bidiagonal.cpp

The document describes an AP library adapted for C++. The AP library for C++ contains a basic set of

/*************************************************************************
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.

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modification, are permitted provided that the following conditions are
met:

- Redistributions of source code must retain the above copyright
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- Redistributions in binary form must reproduce the above copyright
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  in this license in the documentation and/or other materials
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  contributors may be used to endorse or promote products derived from
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*************************************************************************/

#include <stdafx.h>
#include "bidiagonal.h"

/*************************************************************************
Reduction of a rectangular matrix to  bidiagonal form

The algorithm reduces the rectangular matrix A to  bidiagonal form by
orthogonal transformations P and Q: A = Q*B*P.

Input parameters:
    A       -   source matrix. array[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, B, P in compact form (see below).
    TauQ    -   scalar factors which are used to form matrix Q.
    TauP    -   scalar factors which are used to form matrix P.

The main diagonal and one of the  secondary  diagonals  of  matrix  A  are
replaced with bidiagonal  matrix  B.  Other  elements  contain  elementary
reflections which form MxM matrix Q and NxN matrix P, respectively.

If M>=N, B is the upper  bidiagonal  MxN  matrix  and  is  stored  in  the
corresponding  elements  of  matrix  A.  Matrix  Q  is  represented  as  a
product   of   elementary   reflections   Q = H(0)*H(1)*...*H(n-1),  where
H(i) = 1-tau*v*v'. Here tau is a scalar which is stored  in  TauQ[i],  and
vector v has the following  structure:  v(0:i-1)=0, v(i)=1, v(i+1:m-1)  is
stored   in   elements   A(i+1:m-1,i).   Matrix   P  is  as  follows:  P =
G(0)*G(1)*...*G(n-2), where G(i) = 1 - tau*u*u'. Tau is stored in TauP[i],
u(0:i)=0, u(i+1)=1, u(i+2:n-1) is stored in elements A(i,i+2:n-1).

If M<N, B is the  lower  bidiagonal  MxN  matrix  and  is  stored  in  the
corresponding   elements  of  matrix  A.  Q = H(0)*H(1)*...*H(m-2),  where
H(i) = 1 - tau*v*v', tau is stored in TauQ, v(0:i)=0, v(i+1)=1, v(i+2:m-1)
is    stored    in   elements   A(i+2:m-1,i).    P = G(0)*G(1)*...*G(m-1),
G(i) = 1-tau*u*u', tau is stored in  TauP,  u(0:i-1)=0, u(i)=1, u(i+1:n-1)
is stored in A(i,i+1:n-1).

EXAMPLE:

m=6, n=5 (m > n):               m=5, n=6 (m < n):

(  d   e   u1  u1  u1 )         (  d   u1  u1  u1  u1  u1 )
(  v1  d   e   u2  u2 )         (  e   d   u2  u2  u2  u2 )
(  v1  v2  d   e   u3 )         (  v1  e   d   u3  u3  u3 )
(  v1  v2  v3  d   e  )         (  v1  v2  e   d   u4  u4 )
(  v1  v2  v3  v4  d  )         (  v1  v2  v3  e   d   u5 )
(  v1  v2  v3  v4  v5 )

Here vi and ui are vectors which form H(i) and G(i), and d and e -
are the diagonal and off-diagonal elements of matrix B.
*************************************************************************/
void rmatrixbd(ap::real_2d_array& a,
     int m,
     int n,
     ap::real_1d_array& tauq,
     ap::real_1d_array& taup)
{
    ap::real_1d_array work;
    ap::real_1d_array t;
    int minmn;
    int maxmn;
    int i;
    int j;
    double ltau;

    
    //
    // Prepare
    //
    if( n<=0||m<=0 )
    {
        return;
    }
    minmn = ap::minint(m, n);
    maxmn = ap::maxint(m, n);
    work.setbounds(0, maxmn);
    t.setbounds(0, maxmn);
    if( m>=n )
    {
        tauq.setbounds(0, n-1);
        taup.setbounds(0, n-1);
    }
    else
    {
        tauq.setbounds(0, m-1);
        taup.setbounds(0, m-1);
    }
    if( m>=n )
    {
        
        //
        // Reduce to upper bidiagonal form
        //
        for(i = 0; i <= n-1; i++)
        {
            
            //
            // Generate elementary reflector H(i) to annihilate A(i+1:m-1,i)
            //
            ap::vmove(t.getvector(1, m-i), a.getcolumn(i, i, m-1));
            generatereflection(t, m-i, ltau);
            tauq(i) = ltau;
            ap::vmove(a.getcolumn(i, i, m-1), t.getvector(1, m-i));
            t(1) = 1;
            
            //
            // Apply H(i) to A(i:m-1,i+1:n-1) from the left
            //
            applyreflectionfromtheleft(a, ltau, t, i, m-1, i+1, n-1, work);
            if( i<n-1 )
            {
                
                //
                // Generate elementary reflector G(i) to annihilate
                // A(i,i+2:n-1)
                //
                ap::vmove(&t(1), &a(i, i+1), ap::vlen(1,n-i-1));
                generatereflection(t, n-1-i, ltau);
                taup(i) = ltau;
                ap::vmove(&a(i, i+1), &t(1), ap::vlen(i+1,n-1));
                t(1) = 1;
                
                //
                // Apply G(i) to A(i+1:m-1,i+1:n-1) from the right
                //
                applyreflectionfromtheright(a, ltau, t, i+1, m-1, i+1, n-1, work);
            }
            else
            {
                taup(i) = 0;
            }
        }
    }
    else
    {
        
        //
        // Reduce to lower bidiagonal form
        //
        for(i = 0; i <= m-1; i++)
        {
            
            //
            // Generate elementary reflector G(i) to annihilate A(i,i+1:n-1)
            //
            ap::vmove(&t(1), &a(i, i), ap::vlen(1,n-i));
            generatereflection(t, n-i, ltau);
            taup(i) = ltau;
            ap::vmove(&a(i, i), &t(1), ap::vlen(i,n-1));
            t(1) = 1;
            
            //
            // Apply G(i) to A(i+1:m-1,i:n-1) from the right
            //
            applyreflectionfromtheright(a, ltau, t, i+1, m-1, i, n-1, work);
            if( i<m-1 )
            {
                
                //
                // Generate elementary reflector H(i) to annihilate
                // A(i+2:m-1,i)
                //
                ap::vmove(t.getvector(1, m-1-i), a.getcolumn(i, i+1, m-1));
                generatereflection(t, m-1-i, ltau);
                tauq(i) = ltau;
                ap::vmove(a.getcolumn(i, i+1, m-1), t.getvector(1, m-1-i));
                t(1) = 1;
                
                //
                // Apply H(i) to A(i+1:m-1,i+1:n-1) from the left
                //
                applyreflectionfromtheleft(a, ltau, t, i+1, m-1, i+1, n-1, work);
            }
            else
            {
                tauq(i) = 0;
            }
        }
    }
}


/*************************************************************************
Unpacking matrix Q which reduces a matrix to bidiagonal form.

Input parameters:
    QP          -   matrices Q and P in compact form.
                    Output of ToBidiagonal subroutine.
    M           -   number of rows in matrix A.
    N           -   number of columns in matrix A.
    TAUQ        -   scalar factors which are used to form Q.
                    Output of ToBidiagonal subroutine.
    QColumns    -   required number of columns in matrix Q.
                    M>=QColumns>=0.

Output parameters:
    Q           -   first QColumns columns of matrix Q.
                    Array[0..M-1, 0..QColumns-1]
                    If QColumns=0, the array is not modified.

  -- ALGLIB --
     Copyright 2005 by Bochkanov Sergey
*************************************************************************/
void rmatrixbdunpackq(const ap::real_2d_array& qp,
     int m,
     int n,
     const ap::real_1d_array& tauq,
     int qcolumns,
     ap::real_2d_array& q)
{
    int i;
    int j;

    ap::ap_error::make_assertion(qcolumns<=m, "RMatrixBDUnpackQ: QColumns>M!");
    ap::ap_error::make_assertion(qcolumns>=0, "RMatrixBDUnpackQ: QColumns<0!");
    if( m==0||n==0||qcolumns==0 )
    {
        return;
    }
    
    //
    // prepare Q
    //
    q.setbounds(0, m-1, 0, qcolumns-1);
    for(i = 0; i <= m-1; i++)
    {
        for(j = 0; j <= qcolumns-1; j++)
        {
            if( i==j )
            {
                q(i,j) = 1;
            }
            else
            {
                q(i,j) = 0;
            }
        }
    }
    
    //
    // Calculate
    //
    rmatrixbdmultiplybyq(qp, m, n, tauq, q, m, qcolumns, false, false);
}


/*************************************************************************
Multiplication by matrix Q which reduces matrix A to  bidiagonal form.

The algorithm allows pre- or post-multiply by Q or Q'.

Input parameters:
    QP          -   matrices Q and P in compact form.
                    Output of ToBidiagonal subroutine.
    M           -   number of rows in matrix A.
    N           -   number of columns in matrix A.
    TAUQ        -   scalar factors which are used to form Q.
                    Output of ToBidiagonal subroutine.
    Z           -   multiplied matrix.
                    array[0..ZRows-1,0..ZColumns-1]
    ZRows       -   number of rows in matrix Z. If FromTheRight=False,
                    ZRows=M, otherwise ZRows can be arbitrary.
    ZColumns    -   number of columns in matrix Z. If FromTheRight=True,
                    ZColumns=M, otherwise ZColumns can be arbitrary.
    FromTheRight -  pre- or post-multiply.
    DoTranspose -   multiply by Q or Q'.

Output parameters:
    Z           -   product of Z and Q.
                    Array[0..ZRows-1,0..ZColumns-1]
                    If ZRows=0 or ZColumns=0, the array is not modified.

  -- ALGLIB --
     Copyright 2005 by Bochkanov Sergey
*************************************************************************/
void rmatrixbdmultiplybyq(const ap::real_2d_array& qp,
     int m,
     int n,
     const ap::real_1d_array& tauq,
     ap::real_2d_array& z,
     int zrows,
     int zcolumns,
     bool fromtheright,
     bool dotranspose)
{
    int i;
    int i1;
    int i2;
    int istep;
    ap::real_1d_array v;
    ap::real_1d_array work;
    int mx;

    if( m<=0||n<=0||zrows<=0||zcolumns<=0 )
    {
        return;
    }
    ap::ap_error::make_assertion(fromtheright&&zcolumns==m||!fromtheright&&zrows==m, "RMatrixBDMultiplyByQ: incorrect Z size!");
    
    //
    // init
    //
    mx = ap::maxint(m, n);
    mx = ap::maxint(mx, zrows);
    mx = ap::maxint(mx, zcolumns);
    v.setbounds(0, mx);
    work.setbounds(0, mx);
    if( m>=n )
    {
        
        //
        // setup
        //
        if( fromtheright )
        {
            i1 = 0;
            i2 = n-1;
            istep = +1;
        }
        else
        {
            i1 = n-1;
            i2 = 0;
            istep = -1;
        }
        if( dotranspose )
        {
            i = i1;
            i1 = i2;
            i2 = i;
            istep = -istep;
        }
        
        //
        // Process
        //
        i = i1;
        do
        {
            ap::vmove(v.getvector(1, m-i), qp.getcolumn(i, i, m-1));
            v(1) = 1;
            if( fromtheright )
            {
                applyreflectionfromtheright(z, tauq(i), v, 0, zrows-1, i, m-1, work);
            }
            else
            {
                applyreflectionfromtheleft(z, tauq(i), v, i, m-1, 0, zcolumns-1, work);
            }
            i = i+istep;
        }
        while(i!=i2+istep);
    }
    else
    {
        
        //
        // setup
        //
        if( fromtheright )
        {
            i1 = 0;
            i2 = m-2;
            istep = +1;
        }
        else
        {
            i1 = m-2;
            i2 = 0;
            istep = -1;
        }
        if( dotranspose )
        {
            i = i1;
            i1 = i2;
            i2 = i;
            istep = -istep;
        }
        
        //
        // Process
        //
        if( m-1>0 )
        {
            i = i1;
            do
            {
                ap::vmove(v.getvector(1, m-i-1), qp.getcolumn(i, i+1, m-1));
                v(1) = 1;
                if( fromtheright )
                {
                    applyreflectionfromtheright(z, tauq(i), v, 0, zrows-1, i+1, m-1, work);
                }
                else
                {
                    applyreflectionfromtheleft(z, tauq(i), v, i+1, m-1, 0, zcolumns-1, work);
                }
                i = i+istep;
            }
            while(i!=i2+istep);
        }
    }
}