/* linalg/cod.c

 * 

 * Copyright (C) 2016, 2017 Patrick Alken

 * 

 * 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.

 */

#include <config.h>

#include <stdlib.h>

#include <string.h>

#include <gsl/gsl_math.h>

#include <gsl/gsl_vector.h>

#include <gsl/gsl_matrix.h>

#include <gsl/gsl_permute_vector.h>

#include <gsl/gsl_blas.h>

#include <gsl/gsl_linalg.h>

/*

 * This module contains routines for factoring an M-by-N matrix A as:

 *

 * A P = Q R Z^T

 *

 * known as the Complete Orthogonal Decomposition, where:

 *

 * P is a N-by-N permutation matrix

 * Q is M-by-M orthogonal

 * R has an r-by-r upper triangular block

 * Z is N-by-N orthogonal

 *

 * When A is full rank, Z = I and this becomes the QR decomposition

 * with column pivoting. When A is rank deficient, then

 *

 * R = [ R11 0 ] where R11 is r-by-r and r = rank(A)

 *     [  0  0 ]

 */

static int cod_RZ(gsl_matrix * A, gsl_vector * tau);

static double cod_householder_transform(double *alpha, gsl_vector * v);

static int cod_householder_mh(const double tau, const gsl_vector * v,

                              gsl_matrix * A, gsl_vector * work);

static int cod_householder_hv(const double tau, const gsl_vector * v, gsl_vector * w);

static int cod_householder_Zvec(const gsl_matrix * QRZT, const gsl_vector * tau_Z, const size_t rank,

                                gsl_vector * v);

static int cod_trireg_solve(const gsl_matrix * R, const double lambda, const gsl_vector * b,

                            gsl_matrix * S, gsl_vector * x, gsl_vector * work);

int

gsl_linalg_COD_decomp_e(gsl_matrix * A, gsl_vector * tau_Q, gsl_vector * tau_Z,

                        gsl_permutation * p, double tol, size_t * rank, gsl_vector * work)

{

  const size_t M = A->size1;

  const size_t N = A->size2;

  if (tau_Q->size != GSL_MIN (M, N))

    {

      GSL_ERROR ("size of tau_Q must be MIN(M,N)", GSL_EBADLEN);

    }

  else if (tau_Z->size != GSL_MIN (M, N))

    {

      GSL_ERROR ("size of tau_Z must be MIN(M,N)", GSL_EBADLEN);

    }

  else if (p->size != N)

    {

      GSL_ERROR ("permutation size must be N", GSL_EBADLEN);

    }

  else if (work->size != N)

    {

      GSL_ERROR ("work size must be N", GSL_EBADLEN);

    }

  else

    {

      int status, signum;

      size_t r;

      /* decompose: A P = Q R */

      status = gsl_linalg_QRPT_decomp(A, tau_Q, p, &signum, work);

      if (status)

        return status;

      /* estimate rank of A */

      r = gsl_linalg_QRPT_rank(A, tol);

      if (r < N)

        {

          /*

           * matrix is rank-deficient, so that the R factor is

           *

           * R = [ R11 R12 ] =~ [ R11 R12 ]

           *     [  0  R22 ]    [  0   0  ]

           *

           * compute RZ decomposition of upper trapezoidal matrix

           * [ R11 R12 ] = [ R11~ 0 ] Z

           */

          gsl_matrix_view R_upper = gsl_matrix_submatrix(A, 0, 0, r, N);

          gsl_vector_view t = gsl_vector_subvector(tau_Z, 0, r);

          cod_RZ(&R_upper.matrix, &t.vector);

        }

      *rank = r;

      return GSL_SUCCESS;

    }

}

int

gsl_linalg_COD_decomp(gsl_matrix * A, gsl_vector * tau_Q, gsl_vector * tau_Z,

                      gsl_permutation * p, size_t * rank, gsl_vector * work)

{

  return gsl_linalg_COD_decomp_e(A, tau_Q, tau_Z, p, -1.0, rank, work);

}

/*

gsl_linalg_COD_lssolve()

  Find the least squares solution to the overdetermined system 

min ||b - A x||^2

for M >= N using the COD factorization A P = Q R Z

Inputs: QRZT     - matrix A, in COD compressed format, M-by-N

        tau_Q    - Householder scalars for Q, length min(M,N)

        tau_Z    - Householder scalars for Z, length min(M,N)

        perm     - permutation matrix

        rank     - rank of A

        b        - rhs vector, length M

        x        - (output) solution vector, length N

        residual - (output) residual vector, b - A x, length M

*/

int

gsl_linalg_COD_lssolve (const gsl_matrix * QRZT, const gsl_vector * tau_Q, const gsl_vector * tau_Z,

                        const gsl_permutation * perm, const size_t rank, const gsl_vector * b,

                        gsl_vector * x, gsl_vector * residual)

{

  const size_t M = QRZT->size1;

  const size_t N = QRZT->size2;

  if (M < N)

    {

      GSL_ERROR ("QRZT matrix must have M>=N", GSL_EBADLEN);

    }

  else if (M != b->size)

    {

      GSL_ERROR ("matrix size must match b size", GSL_EBADLEN);

    }

  else if (rank > GSL_MIN (M, N))

    {

      GSL_ERROR ("rank must be <= MIN(M,N)", GSL_EBADLEN);

    }

  else if (N != x->size)

    {

      GSL_ERROR ("matrix size must match solution size", GSL_EBADLEN);

    }

  else if (M != residual->size)

    {

      GSL_ERROR ("matrix size must match residual size", GSL_EBADLEN);

    }

  else

    {

      gsl_matrix_const_view R11 = gsl_matrix_const_submatrix (QRZT, 0, 0, rank, rank);

      gsl_vector_view QTb1 = gsl_vector_subvector(residual, 0, rank);

      gsl_vector_view x1 = gsl_vector_subvector(x, 0, rank);

      gsl_vector_set_zero(x);

      /* compute residual = Q^T b = [ c1 ; c2 ] */

      gsl_vector_memcpy(residual, b);

      gsl_linalg_QR_QTvec (QRZT, tau_Q, residual);

      /* solve x1 := R11^{-1} (Q^T b)(1:r) */

      gsl_vector_memcpy(&(x1.vector), &(QTb1.vector));

      gsl_blas_dtrsv(CblasUpper, CblasNoTrans, CblasNonUnit, &(R11.matrix), &(x1.vector));

      /* compute Z ( R11^{-1} x1; 0 ) */

      cod_householder_Zvec(QRZT, tau_Z, rank, x);

      /* compute x = P Z^T ( R11^{-1} x1; 0 ) */

      gsl_permute_vector_inverse(perm, x);

      /* compute residual = b - A x = Q (Q^T b - R [ R11^{-1} x1; 0 ]) = Q [ 0 ; c2 ] */

      gsl_vector_set_zero(&(QTb1.vector));

      gsl_linalg_QR_Qvec(QRZT, tau_Q, residual);

      return GSL_SUCCESS;

    }

}

/*

gsl_linalg_COD_lssolve2()

  Find the least squares solution to the Tikhonov regularized

system in standard form:

min ||b - A x||^2 + lambda^2 ||x||^2

for M >= N using the COD factorization A P = Q R Z

Inputs: lambda   - parameter

        QRZT     - matrix A, in COD compressed format, M-by-N

        tau_Q    - Householder scalars for Q, length min(M,N)

        tau_Z    - Householder scalars for Z, length min(M,N)

        perm     - permutation matrix

        rank     - rank of A

        b        - rhs vector, length M

        x        - (output) solution vector, length N

        residual - (output) residual vector, b - A x, length M

        S        - workspace, rank-by-rank

        work     - workspace, length rank

*/

int

gsl_linalg_COD_lssolve2 (const double lambda, const gsl_matrix * QRZT, const gsl_vector * tau_Q, const gsl_vector * tau_Z,

                         const gsl_permutation * perm, const size_t rank, const gsl_vector * b,

                         gsl_vector * x, gsl_vector * residual, gsl_matrix * S, gsl_vector * work)

{

  const size_t M = QRZT->size1;

  const size_t N = QRZT->size2;

  if (M < N)

    {

      GSL_ERROR ("QRZT matrix must have M>=N", GSL_EBADLEN);

    }

  else if (M != b->size)

    {

      GSL_ERROR ("matrix size must match b size", GSL_EBADLEN);

    }

  else if (rank > GSL_MIN (M, N))

    {

      GSL_ERROR ("rank must be <= MIN(M,N)", GSL_EBADLEN);

    }

  else if (N != x->size)

    {

      GSL_ERROR ("matrix size must match solution size", GSL_EBADLEN);

    }

  else if (M != residual->size)

    {

      GSL_ERROR ("matrix size must match residual size", GSL_EBADLEN);

    }

  else if (S->size1 != rank || S->size2 != rank)

    {

      GSL_ERROR ("S must be rank-by-rank", GSL_EBADLEN);

    }

  else if (work->size != rank)

    {

      GSL_ERROR ("work must be length rank", GSL_EBADLEN);

    }

  else

    {

      gsl_matrix_const_view R11 = gsl_matrix_const_submatrix (QRZT, 0, 0, rank, rank);

      gsl_vector_view c1 = gsl_vector_subvector(residual, 0, rank);

      gsl_vector_view y1 = gsl_vector_subvector(x, 0, rank);

      gsl_vector_set_zero(x);

      /* compute residual = Q^T b = [ c1 ; c2 ]*/

      gsl_vector_memcpy(residual, b);

      gsl_linalg_QR_QTvec (QRZT, tau_Q, residual);

      /* solve [ R11 ; lambda*I ] y1 = [ (Q^T b)(1:r) ; 0 ] */

      cod_trireg_solve(&(R11.matrix), lambda, &(c1.vector), S, &(y1.vector), work);

      /* save y1 for later residual calculation */

      gsl_vector_memcpy(work, &(y1.vector));

      /* compute Z [ y1; 0 ] */

      cod_householder_Zvec(QRZT, tau_Z, rank, x);

      /* compute x = P Z^T ( y1; 0 ) */

      gsl_permute_vector_inverse(perm, x);

      /* compute residual = b - A x = Q (Q^T b - [ R11 y1; 0 ]) = Q [ c1 - R11*y1 ; c2 ] */

      /* work = R11*y1 */

      gsl_blas_dtrmv(CblasUpper, CblasNoTrans, CblasNonUnit, &(R11.matrix), work);

      gsl_vector_sub(&(c1.vector), work);

      gsl_linalg_QR_Qvec(QRZT, tau_Q, residual);

      return GSL_SUCCESS;

    }

}

/*

gsl_linalg_COD_unpack()

  Unpack encoded COD decomposition into the matrices Q,R,Z,P

Inputs: QRZT  - encoded COD decomposition

        tau_Q - Householder scalars for Q

        tau_Z - Householder scalars for Z

        rank  - rank of matrix (as determined from gsl_linalg_COD_decomp)

        Q     - (output) M-by-M matrix Q

        R     - (output) M-by-N matrix R

        Z     - (output) N-by-N matrix Z

*/

int

gsl_linalg_COD_unpack(const gsl_matrix * QRZT, const gsl_vector * tau_Q,

                      const gsl_vector * tau_Z, const size_t rank, gsl_matrix * Q,

                      gsl_matrix * R, gsl_matrix * Z)

{

  const size_t M = QRZT->size1;

  const size_t N = QRZT->size2;

  if (tau_Q->size != GSL_MIN (M, N))

    {

      GSL_ERROR ("size of tau_Q must be MIN(M,N)", GSL_EBADLEN);

    }

  else if (tau_Z->size != GSL_MIN (M, N))

    {

      GSL_ERROR ("size of tau_Z must be MIN(M,N)", GSL_EBADLEN);

    }

  else if (rank > GSL_MIN (M, N))

    {

      GSL_ERROR ("rank must be <= MIN(M,N)", GSL_EBADLEN);

    }

  else if (Q->size1 != M || Q->size2 != M)

    {

      GSL_ERROR ("Q must by M-by-M", GSL_EBADLEN);

    }

  else if (R->size1 != M || R->size2 != N)

    {

      GSL_ERROR ("R must by M-by-N", GSL_EBADLEN);

    }

  else if (Z->size1 != N || Z->size2 != N)

    {

      GSL_ERROR ("Z must by N-by-N", GSL_EBADLEN);

    }

  else

    {

      size_t i;

      gsl_matrix_view R11 = gsl_matrix_submatrix(R, 0, 0, rank, rank);

      gsl_matrix_const_view QRZT11 = gsl_matrix_const_submatrix(QRZT, 0, 0, rank, rank);

      /* form Q matrix */

      gsl_matrix_set_identity(Q);

      for (i = GSL_MIN (M, N); i-- > 0;)

        {

          gsl_vector_const_view h = gsl_matrix_const_subcolumn (QRZT, i, i, M - i);

          gsl_matrix_view m = gsl_matrix_submatrix (Q, i, i, M - i, M - i);

          double ti = gsl_vector_get (tau_Q, i);

          gsl_linalg_householder_hm (ti, &h.vector, &m.matrix);

        }

      /* form Z matrix */

      gsl_matrix_set_identity(Z);

      if (rank < N)

        {

          gsl_vector_view work = gsl_matrix_row(R, 0); /* temporary workspace, size N */

          /* multiply I by Z from the right */

          gsl_linalg_COD_matZ(QRZT, tau_Z, rank, Z, &work.vector);

        }

      /* copy rank-by-rank upper triangle of QRZT into R and zero the rest */

      gsl_matrix_set_zero(R);

      gsl_matrix_tricpy('U', 1, &R11.matrix, &QRZT11.matrix);

      return GSL_SUCCESS;

    }

}

/*

gsl_linalg_COD_matZ

  Multiply an M-by-N matrix A on the right by Z (N-by-N)

Inputs: QRZT  - encoded COD matrix

        tau_Z - Householder scalars for Z

        rank  - matrix rank

        A     - on input, M-by-N matrix

                on output, A * Z

        work  - workspace of length M

*/

int

gsl_linalg_COD_matZ(const gsl_matrix * QRZT, const gsl_vector * tau_Z, const size_t rank,

                    gsl_matrix * A, gsl_vector * work)

{

  const size_t M = A->size1;

  const size_t N = A->size2;

  if (tau_Z->size != GSL_MIN (QRZT->size1, QRZT->size2))

    {

      GSL_ERROR("tau_Z must be GSL_MIN(M,N)", GSL_EBADLEN);

    }

  else if (QRZT->size2 != N)

    {

      GSL_ERROR("QRZT must have N columns", GSL_EBADLEN);

    }

  else if (work->size != M)

    {

      GSL_ERROR("workspace must be length M", GSL_EBADLEN);

    }

  else

    {

      /* if rank == N, then Z = I and there is nothing to do */

      if (rank < N)

        {

          size_t i;

          for (i = rank; i > 0 && i--; )

            {

              gsl_vector_const_view h = gsl_matrix_const_subrow (QRZT, i, rank, N - rank);

              gsl_matrix_view m = gsl_matrix_submatrix (A, 0, i, M, N - i);

              double ti = gsl_vector_get (tau_Z, i);

              cod_householder_mh (ti, &h.vector, &m.matrix, work);

            }

        }

      return GSL_SUCCESS;

    }

}

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

 * INTERNAL ROUTINES                         *

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

/*

cod_RZ()

  Perform RZ decomposition of an upper trapezoidal matrix,

A = [ A11 A12 ] = [ R 0 ] Z

where A is M-by-N with N >= M, A11 is M-by-M upper triangular,

and A12 is M-by-(N-M). On output, Z is stored as Householder

reflectors in the A12 portion of A,

Z = Z(1) Z(2) ... Z(M)

Inputs: A   - M-by-N matrix with N >= M

              On input, upper trapezoidal matrix [ A11 A12 ]

              On output, A11 is overwritten by R (subdiagonal elements

              are not touched), and A12 is overwritten by Z in packed storage

        tau - (output) Householder scalars, size M

*/

static int

cod_RZ(gsl_matrix * A, gsl_vector * tau)

{

  const size_t M = A->size1;

  const size_t N = A->size2;

  if (tau->size != M)

    {

      GSL_ERROR("tau has wrong size", GSL_EBADLEN);

    }

  else if (N < M)

    {

      GSL_ERROR("N must be >= M", GSL_EINVAL);

    }

  else if (M == N)

    {

      /* quick return */

      gsl_vector_set_all(tau, 0.0);

      return GSL_SUCCESS;

    }

  else

    {

      size_t k;

      for (k = M; k > 0 && k--; )

        {

          double *alpha = gsl_matrix_ptr(A, k, k);

          gsl_vector_view z = gsl_matrix_subrow(A, k, M, N - M);

          double tauk;

          /* compute Householder reflection to zero [ A(k,k) A(k,M+1:N) ] */

          tauk = cod_householder_transform(alpha, &z.vector);

          gsl_vector_set(tau, k, tauk);

          if ((tauk != 0) && (k > 0))

            {

              gsl_vector_view w = gsl_vector_subvector(tau, 0, k);

              gsl_matrix_view B = gsl_matrix_submatrix(A, 0, k, k, N - k);

              cod_householder_mh(tauk, &z.vector, &B.matrix, &w.vector);

            }

        }

      return GSL_SUCCESS;

    }

}

static double

cod_householder_transform(double *alpha, gsl_vector * v)

{

  double beta, tau;

  double xnorm = gsl_blas_dnrm2(v);

  if (xnorm == 0)

    {

      return 0.0; /* tau = 0 */

    }

  beta = - (*alpha >= 0.0 ? +1.0 : -1.0) * gsl_hypot(*alpha, xnorm);

  tau = (beta - *alpha) / beta;

  {

    double s = (*alpha - beta);

    if (fabs(s) > GSL_DBL_MIN) 

      {

        gsl_blas_dscal (1.0 / s, v);

      }

    else

      {

        gsl_blas_dscal (GSL_DBL_EPSILON / s, v);

        gsl_blas_dscal (1.0 / GSL_DBL_EPSILON, v);

      }

    *alpha = beta;

  }

  return tau;

}

/*

cod_householder_hv

  Apply Householder reflection H = (I - tau*v*v') to vector v from the left,

w' = H * w

Inputs: tau  - Householder scalar

        v    - Householder vector, size M

        w    - on input, w vector, size M

               on output, H * w

Notes:

1) Based on LAPACK routine DLARZ

*/

static int

cod_householder_hv(const double tau, const gsl_vector * v, gsl_vector * w)

{

  if (tau == 0)

    {

      return GSL_SUCCESS; /* H = I */

    }

  else

    {

      const size_t M = w->size;

      const size_t L = v->size;

      double w0 = gsl_vector_get(w, 0);

      gsl_vector_view w1 = gsl_vector_subvector(w, M - L, L);

      double d1, d;

      /* d1 := v . w(M-L:M) */

      gsl_blas_ddot(v, &w1.vector, &d1;;

      /* d := w(1) + v . w(M-L:M) */

      d = w0 + d1;

      /* w(1) = w(1) - tau * d */

      gsl_vector_set(w, 0, w0 - tau * d);

      /* w(M-L:M) = w(M-L:M) - tau * d * v */

      gsl_blas_daxpy(-tau * d, v, &w1.vector);

      return GSL_SUCCESS;

    }

}

/*

cod_householder_mh

  Apply Householder reflection H = (I - tau*v*v') to matrix A from the right

Inputs: tau  - Householder scalar

        v    - Householder vector, size N-M

        A    - matrix, size M-by-N

        work - workspace, size M

Notes:

1) Based on LAPACK routine DLARZ

*/

static int

cod_householder_mh(const double tau, const gsl_vector * v, gsl_matrix * A,

                   gsl_vector * work)

{

  if (tau == 0)

    {

      return GSL_SUCCESS; /* H = I */

    }

  else

    {

      const size_t M = A->size1;

      const size_t N = A->size2;

      const size_t L = v->size;

      gsl_vector_view A1 = gsl_matrix_subcolumn(A, 0, 0, M);

      gsl_matrix_view C = gsl_matrix_submatrix(A, 0, N - L, M, L);

      /* work(1:M) = A(1:M,1) */

      gsl_vector_memcpy(work, &A1.vector);

      /* work(1:M) = work(1:M) + A(1:M,M+1:N) * v(1:N-M) */

      gsl_blas_dgemv(CblasNoTrans, 1.0, &C.matrix, v, 1.0, work);

      /* A(1:M,1) = A(1:M,1) - tau * work(1:M) */

      gsl_blas_daxpy(-tau, work, &A1.vector);

      /* A(1:M,M+1:N) = A(1:M,M+1:N) - tau * work(1:M) * v(1:N-M)' */

      gsl_blas_dger(-tau, work, v, &C.matrix);

      return GSL_SUCCESS;

    }

}

/*

cod_householder_Zvec

  Multiply a vector by Z

Inputs: QRZT  - encoded COD matrix

        tau_Z - Householder scalars for Z

        rank  - matrix rank

        v     - on input, vector of length N

                on output, Z^T * v

*/

static int

cod_householder_Zvec(const gsl_matrix * QRZT, const gsl_vector * tau_Z, const size_t rank,

                     gsl_vector * v)

{

  const size_t M = QRZT->size1;

  const size_t N = QRZT->size2;

  if (tau_Z->size != GSL_MIN (M, N))

    {

      GSL_ERROR("tau_Z must be GSL_MIN(M,N)", GSL_EBADLEN);

    }

  else if (v->size != N)

    {

      GSL_ERROR("v must be length N", GSL_EBADLEN);

    }

  else

    {

      if (rank < N)

        {

          size_t i;

          for (i = 0; i < rank; ++i)

            {

              gsl_vector_const_view h = gsl_matrix_const_subrow (QRZT, i, rank, N - rank);

              gsl_vector_view w = gsl_vector_subvector (v, i, N - i);

              double ti = gsl_vector_get (tau_Z, i);

              cod_householder_hv(ti, &h.vector, &w.vector);

            }

        }

      return GSL_SUCCESS;

    }

}

/*

cod_trireg_solve()

  This function computes the solution to the least squares system

  [    R     ] x = [ b ]

  [ lambda*I ]     [ 0 ]

where R is an N-by-N upper triangular matrix, lambda is a scalar parameter,

and b is a vector of length N. This is done by computing the QR factorization

[    R     ] = W S^T

[ lambda*I ]

where S^T is upper triangular, and solving

S^T x = W^T [ b ]

            [ 0 ]

Inputs: R      - full rank upper triangular matrix; the diagonal

                 elements are modified but restored on output

        lambda - scalar parameter lambda

        b      - right hand side vector b

        S      - workspace, N-by-N

        x      - (output) least squares solution of the system

        work   - workspace of length N

*/

static int

cod_trireg_solve (const gsl_matrix * R, const double lambda, const gsl_vector * b,

                  gsl_matrix * S, gsl_vector * x, gsl_vector * work)

{

  const size_t N = R->size2;

  gsl_vector_const_view diag = gsl_matrix_const_diagonal(R);

  size_t i, j, k;

  if (lambda <= 0.0)

    {

      GSL_ERROR("lambda must be positive", GSL_EINVAL);

    }

  /* copy R and b to preserve input and initialise S; store diag(R) in work */

  gsl_matrix_transpose_tricpy('U', 0, S, R);

  gsl_vector_memcpy(work, &diag.vector);

  gsl_vector_memcpy(x, b);

  /* eliminate the diagonal matrix lambda*I using Givens rotations */

  for (j = 0; j < N; j++)

    {

      double bj = 0.0;

      gsl_matrix_set (S, j, j, lambda);

      for (k = j + 1; k < N; k++)

        {

          gsl_matrix_set (S, k, k, 0.0);

        }

      /* the transformations to eliminate the row of lambda*I modify only a

         single element of b beyond the first n, which is initially

         zero */

      for (k = j; k < N; k++)

        {

          /* determine a Givens rotation which eliminates the

             appropriate element in the current row of lambda*I */

          double sine, cosine;

          double xk = gsl_vector_get (x, k);

          double rkk = gsl_vector_get (work, k);

          double skk = gsl_matrix_get (S, k, k);

          if (skk == 0)

            {

              continue;

            }

          if (fabs (rkk) < fabs (skk))

            {

              double cotangent = rkk / skk;

              sine = 0.5 / sqrt (0.25 + 0.25 * cotangent * cotangent);

              cosine = sine * cotangent;

            }

          else

            {

              double tangent = skk / rkk;

              cosine = 0.5 / sqrt (0.25 + 0.25 * tangent * tangent);

              sine = cosine * tangent;

            }

          /* Compute the modified diagonal element of r and the

             modified element of [b,0] */

          {

            double new_rkk = cosine * rkk + sine * skk;

            double new_xk = cosine * xk + sine * bj;

            bj = -sine * xk + cosine * bj;

            gsl_vector_set(work, k, new_rkk);

            gsl_matrix_set(S, k, k, new_rkk);

            gsl_vector_set(x, k, new_xk);

          }

          /* Accumulate the transformation in the row of s */

          for (i = k + 1; i < N; i++)

            {

              double sik = gsl_matrix_get (S, i, k);

              double sii = gsl_matrix_get (S, i, i);

              double new_sik = cosine * sik + sine * sii;

              double new_sii = -sine * sik + cosine * sii;

              gsl_matrix_set(S, i, k, new_sik);

              gsl_matrix_set(S, i, i, new_sii);

            }

        }

    }

  /* solve: S^T x = rhs in place */

  gsl_blas_dtrsv(CblasLower, CblasTrans, CblasNonUnit, S, x);

  return GSL_SUCCESS;

}