text: /* linalg/qr.c
   * 
   * Copyright (C) 1996, 1997, 1998, 1999, 2000, 2007 Gerard Jungman, 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.
   */
  
  /* Author:  G. Jungman */
  
  #include <config.h>
  #include <stdlib.h>
  #include <string.h>
  #include <gsl/gsl_linalg.h>
  #include <gsl/gsl_math.h>
  #include <gsl/gsl_vector.h>
  #include <gsl/gsl_matrix.h>
  #include <gsl/gsl_blas.h>
  
  #include "apply_givens.c"
  
  /* Factorise a general M x N matrix A into
   *  
   *   A = Q R
   *
   * where Q is orthogonal (M x M) and R is upper triangular (M x N).
   *
   * Q is stored as a packed set of Householder transformations in the
   * strict lower triangular part of the input matrix.
   *
   * R is stored in the diagonal and upper triangle of the input matrix.
   *
   * 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+1,i), m(i+2,i), ... , m(M,i)]
   *
   * This storage scheme is the same as in LAPACK.  */
  
  int
  gsl_linalg_QR_decomp (gsl_matrix * A, gsl_vector * tau)
  {
    const size_t M = A->size1;
    const size_t N = A->size2;
  
    if (tau->size != GSL_MIN (M, N))
      {
        GSL_ERROR ("size of tau must be MIN(M,N)", GSL_EBADLEN);
      }
    else
      {
        size_t i;
  
        for (i = 0; i < GSL_MIN (M, N); i++)
          {
            /* 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_column (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 and
               update the norms */
  
            if (i + 1 < N)
              {
                gsl_matrix_view m = gsl_matrix_submatrix (A, i, i + 1, M - i, N - (i + 1));
                gsl_linalg_householder_hm (tau_i, &(c.vector), &(m.matrix));
              }
          }
  
        return GSL_SUCCESS;
      }
  }
  
  /* Solves the system A x = b using the QR factorisation,
  
   *  R x = Q^T b
   *
   * to obtain x. Based on SLATEC code. 
   */
  
  int
  gsl_linalg_QR_solve (const gsl_matrix * QR, const gsl_vector * tau, const gsl_vector * b, gsl_vector * x)
  {
    if (QR->size1 != QR->size2)
      {
        GSL_ERROR ("QR matrix must be square", GSL_ENOTSQR);
      }
    else if (QR->size1 != b->size)
      {
        GSL_ERROR ("matrix size must match b size", GSL_EBADLEN);
      }
    else if (QR->size2 != x->size)
      {
        GSL_ERROR ("matrix size must match solution size", GSL_EBADLEN);
      }
    else
      {
        /* Copy x <- b */
  
        gsl_vector_memcpy (x, b);
  
        /* Solve for x */
  
        gsl_linalg_QR_svx (QR, tau, x);
  
        return GSL_SUCCESS;
      }
  }
  
  /* Solves the system A x = b in place using the QR factorisation,
  
   *  R x = Q^T b
   *
   * to obtain x. Based on SLATEC code. 
   */
  
  int
  gsl_linalg_QR_svx (const gsl_matrix * QR, const gsl_vector * tau, gsl_vector * x)
  {
  
    if (QR->size1 != QR->size2)
      {
        GSL_ERROR ("QR matrix must be square", GSL_ENOTSQR);
      }
    else if (QR->size1 != x->size)
      {
        GSL_ERROR ("matrix size must match x/rhs size", GSL_EBADLEN);
      }
    else
      {
        /* compute rhs = Q^T b */
  
        gsl_linalg_QR_QTvec (QR, tau, x);
  
        /* Solve R x = rhs, storing x in-place */
  
        gsl_blas_dtrsv (CblasUpper, CblasNoTrans, CblasNonUnit, QR, x);
  
        return GSL_SUCCESS;
      }
  }
  
  
  /* Find the least squares solution to the overdetermined system 
   *
   *   A x = b 
   *  
   * for M >= N using the QR factorization A = Q R. 
   */
  
  int
  gsl_linalg_QR_lssolve (const gsl_matrix * QR, const gsl_vector * tau, const gsl_vector * b, gsl_vector * x, gsl_vector * residual)
  {
    const size_t M = QR->size1;
    const size_t N = QR->size2;
  
    if (M < N)
      {
        GSL_ERROR ("QR matrix must have M>=N", GSL_EBADLEN);
      }
    else if (M != b->size)
      {
        GSL_ERROR ("matrix size must match b size", 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 R = gsl_matrix_const_submatrix (QR, 0, 0, N, N);
        gsl_vector_view c = gsl_vector_subvector(residual, 0, N);
  
        gsl_vector_memcpy(residual, b);
  
        /* compute rhs = Q^T b */
  
        gsl_linalg_QR_QTvec (QR, tau, residual);
  
        /* Solve R x = rhs */
  
        gsl_vector_memcpy(x, &(c.vector));
  
        gsl_blas_dtrsv (CblasUpper, CblasNoTrans, CblasNonUnit, &(R.matrix), x);
  
        /* Compute residual = b - A x = Q (Q^T b - R x) */
        
        gsl_vector_set_zero(&(c.vector));
  
        gsl_linalg_QR_Qvec(QR, tau, residual);
  
        return GSL_SUCCESS;
      }
  }
  
  
  int
  gsl_linalg_QR_Rsolve (const gsl_matrix * QR, const gsl_vector * b, gsl_vector * x)
  {
    if (QR->size1 != QR->size2)
      {
        GSL_ERROR ("QR matrix must be square", GSL_ENOTSQR);
      }
    else if (QR->size1 != b->size)
      {
        GSL_ERROR ("matrix size must match b size", GSL_EBADLEN);
      }
    else if (QR->size2 != x->size)
      {
        GSL_ERROR ("matrix size must match x size", GSL_EBADLEN);
      }
    else
      {
        /* Copy x <- b */
  
        gsl_vector_memcpy (x, b);
  
        /* Solve R x = b, storing x in-place */
  
        gsl_blas_dtrsv (CblasUpper, CblasNoTrans, CblasNonUnit, QR, x);
  
        return GSL_SUCCESS;
      }
  }
  
  
  int
  gsl_linalg_QR_Rsvx (const gsl_matrix * QR, gsl_vector * x)
  {
    if (QR->size1 != QR->size2)
      {
        GSL_ERROR ("QR matrix must be square", GSL_ENOTSQR);
      }
    else if (QR->size1 != x->size)
      {
        GSL_ERROR ("matrix size must match rhs size", GSL_EBADLEN);
      }
    else
      {
        /* Solve R x = b, storing x in-place */
  
        gsl_blas_dtrsv (CblasUpper, CblasNoTrans, CblasNonUnit, QR, x);
  
        return GSL_SUCCESS;
      }
  }
  
  int
  gsl_linalg_R_solve (const gsl_matrix * R, const gsl_vector * b, gsl_vector * x)
  {
    if (R->size1 != R->size2)
      {
        GSL_ERROR ("R matrix must be square", GSL_ENOTSQR);
      }
    else if (R->size1 != b->size)
      {
        GSL_ERROR ("matrix size must match b size", GSL_EBADLEN);
      }
    else if (R->size2 != x->size)
      {
        GSL_ERROR ("matrix size must match solution size", GSL_EBADLEN);
      }
    else
      {
        /* Copy x <- b */
  
        gsl_vector_memcpy (x, b);
  
        /* Solve R x = b, storing x inplace in b */
  
        gsl_blas_dtrsv (CblasUpper, CblasNoTrans, CblasNonUnit, R, x);
  
        return GSL_SUCCESS;
      }
  }
  
  int
  gsl_linalg_R_svx (const gsl_matrix * R, gsl_vector * x)
  {
    if (R->size1 != R->size2)
      {
        GSL_ERROR ("R matrix must be square", GSL_ENOTSQR);
      }
    else if (R->size2 != x->size)
      {
        GSL_ERROR ("matrix size must match solution size", GSL_EBADLEN);
      }
    else
      {
        /* Solve R x = b, storing x inplace in b */
  
        gsl_blas_dtrsv (CblasUpper, CblasNoTrans, CblasNonUnit, R, x);
  
        return GSL_SUCCESS;
      }
  }
  
  
  
  /* Form the product Q^T v  from a QR factorized matrix 
   */
  
  int
  gsl_linalg_QR_QTvec (const gsl_matrix * QR, const gsl_vector * tau, gsl_vector * v)
  {
    const size_t M = QR->size1;
    const size_t N = QR->size2;
  
    if (tau->size != GSL_MIN (M, N))
      {
        GSL_ERROR ("size of tau must be MIN(M,N)", GSL_EBADLEN);
      }
    else if (v->size != M)
      {
        GSL_ERROR ("vector size must be M", GSL_EBADLEN);
      }
    else
      {
        size_t i;
  
        /* compute Q^T v */
  
        for (i = 0; i < GSL_MIN (M, N); i++)
          {
            gsl_vector_const_view c = gsl_matrix_const_column (QR, i);
            gsl_vector_const_view h = gsl_vector_const_subvector (&(c.vector), i, M - i);
            gsl_vector_view w = gsl_vector_subvector (v, i, M - i);
            double ti = gsl_vector_get (tau, i);
            gsl_linalg_householder_hv (ti, &(h.vector), &(w.vector));
          }
        return GSL_SUCCESS;
      }
  }
  
  
  int
  gsl_linalg_QR_Qvec (const gsl_matrix * QR, const gsl_vector * tau, gsl_vector * v)
  {
    const size_t M = QR->size1;
    const size_t N = QR->size2;
  
    if (tau->size != GSL_MIN (M, N))
      {
        GSL_ERROR ("size of tau must be MIN(M,N)", GSL_EBADLEN);
      }
    else if (v->size != M)
      {
        GSL_ERROR ("vector size must be M", GSL_EBADLEN);
      }
    else
      {
        size_t i;
  
        /* compute Q v */
  
        for (i = GSL_MIN (M, N); i-- > 0;)
          {
            gsl_vector_const_view c = gsl_matrix_const_column (QR, i);
            gsl_vector_const_view h = gsl_vector_const_subvector (&(c.vector), 
                                                                  i, M - i);
            gsl_vector_view w = gsl_vector_subvector (v, i, M - i);
            double ti = gsl_vector_get (tau, i);
            gsl_linalg_householder_hv (ti, &h.vector, &w.vector);
          }
        return GSL_SUCCESS;
      }
  }
  
  /* Form the product Q^T A from a QR factorized matrix */
  
  int
  gsl_linalg_QR_QTmat (const gsl_matrix * QR, const gsl_vector * tau, gsl_matrix * A)
  {
    const size_t M = QR->size1;
    const size_t N = QR->size2;
  
    if (tau->size != GSL_MIN (M, N))
      {
        GSL_ERROR ("size of tau must be MIN(M,N)", GSL_EBADLEN);
      }
    else if (A->size1 != M)
      {
        GSL_ERROR ("matrix must have M rows", GSL_EBADLEN);
      }
    else
      {
        size_t i;
  
        /* compute Q^T A */
  
        for (i = 0; i < GSL_MIN (M, N); i++)
          {
            gsl_vector_const_view c = gsl_matrix_const_column (QR, i);
            gsl_vector_const_view h = gsl_vector_const_subvector (&(c.vector), i, M - i);
            gsl_matrix_view m = gsl_matrix_submatrix(A, i, 0, M - i, A->size2);
            double ti = gsl_vector_get (tau, i);
            gsl_linalg_householder_hm (ti, &(h.vector), &(m.matrix));
          }
        return GSL_SUCCESS;
      }
  }
  
  /* Form the product A Q from a QR factorized matrix */
  int
  gsl_linalg_QR_matQ (const gsl_matrix * QR, const gsl_vector * tau, gsl_matrix * A)
  {
    const size_t M = QR->size1;
    const size_t N = QR->size2;
  
    if (tau->size != GSL_MIN (M, N))
      {
        GSL_ERROR ("size of tau must be MIN(M,N)", GSL_EBADLEN);
      }
    else if (A->size2 != M)
      {
        GSL_ERROR ("matrix must have M columns", GSL_EBADLEN);
      }
    else
      {
        size_t i;
  
        /* compute A Q */
  
        for (i = 0; i < GSL_MIN (M, N); i++)
          {
            gsl_vector_const_view c = gsl_matrix_const_column (QR, i);
            gsl_vector_const_view h = gsl_vector_const_subvector (&(c.vector), i, M - i);
            gsl_matrix_view m = gsl_matrix_submatrix(A, 0, i, A->size1, M - i);
            double ti = gsl_vector_get (tau, i);
            gsl_linalg_householder_mh (ti, &(h.vector), &(m.matrix));
          }
        return GSL_SUCCESS;
      }
  }
  
  /*  Form the orthogonal matrix Q from the packed QR matrix */
  
  int
  gsl_linalg_QR_unpack (const gsl_matrix * QR, const gsl_vector * tau, gsl_matrix * Q, gsl_matrix * R)
  {
    const size_t M = QR->size1;
    const size_t N = QR->size2;
  
    if (Q->size1 != M || Q->size2 != M)
      {
        GSL_ERROR ("Q matrix must be M x M", GSL_ENOTSQR);
      }
    else if (R->size1 != M || R->size2 != N)
      {
        GSL_ERROR ("R matrix must be M x N", GSL_ENOTSQR);
      }
    else if (tau->size != GSL_MIN (M, N))
      {
        GSL_ERROR ("size of tau must be MIN(M,N)", GSL_EBADLEN);
      }
    else
      {
        size_t i, j;
  
        /* Initialize Q to the identity */
  
        gsl_matrix_set_identity (Q);
  
        for (i = GSL_MIN (M, N); i-- > 0;)
          {
            gsl_vector_const_view c = gsl_matrix_const_column (QR, i);
            gsl_vector_const_view h = gsl_vector_const_subvector (&c.vector,
                                                                  i, M - i);
            gsl_matrix_view m = gsl_matrix_submatrix (Q, i, i, M - i, M - i);
            double ti = gsl_vector_get (tau, i);
            gsl_linalg_householder_hm (ti, &h.vector, &m.matrix);
          }
  
        /*  Form the right triangular matrix R from a packed QR matrix */
  
        for (i = 0; i < M; i++)
          {
            for (j = 0; j < i && j < N; j++)
              gsl_matrix_set (R, i, j, 0.0);
  
            for (j = i; j < N; j++)
              gsl_matrix_set (R, i, j, gsl_matrix_get (QR, i, j));
          }
  
        return GSL_SUCCESS;
      }
  }
  
  
  /* Update a QR factorisation for A= Q R ,  A' = A + u v^T,
  
   * Q' R' = QR + u v^T
   *       = Q (R + Q^T u v^T)
   *       = Q (R + w v^T)
   *
   * 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_QR_update (gsl_matrix * Q, gsl_matrix * R,
                        gsl_vector * w, const gsl_vector * v)
  {
    const size_t M = R->size1;
    const size_t N = R->size2;
  
    if (Q->size1 != M || Q->size2 != M)
      {
        GSL_ERROR ("Q matrix must be M x M if R is M x N", GSL_ENOTSQR);
      }
    else if (w->size != M)
      {
        GSL_ERROR ("w must be length M if R is M x N", GSL_EBADLEN);
      }
    else if (v->size != N)
      {
        GSL_ERROR ("v must be length N if R is M x N", GSL_EBADLEN);
      }
    else
      {
        size_t j, k;
        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 R,  H = J_1^T ... J^T_(n-1) R
           so that H is upper Hessenberg.  (12.5.2) */
  
        for (k = M - 1; k > 0; k--)  /* loop from k = M-1 to 1 */
          {
            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_qr (M, N, Q, R, k - 1, k, c, s);
          }
  
        w0 = gsl_vector_get (w, 0);
  
        /* Add in w v^T  (Equation 12.5.3) */
  
        for (j = 0; j < N; j++)
          {
            double r0j = gsl_matrix_get (R, 0, j);
            double vj = gsl_vector_get (v, j);
            gsl_matrix_set (R, 0, j, r0j + w0 * vj);
          }
  
        /* Apply Givens transformations R' = G_(n-1)^T ... G_1^T H
           Equation 12.5.4 */
  
        for (k = 1; k < GSL_MIN(M,N+1); k++)
          {
            double c, s;
            double diag = gsl_matrix_get (R, k - 1, k - 1);
            double offdiag = gsl_matrix_get (R, k, k - 1);
  
            gsl_linalg_givens (diag, offdiag, &c, &s);
            apply_givens_qr (M, N, Q, R, k - 1, k, c, s);
  
            gsl_matrix_set (R, k, k - 1, 0.0);    /* exact zero of G^T */
          }
  
        return GSL_SUCCESS;
      }
  }
  
  int
  gsl_linalg_QR_QRsolve (gsl_matrix * Q, gsl_matrix * R, const gsl_vector * b, gsl_vector * x)
  {
    const size_t M = R->size1;
    const size_t N = R->size2;
  
    if (M != N)
      {
        return GSL_ENOTSQR;
      }
    else if (Q->size1 != M || b->size != M || x->size != M)
      {
        return GSL_EBADLEN;
      }
    else
      {
        /* compute sol = Q^T b */
  
        gsl_blas_dgemv (CblasTrans, 1.0, Q, b, 0.0, x);
  
        /* Solve R x = sol, storing x in-place */
  
        gsl_blas_dtrsv (CblasUpper, CblasNoTrans, CblasNonUnit, R, x);
  
        return GSL_SUCCESS;
      }
  }
  