/* linalg/ptlq.c

 * 

 * Copyright (C) 1996, 1997, 1998, 1999, 2000, 2007 Gerard Jungman, Brian Gough

 * Copyright (C) 2004 Joerg Wensch, modifications for LQ. 

 * 

 * 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_blas.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_linalg.h>

#include "apply_givens.c"

/* The purpose of this package is to speed up QR-decomposition for

   large matrices.  Because QR-decomposition is column oriented, but

   GSL uses a row-oriented matrix format, there can considerable

   speedup obtained by computing the LQ-decomposition of the

   transposed matrix instead.  This package provides LQ-decomposition

   and related algorithms.  */

/* Factorise a general N x M matrix A into

 *

 *   P A = L Q

 *

 * where Q is orthogonal (M x M) and L is lower triangular (N x M).

 * When A is rank deficient, r = rank(A) < n, then the permutation is

 * used to ensure that the lower n - r columns of L are zero and the first

 * l rows of Q form an orthonormal basis for the rows of A.

 *

 * Q is stored as a packed set of Householder transformations in the

 * strict upper triangular part of the input matrix.

 *

 * L is stored in the diagonal and lower triangle of the input matrix.

 *

 * P: column j of P is column k of the identity matrix, where k =

 * permutation->data[j]

 *

 * The full matrix for Q can be obtained as the product

 *

 *       Q = Q_k .. Q_2 Q_1

 *

 * where k = MIN(M,N) and

 *

 *       Q_i = (I - tau_i * v_i * v_i')

 *

 * and where v_i is a Householder vector

 *

 *       v_i = [1, m(i,i+1), m(i,i+2), ... , m(i,M)]

 *

 * This storage scheme is the same as in LAPACK.  See LAPACK's

 * dgeqpf.f for details.

 * 

 */

int

gsl_linalg_PTLQ_decomp (gsl_matrix * A, gsl_vector * tau, gsl_permutation * p, int *signum, gsl_vector * norm)

{

  const size_t N = A->size1;

  const size_t M = A->size2;

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

    {

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

    }

  else if (p->size != N)

    {

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

    }

  else if (norm->size != N)

    {

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

    }

  else

    {

      size_t i;

      *signum = 1;

      gsl_permutation_init (p); /* set to identity */

      /* Compute column norms and store in workspace */

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

        {

          gsl_vector_view c = gsl_matrix_row (A, i);

          double x = gsl_blas_dnrm2 (&c.vector);

          gsl_vector_set (norm, i, x);

        }

      for (i = 0; i < GSL_MIN (M, N); i++)

        {

          /* Bring the column of largest norm into the pivot position */

          double max_norm = gsl_vector_get(norm, i);

          size_t j, kmax = i;

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

            {

              double x = gsl_vector_get (norm, j);

              if (x > max_norm)

                {

                  max_norm = x;

                  kmax = j;

                }

            }

          if (kmax != i)

            {

              gsl_matrix_swap_rows (A, i, kmax);

              gsl_permutation_swap (p, i, kmax);

              gsl_vector_swap_elements(norm,i,kmax);

              (*signum) = -(*signum);

            }

          /* Compute the Householder transformation to reduce the j-th

             column of the matrix to a multiple of the j-th unit vector */

          {

            gsl_vector_view c_full = gsl_matrix_row (A, i);

            gsl_vector_view c = gsl_vector_subvector (&c_full.vector, 

                                                      i, M - i);

            double tau_i = gsl_linalg_householder_transform (&c.vector);

            gsl_vector_set (tau, i, tau_i);

            /* Apply the transformation to the remaining columns */

            if (i + 1 < N)

              {

                gsl_matrix_view m = gsl_matrix_submatrix (A, i +1, i, N - (i+1), M - i);

                gsl_linalg_householder_mh (tau_i, &c.vector, &m.matrix);

              }

          }

          /* Update the norms of the remaining columns too */

          if (i + 1 < M) 

            {

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

                {

                  double x = gsl_vector_get (norm, j);

                  if (x > 0.0)

                    {

                      double y = 0;

                      double temp= gsl_matrix_get (A, j, i) / x;

                      if (fabs (temp) >= 1)

                        y = 0.0;

                      else

                        y = x * sqrt (1 - temp * temp);

                      /* recompute norm to prevent loss of accuracy */

                      if (fabs (y / x) < sqrt (20.0) * GSL_SQRT_DBL_EPSILON)

                        {

                          gsl_vector_view c_full = gsl_matrix_row (A, j);

                          gsl_vector_view c = 

                            gsl_vector_subvector(&c_full.vector,

                                                 i+1, M - (i+1));

                          y = gsl_blas_dnrm2 (&c.vector);

                        }

                      gsl_vector_set (norm, j, y);

                    }

                }

            }

        }

      return GSL_SUCCESS;

    }

}

int

gsl_linalg_PTLQ_decomp2 (const gsl_matrix * A, gsl_matrix * q, gsl_matrix * r, gsl_vector * tau, gsl_permutation * p, int *signum, gsl_vector * norm)

{

  const size_t N = A->size1;

  const size_t M = A->size2;

  if (q->size1 != M || q->size2 !=M) 

    {

      GSL_ERROR ("q must be M x M", GSL_EBADLEN);

    }

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

    {

      GSL_ERROR ("r must be N x M", GSL_EBADLEN);

    }

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

    {

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

    }

  else if (p->size != N)

    {

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

    }

  else if (norm->size != N)

    {

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

    }

  gsl_matrix_memcpy (r, A);

  gsl_linalg_PTLQ_decomp (r, tau, p, signum, norm);

  /* FIXME:  aliased arguments depends on behavior of unpack routine! */

  gsl_linalg_LQ_unpack (r, tau, q, r);

  return GSL_SUCCESS;

}

/* Solves the system x^T A = b^T using the P^T L Q  factorisation,

   z^T L = b^T Q^T 

   x = P z;

   to obtain x. Based on SLATEC code. */

int

gsl_linalg_PTLQ_solve_T (const gsl_matrix * QR,

                         const gsl_vector * tau,

                         const gsl_permutation * p,

                         const gsl_vector * b,

                         gsl_vector * x)

{

  if (QR->size1 != QR->size2)

    {

      GSL_ERROR ("QR matrix must be square", GSL_ENOTSQR);

    }

  else if (QR->size2 != p->size)

    {

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

    }

  else if (QR->size2 != b->size)

    {

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

    }

  else if (QR->size1 != x->size)

    {

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

    }

  else

    {

      gsl_vector_memcpy (x, b);

      gsl_linalg_PTLQ_svx_T (QR, tau, p, x);

      return GSL_SUCCESS;

    }

}

int

gsl_linalg_PTLQ_svx_T (const gsl_matrix * LQ,

                       const gsl_vector * tau,

                       const gsl_permutation * p,

                       gsl_vector * x)

{

  if (LQ->size1 != LQ->size2)

    {

      GSL_ERROR ("LQ matrix must be square", GSL_ENOTSQR);

    }

  else if (LQ->size2 != p->size)

    {

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

    }

  else if (LQ->size1 != x->size)

    {

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

    }

  else

    {

      /* compute sol = b^T Q^T */

      gsl_linalg_LQ_vecQT (LQ, tau, x);

      /* Solve  L^T x = sol, storing x inplace in sol */

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

      gsl_permute_vector_inverse (p, x);

      return GSL_SUCCESS;

    }

}

int

gsl_linalg_PTLQ_LQsolve_T (const gsl_matrix * Q, const gsl_matrix * L,

                           const gsl_permutation * p,

                           const gsl_vector * b,

                           gsl_vector * x)

{

  if (Q->size1 != Q->size2 || L->size1 != L->size2)

    {

      return GSL_ENOTSQR;

    }

  else if (Q->size1 != p->size || Q->size1 != L->size1

           || Q->size1 != b->size)

    {

      return GSL_EBADLEN;

    }

  else

    {

      /* compute b' = Q b */

      gsl_blas_dgemv (CblasNoTrans, 1.0, Q, b, 0.0, x);

      /* Solve L^T x = b', storing x inplace */

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

      /* Apply permutation to solution in place */

      gsl_permute_vector_inverse (p, x);

      return GSL_SUCCESS;

    }

}

int

gsl_linalg_PTLQ_Lsolve_T (const gsl_matrix * LQ,

                        const gsl_permutation * p,

                        const gsl_vector * b,

                        gsl_vector * x)

{

  if (LQ->size1 != LQ->size2)

    {

      GSL_ERROR ("LQ matrix must be square", GSL_ENOTSQR);

    }

  else if (LQ->size1 != b->size)

    {

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

    }

  else if (LQ->size2 != x->size)

    {

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

    }

  else if (p->size != x->size)

    {

      GSL_ERROR ("permutation size must match x size", GSL_EBADLEN);

    }

  else

    {

      /* Copy x <- b */

      gsl_vector_memcpy (x, b);

      /* Solve L^T x = b, storing x inplace */

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

      gsl_permute_vector_inverse (p, x);

      return GSL_SUCCESS;

    }

}

int

gsl_linalg_PTLQ_Lsvx_T (const gsl_matrix * LQ,

                        const gsl_permutation * p,

                        gsl_vector * x)

{

  if (LQ->size1 != LQ->size2)

    {

      GSL_ERROR ("LQ matrix must be square", GSL_ENOTSQR);

    }

  else if (LQ->size2 != x->size)

    {

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

    }

  else if (p->size != x->size)

    {

      GSL_ERROR ("permutation size must match x size", GSL_EBADLEN);

    }

  else

    {

      /* Solve L^T x = b, storing x inplace */

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

      gsl_permute_vector_inverse (p, x);

      return GSL_SUCCESS;

    }

}

/* Update a P^T L Q factorisation for P A= L Q ,  A' =  A +  v u^T,

                                                 PA' = PA + Pv u^T

 * P^T L' Q' = P^T LQ + v u^T

 *       = P^T (L + (P v) u^T Q^T) Q

 *       = P^T (L + (P v) w^T) Q

 *

 * where w = Q^T u.

 *

 * Algorithm from Golub and Van Loan, "Matrix Computations", Section

 * 12.5 (Updating Matrix Factorizations, Rank-One Changes)

 */

int

gsl_linalg_PTLQ_update (gsl_matrix * Q, gsl_matrix * L,

                        const gsl_permutation * p,

                        const gsl_vector * v, gsl_vector * w)

{

  if (Q->size1 != Q->size2 || L->size1 != L->size2)

    {

      return GSL_ENOTSQR;

    }

  else if (L->size1 != Q->size2 || v->size != Q->size2 || w->size != Q->size2)

    {

      return GSL_EBADLEN;

    }

  else

    {

      size_t j, k;

      const size_t N = Q->size1;

      const size_t M = Q->size2;

      double w0;

      /* Apply Given's rotations to reduce w to (|w|, 0, 0, ... , 0) 

         J_1^T .... J_(n-1)^T w = +/- |w| e_1

         simultaneously applied to L,  H = J_1^T ... J^T_(n-1) L

         so that H is upper Hessenberg.  (12.5.2) */

      for (k = M - 1; k > 0; k--)

        {

          double c, s;

          double wk = gsl_vector_get (w, k);

          double wkm1 = gsl_vector_get (w, k - 1);

          gsl_linalg_givens (wkm1, wk, &c, &s);

          gsl_linalg_givens_gv (w, k - 1, k, c, s);

          apply_givens_lq (M, N, Q, L, k - 1, k, c, s);

        }

      w0 = gsl_vector_get (w, 0);

      /* Add in v w^T  (Equation 12.5.3) */

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

        {

          double lj0 = gsl_matrix_get (L, j, 0);

          size_t p_j = gsl_permutation_get (p, j);

          double vj = gsl_vector_get (v, p_j);

          gsl_matrix_set (L, j, 0, lj0 + w0 * vj);

        }

      /* Apply Givens transformations L' = G_(n-1)^T ... G_1^T H  

         Equation 12.5.4 */

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

        {

          double c, s;

          double diag = gsl_matrix_get (L, k - 1, k - 1);

          double offdiag = gsl_matrix_get (L, k - 1, k );

          gsl_linalg_givens (diag, offdiag, &c, &s);

          apply_givens_lq (M, N, Q, L, k - 1, k, c, s);

        }

      return GSL_SUCCESS;

    }

}