text: /*************************************************************************
  Copyright (c) 1992-2007 The University of Tennessee.  All rights reserved.
  
  Contributors:
      * Sergey Bochkanov (ALGLIB project). Translation from FORTRAN to
        pseudocode.
  
  See subroutines comments for additional copyrights.
  
  Redistribution and use in source and binary forms, with or without
  modification, are permitted provided that the following conditions are
  met:
  
  - Redistributions of source code must retain the above copyright
    notice, this list of conditions and the following disclaimer.
  
  - Redistributions in binary form must reproduce the above copyright
    notice, this list of conditions and the following disclaimer listed
    in this license in the documentation and/or other materials
    provided with the distribution.
  
  - Neither the name of the copyright holders nor the names of its
    contributors may be used to endorse or promote products derived from
    this software without specific prior written permission.
  
  THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
  "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
  LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
  A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
  OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
  SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
  LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
  DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
  THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
  (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
  OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
  *************************************************************************/
  
  #ifndef _qr_h
  #define _qr_h
  
  #include "ap.h"
  
  #include "reflections.h"
  
  
  /*************************************************************************
  QR decomposition of a rectangular matrix of size MxN
  
  Input parameters:
      A   -   matrix A whose indexes range within [0..M-1, 0..N-1].
      M   -   number of rows in matrix A.
      N   -   number of columns in matrix A.
  
  Output parameters:
      A   -   matrices Q and R in compact form (see below).
      Tau -   array of scalar factors which are used to form
              matrix Q. Array whose index ranges within [0.. Min(M-1,N-1)].
  
  Matrix A is represented as A = QR, where Q is an orthogonal matrix of size
  MxM, R - upper triangular (or upper trapezoid) matrix of size M x N.
  
  The elements of matrix R are located on and above the main diagonal of
  matrix A. The elements which are located in Tau array and below the main
  diagonal of matrix A are used to form matrix Q as follows:
  
  Matrix Q is represented as a product of elementary reflections
  
  Q = H(0)*H(2)*...*H(k-1),
  
  where k = min(m,n), and each H(i) is in the form
  
  H(i) = 1 - tau * v * (v^T)
  
  where tau is a scalar stored in Tau[I]; v - real vector,
  so that v(0:i-1) = 0, v(i) = 1, v(i+1:m-1) stored in A(i+1:m-1,i).
  
    -- LAPACK routine (version 3.0) --
       Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
       Courant Institute, Argonne National Lab, and Rice University
       February 29, 1992.
       Translation from FORTRAN to pseudocode (AlgoPascal)
       by Sergey Bochkanov, ALGLIB project, 2005-2007.
  *************************************************************************/
  void rmatrixqr(ap::real_2d_array& a, int m, int n, ap::real_1d_array& tau);
  
  
  /*************************************************************************
  Partial unpacking of matrix Q from the QR decomposition of a matrix A
  
  Input parameters:
      A       -   matrices Q and R in compact form.
                  Output of RMatrixQR subroutine.
      M       -   number of rows in given matrix A. M>=0.
      N       -   number of columns in given matrix A. N>=0.
      Tau     -   scalar factors which are used to form Q.
                  Output of the RMatrixQR subroutine.
      QColumns -  required number of columns of matrix Q. M>=QColumns>=0.
  
  Output parameters:
      Q       -   first QColumns columns of matrix Q.
                  Array whose indexes range within [0..M-1, 0..QColumns-1].
                  If QColumns=0, the array remains unchanged.
  
    -- ALGLIB --
       Copyright 2005 by Bochkanov Sergey
  *************************************************************************/
  void rmatrixqrunpackq(const ap::real_2d_array& a,
       int m,
       int n,
       const ap::real_1d_array& tau,
       int qcolumns,
       ap::real_2d_array& q);
  
  
  /*************************************************************************
  Unpacking of matrix R from the QR decomposition of a matrix A
  
  Input parameters:
      A       -   matrices Q and R in compact form.
                  Output of RMatrixQR subroutine.
      M       -   number of rows in given matrix A. M>=0.
      N       -   number of columns in given matrix A. N>=0.
  
  Output parameters:
      R       -   matrix R, array[0..M-1, 0..N-1].
  
    -- ALGLIB --
       Copyright 2005 by Bochkanov Sergey
  *************************************************************************/
  void rmatrixqrunpackr(const ap::real_2d_array& a,
       int m,
       int n,
       ap::real_2d_array& r);
  
  
  /*************************************************************************
  Obsolete 1-based subroutine. See RMatrixQR for 0-based replacement.
  *************************************************************************/
  void qrdecomposition(ap::real_2d_array& a,
       int m,
       int n,
       ap::real_1d_array& tau);
  
  
  /*************************************************************************
  Obsolete 1-based subroutine. See RMatrixQRUnpackQ for 0-based replacement.
  *************************************************************************/
  void unpackqfromqr(const ap::real_2d_array& a,
       int m,
       int n,
       const ap::real_1d_array& tau,
       int qcolumns,
       ap::real_2d_array& q);
  
  
  /*************************************************************************
  Obsolete 1-based subroutine. See RMatrixQR for 0-based replacement.
  *************************************************************************/
  void qrdecompositionunpacked(ap::real_2d_array a,
       int m,
       int n,
       ap::real_2d_array& q,
       ap::real_2d_array& r);
  
  
  #endif
  