 0);
  double f0 = gsl_vector_get (f, 0);
  
  double d1 = gsl_vector_get (d, 1);

  if (d0 == 0.0)
    {
      /* Eliminate off-diagonal element in [0,f0;0,d1] to make [d,0;0,0] */

      create_givens (f0, d1, &c, &s);

      /* compute B <= G^T B X,  where X = [0,1;1,0] */

      gsl_vector_set (d, 0, c * f0 - s * d1);
      gsl_vector_set (f, 0, s * f0 + c * d1);
      gsl_vector_set (d, 1, 0.0);

      /* Compute U <= U G */

      for (i = 0; i < M; i++)
        {
          double Uip = gsl_matrix_get (U, i, 0);
          double Uiq = gsl_matrix_get (U, i, 1);
          gsl_matrix_set (U, i, 0, c * Uip - s * Uiq);
          gsl_matrix_set (U, i, 1, s * Uip + c * Uiq);
        }

      /* Compute V <= V X */

      gsl_matrix_swap_columns (V, 0, 1);

      return;
    }
  else if (d1 == 0.0)
    {
      /* Eliminate off-diagonal element in [d0,f0;0,0] */

      create_givens (d0, f0, &c, &s);

      /* compute B <= B G */

      gsl_vector_set (d, 0, d0 * c - f0 * s);
      gsl_vector_set (f, 0, 0.0);

      /* Compute V <= V G */

      for (i = 0; i < N; i++)
        {
          double Vip = gsl_matrix_get (V, i, 0);
          double Viq = gsl_matrix_get (V, i, 1);
          gsl_matrix_set (V, i, 0, c * Vip - s * Viq);
          gsl_matrix_set (V, i, 1, s * Vip + c * Viq);
        }

      return;
    }
  else
    {
      /* Make columns orthogonal, A = [d0, f0; 0, d1] * G */

      create_schur (d0, f0, d1, &c, &s);

      /* compute B <= B G */
      
      a11 = c * d0 - s * f0;
      a21 = - s * d1;
      
      a12 = s * d0 + c * f0;
      a22 = c * d1;
      
      /* Compute V <= V G */
      
      for (i = 0; i < N; i++)
        {
          double Vip = gsl_matrix_get (V, i, 0);
          double Viq = gsl_matrix_get (V, i, 1);
          gsl_matrix_set (V, i, 0, c * Vip - s * Viq);
          gsl_matrix_set (V, i, 1, s * Vip + c * Viq);
        }
      
      /* Eliminate off-diagonal elements, bring column with largest
         norm to first column */
      
      if (hypot(a11, a21) < hypot(a12,a22))
        {
          double t1, t2;

          /* B <= B X */

          t1 = a11; a11 = a12; a12 = t1;
          t2 = a21; a21 = a22; a22 = t2;

          /* V <= V X */

          gsl_matrix_swap_columns(V, 0, 1);
        } 

      create_givens (a11, a21, &c, &s);
      
      /* compute B <= G^T B */
      
      gsl_vector_set (d, 0, c * a11 - s * a21);
      gsl_vector_set (f, 0, c * a12 - s * a22);
      gsl_vector_set (d, 1, s * a12 + c * a22);
      
      /* Compute U <= U G */
      
      for (i = 0; i < M; i++)
        {
          double Uip = gsl_matrix_get (U, i, 0);
          double Uiq = gsl_matrix_get (U, i, 1);
          gsl_matrix_set (U, i, 0, c * Uip - s * Uiq);
          gsl_matrix_set (U, i, 1, s * Uip + c * Uiq);
        }

      return;
    }
}


static void
chase_out_intermediate_zero (gsl_vector * d, gsl_vector * f, gsl_matrix * U, size_t k0)
{
#if !USE_BLAS
  const size_t M = U->size1;
#endif
  const size_t n = d->size;
  double c, s;
  double x, y;
  size_t k;

  x = gsl_vector_get (f, k0);
  y = gsl_vector_get (d, k0+1);

  for (k = k0; k < n - 1; k++)
    {
      create_givens (y, -x, &c, &s);
      
      /* Compute U <= U G */

#ifdef USE_BLAS
      {
        gsl_vector_view Uk0 = gsl_matrix_column(U,k0);
        gsl_vector_view Ukp1 = gsl_matrix_column(U,k+1);
        gsl_blas_drot(&Uk0.vector, &Ukp1.vector, c, -s);
      }
#else
      {
        size_t i;

        for (i = 0; i < M; i++)
          {
            double Uip = gsl_matrix_get (U, i, k0);
            double Uiq = gsl_matrix_get (U, i, k + 1);
            gsl_matrix_set (U, i, k0, c * Uip - s * Uiq);
            gsl_matrix_set (U, i, k + 1, s * Uip + c * Uiq);
          }
      }
#endif
      
      /* compute B <= G^T B */
      
      gsl_vector_set (d, k + 1, s * x + c * y);

      if (k == k0)
        gsl_vector_set (f, k, c * x - s * y );

      if (k < n - 2) 
        {
          double z = gsl_vector_get (f, k + 1);
          gsl_vector_set (f, k + 1, c * z); 

          x = -s * z ;
          y = gsl_vector_get (d, k + 2); 
        }
    }
}

static void
chase_out_trailing_zero (gsl_vector * d, gsl_vector * f, gsl_matrix * V)
{
#if !USE_BLAS
  const size_t N = V->size1;
#endif
  const size_t n = d->size;
  double c, s;
  double x, y;
  size_t k;

  x = gsl_vector_get (d, n - 2);
  y = gsl_vector_get (f, n - 2);

  for (k = n - 1; k-- > 0;)
    {
      create_givens (x, y, &c, &s);

      /* Compute V <= V G where G = [c, s ; -s, c] */

#ifdef USE_BLAS
      {
        gsl_vector_view Vp = gsl_matrix_column(V,k);
        gsl_vector_view Vq = gsl_matrix_column(V,n-1);
        gsl_blas_drot(&Vp.vector, &Vq.vector, c, -s);
      }
#else
      {
        size_t i;
   
        for (i = 0; i < N; i++)
          {
            double Vip = gsl_matrix_get (V, i, k);
            double Viq = gsl_matrix_get (V, i, n - 1);
            gsl_matrix_set (V, i, k, c * Vip - s * Viq);
            gsl_matrix_set (V, i, n - 1, s * Vip + c * Viq);
          }
      }
#endif

      /* compute B <= B G */
      
      gsl_vector_set (d, k, c * x - s * y);

      if (k == n - 2)
        gsl_vector_set (f, k, s * x + c * y );

      if (k > 0) 
        {
          double z = gsl_vector_get (f, k - 1);
          gsl_vector_set (f, k - 1, c * z); 

          x = gsl_vector_get (d, k - 1); 
          y = s * z ;
        }
    }
}

static void
qrstep (gsl_vector * d, gsl_vector * f, gsl_matrix * U, gsl_matrix * V)
{
#if !USE_BLAS
  const size_t M = U->size1;
  const size_t N = V->size1;
#endif
  const size_t n = d->size;
  double y, z;
  double ak, bk, zk, ap, bp, aq, bq;
  size_t i, k;

  if (n == 1)
    return;  /* shouldn't happen */

  /* Compute 2x2 svd directly */

  if (n == 2)
    {
      svd2 (d, f, U, V);
      return;
    }

  /* Chase out any zeroes on the diagonal */

  for (i = 0; i < n - 1; i++)
    {
      double d_i = gsl_vector_get (d, i);
      
      if (d_i == 0.0)
        {
          chase_out_intermediate_zero (d, f, U, i);
          return;
        }
    }

  /* Chase out any zero at the end of the diagonal */

  {
    double d_nm1 = gsl_vector_get (d, n - 1);

    if (d_nm1 == 0.0) 
      {
        chase_out_trailing_zero (d, f, V);
        return;
      }
  }


  /* Apply QR reduction steps to the diagonal and offdiagonal */

  {
    double d0 = gsl_vector_get (d, 0);
    double f0 = gsl_vector_get (f, 0);
    
    double d1 = gsl_vector_get (d, 1);
    double f1 = gsl_vector_get (f, 1);
    
    {
      double mu = trailing_eigenvalue (d, f);
    
      y = d0 * d0 - mu;
      z = d0 * f0;
    }
    
    /* Set up the recurrence for Givens rotations on a bidiagonal matrix */
    
    ak = 0;
    bk = 0;
    
    ap = d0;
    bp = f0;
    
    aq = d1;
    bq = f1;
  }

  for (k = 0; k < n - 1; k++)
    {
      double c, s;
      create_givens (y, z, &c, &s);

      /* Compute V <= V G */

#ifdef USE_BLAS
      {
        gsl_vector_view Vk = gsl_matrix_column(V,k);
        gsl_vector_view Vkp1 = gsl_matrix_column(V,k+1);
        gsl_blas_drot(&Vk.vector, &Vkp1.vector, c, -s);
      }
#else
      for (i = 0; i < N; i++)
        {
          double Vip = gsl_matrix_get (V, i, k);
          double Viq = gsl_matrix_get (V, i, k + 1);
          gsl_matrix_set (V, i, k, c * Vip - s * Viq);
          gsl_matrix_set (V, i, k + 1, s * Vip + c * Viq);
        }
#endif

      /* compute B <= B G */

      {
        double bk1 = c * bk - s * z;

        double ap1 = c * ap - s * bp;
        double bp1 = s * ap + c * bp;
        double zp1 = -s * aq;

        double aq1 = c * aq;

        if (k > 0)
          {
            gsl_vector_set (f, k - 1, bk1);
          }

        ak = ap1;
        bk = bp1;
        zk = zp1;

        ap = aq1;

        if (k < n - 2)
          {
            bp = gsl_vector_get (f, k + 1);
          }
        else
          {
            bp = 0.0;
          }

        y = ak;
        z = zk;
      }

      create_givens (y, z, &c, &s);

      /* Compute U <= U G */

#ifdef USE_BLAS
      {
        gsl_vector_view Uk = gsl_matrix_column(U,k);
        gsl_vector_view Ukp1 = gsl_matrix_column(U,k+1);
        gsl_blas_drot(&Uk.vector, &Ukp1.vector, c, -s);
      }
#else
      for (i = 0; i < M; i++)
        {
          double Uip = gsl_matrix_get (U, i, k);
          double Uiq = gsl_matrix_get (U, i, k + 1);
          gsl_matrix_set (U, i, k, c * Uip - s * Uiq);
          gsl_matrix_set (U, i, k + 1, s * Uip + c * Uiq);
        }
#endif

      /* compute B <= G^T B */

      {
        double ak1 = c * ak - s * zk;
        double bk1 = c * bk - s * ap;
        double zk1 = -s * bp;

        double ap1 = s * bk + c * ap;
        double bp1 = c * bp;

        gsl_vector_set (d, k, ak1);

        ak = ak1;
        bk = bk1;
        zk = zk1;

        ap = ap1;
        bp = bp1;

        if (k < n - 2)
          {
            aq = gsl_vector_get (d, k + 2);
          }
        else
          {
            aq = 0.0;
          }

        y = bk;
        z = zk;
      }
    }

  gsl_vector_set (f, n - 2, bk);
  gsl_vector_set (d, n - 1, ap);
}



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