/* linalg/qr.c *  * Copyright (C) 1996, 1997, 1998, 1999, 2000 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 2 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., 675 Mass Ave, Cambridge, MA 02139, USA. *//* Author:  G. Jungman */#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_blas.h>#include <gsl/gsl_linalg.h>#define REAL double#include "givens.c"#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.  */intgsl_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.  */intgsl_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.  */intgsl_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.  */intgsl_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;    }}intgsl_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;    }}intgsl_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;    }}intgsl_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;    }}/* Form the product Q^T v  from a QR factorized matrix  */intgsl_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 N", 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;    }}intgsl_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 N", GSL_EBADLEN);    }  else    {      size_t i;      /* compute Q^T v */      for (i = GSL_MIN (M, N); i > 0 && 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;    }}/*  Form the orthogonal matrix Q from the packed QR matrix */intgsl_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 && 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 (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)   */intgsl_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--)        {          double c, s;          double wk = gsl_vector_get (w, k);          double wkm1 = gsl_vector_get (w, k - 1);          create_givens (wkm1, wk, &c, &s);          apply_givens_vec (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);          create_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;    }}intgsl_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;    }}