source: branches/CEDMA-Exporter-715/sources/HeuristicLab.LinearRegression/3.2/alglib/bidiagonal.cs @ 2273

/*************************************************************************
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,
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THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
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OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
*************************************************************************/

using System;

class bidiagonal
{
    /*************************************************************************
    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.
    *************************************************************************/
    public static void rmatrixbd(ref double[,] a,
        int m,
        int n,
        ref double[] tauq,
        ref double[] taup)
    {
        double[] work = new double[0];
        double[] t = new double[0];
        int minmn = 0;
        int maxmn = 0;
        int i = 0;
        int j = 0;
        double ltau = 0;
        int i_ = 0;
        int i1_ = 0;

        
        //
        // Prepare
        //
        if( n<=0 | m<=0 )
        {
            return;
        }
        minmn = Math.Min(m, n);
        maxmn = Math.Max(m, n);
        work = new double[maxmn+1];
        t = new double[maxmn+1];
        if( m>=n )
        {
            tauq = new double[n-1+1];
            taup = new double[n-1+1];
        }
        else
        {
            tauq = new double[m-1+1];
            taup = new double[m-1+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)
                //
                i1_ = (i) - (1);
                for(i_=1; i_<=m-i;i_++)
                {
                    t[i_] = a[i_+i1_,i];
                }
                reflections.generatereflection(ref t, m-i, ref ltau);
                tauq[i] = ltau;
                i1_ = (1) - (i);
                for(i_=i; i_<=m-1;i_++)
                {
                    a[i_,i] = t[i_+i1_];
                }
                t[1] = 1;
                
                //
                // Apply H(i) to A(i:m-1,i+1:n-1) from the left
                //
                reflections.applyreflectionfromtheleft(ref a, ltau, ref t, i, m-1, i+1, n-1, ref work);
                if( i<n-1 )
                {
                    
                    //
                    // Generate elementary reflector G(i) to annihilate
                    // A(i,i+2:n-1)
                    //
                    i1_ = (i+1) - (1);
                    for(i_=1; i_<=n-i-1;i_++)
                    {
                        t[i_] = a[i,i_+i1_];
                    }
                    reflections.generatereflection(ref t, n-1-i, ref ltau);
                    taup[i] = ltau;
                    i1_ = (1) - (i+1);
                    for(i_=i+1; i_<=n-1;i_++)
                    {
                        a[i,i_] = t[i_+i1_];
                    }
                    t[1] = 1;
                    
                    //
                    // Apply G(i) to A(i+1:m-1,i+1:n-1) from the right
                    //
                    reflections.applyreflectionfromtheright(ref a, ltau, ref t, i+1, m-1, i+1, n-1, ref 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)
                //
                i1_ = (i) - (1);
                for(i_=1; i_<=n-i;i_++)
                {
                    t[i_] = a[i,i_+i1_];
                }
                reflections.generatereflection(ref t, n-i, ref ltau);
                taup[i] = ltau;
                i1_ = (1) - (i);
                for(i_=i; i_<=n-1;i_++)
                {
                    a[i,i_] = t[i_+i1_];
                }
                t[1] = 1;
                
                //
                // Apply G(i) to A(i+1:m-1,i:n-1) from the right
                //
                reflections.applyreflectionfromtheright(ref a, ltau, ref t, i+1, m-1, i, n-1, ref work);
                if( i<m-1 )
                {
                    
                    //
                    // Generate elementary reflector H(i) to annihilate
                    // A(i+2:m-1,i)
                    //
                    i1_ = (i+1) - (1);
                    for(i_=1; i_<=m-1-i;i_++)
                    {
                        t[i_] = a[i_+i1_,i];
                    }
                    reflections.generatereflection(ref t, m-1-i, ref ltau);
                    tauq[i] = ltau;
                    i1_ = (1) - (i+1);
                    for(i_=i+1; i_<=m-1;i_++)
                    {
                        a[i_,i] = t[i_+i1_];
                    }
                    t[1] = 1;
                    
                    //
                    // Apply H(i) to A(i+1:m-1,i+1:n-1) from the left
                    //
                    reflections.applyreflectionfromtheleft(ref a, ltau, ref t, i+1, m-1, i+1, n-1, ref 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
    *************************************************************************/
    public static void rmatrixbdunpackq(ref double[,] qp,
        int m,
        int n,
        ref double[] tauq,
        int qcolumns,
        ref double[,] q)
    {
        int i = 0;
        int j = 0;

        System.Diagnostics.Debug.Assert(qcolumns<=m, "RMatrixBDUnpackQ: QColumns>M!");
        System.Diagnostics.Debug.Assert(qcolumns>=0, "RMatrixBDUnpackQ: QColumns<0!");
        if( m==0 | n==0 | qcolumns==0 )
        {
            return;
        }
        
        //
        // prepare Q
        //
        q = new double[m-1+1, qcolumns-1+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(ref qp, m, n, ref tauq, ref 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
    *************************************************************************/
    public static void rmatrixbdmultiplybyq(ref double[,] qp,
        int m,
        int n,
        ref double[] tauq,
        ref double[,] z,
        int zrows,
        int zcolumns,
        bool fromtheright,
        bool dotranspose)
    {
        int i = 0;
        int i1 = 0;
        int i2 = 0;
        int istep = 0;
        double[] v = new double[0];
        double[] work = new double[0];
        int mx = 0;
        int i_ = 0;
        int i1_ = 0;

        if( m<=0 | n<=0 | zrows<=0 | zcolumns<=0 )
        {
            return;
        }
        System.Diagnostics.Debug.Assert(fromtheright & zcolumns==m | !fromtheright & zrows==m, "RMatrixBDMultiplyByQ: incorrect Z size!");
        
        //
        // init
        //
        mx = Math.Max(m, n);
        mx = Math.Max(mx, zrows);
        mx = Math.Max(mx, zcolumns);
        v = new double[mx+1];
        work = new double[mx+1];
        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
            {
                i1_ = (i) - (1);
                for(i_=1; i_<=m-i;i_++)
                {
                    v[i_] = qp[i_+i1_,i];
                }
                v[1] = 1;
                if( fromtheright )
                {
                    reflections.applyreflectionfromtheright(ref z, tauq[i], ref v, 0, zrows-1, i, m-1, ref work);
                }
                else
                {
                    reflections.applyreflectionfromtheleft(ref z, tauq[i], ref v, i, m-1, 0, zcolumns-1, ref 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
                {
                    i1_ = (i+1) - (1);
                    for(i_=1; i_<=m-i-1;i_++)
                    {
                        v[i_] = qp[i_+i1_,i];
                    }
                    v[1] = 1;
                    if( fromtheright )
                    {
                        reflections.applyreflectionfromtheright(ref z, tauq[i], ref v, 0, zrows-1, i+1, m-1, ref work);
                    }
                    else
                    {
                        reflections.applyreflectionfromtheleft(ref z, tauq[i], ref v, i+1, m-1, 0, zcolumns-1, ref work);
                    }
                    i = i+istep;
                }
                while( i!=i2+istep );
            }
        }
    }


    /*************************************************************************
    Unpacking matrix P which reduces matrix A to bidiagonal form.
    The subroutine returns transposed matrix P.

    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.
        TAUP    -   scalar factors which are used to form P.
                    Output of ToBidiagonal subroutine.
        PTRows  -   required number of rows of matrix P^T. N >= PTRows >= 0.

    Output parameters:
        PT      -   first PTRows columns of matrix P^T
                    Array[0..PTRows-1, 0..N-1]
                    If PTRows=0, the array is not modified.

      -- ALGLIB --
         Copyright 2005-2007 by Bochkanov Sergey
    *************************************************************************/
    public static void rmatrixbdunpackpt(ref double[,] qp,
        int m,
        int n,
        ref double[] taup,
        int ptrows,
        ref double[,] pt)
    {
        int i = 0;
        int j = 0;

        System.Diagnostics.Debug.Assert(ptrows<=n, "RMatrixBDUnpackPT: PTRows>N!");
        System.Diagnostics.Debug.Assert(ptrows>=0, "RMatrixBDUnpackPT: PTRows<0!");
        if( m==0 | n==0 | ptrows==0 )
        {
            return;
        }
        
        //
        // prepare PT
        //
        pt = new double[ptrows-1+1, n-1+1];
        for(i=0; i<=ptrows-1; i++)
        {
            for(j=0; j<=n-1; j++)
            {
                if( i==j )
                {
                    pt[i,j] = 1;
                }
                else
                {
                    pt[i,j] = 0;
                }
            }
        }
        
        //
        // Calculate
        //
        rmatrixbdmultiplybyp(ref qp, m, n, ref taup, ref pt, ptrows, n, true, true);
    }


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

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

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

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

      -- ALGLIB --
         Copyright 2005-2007 by Bochkanov Sergey
    *************************************************************************/
    public static void rmatrixbdmultiplybyp(ref double[,] qp,
        int m,
        int n,
        ref double[] taup,
        ref double[,] z,
        int zrows,
        int zcolumns,
        bool fromtheright,
        bool dotranspose)
    {
        int i = 0;
        double[] v = new double[0];
        double[] work = new double[0];
        int mx = 0;
        int i1 = 0;
        int i2 = 0;
        int istep = 0;
        int i_ = 0;
        int i1_ = 0;

        if( m<=0 | n<=0 | zrows<=0 | zcolumns<=0 )
        {
            return;
        }
        System.Diagnostics.Debug.Assert(fromtheright & zcolumns==n | !fromtheright & zrows==n, "RMatrixBDMultiplyByP: incorrect Z size!");
        
        //
        // init
        //
        mx = Math.Max(m, n);
        mx = Math.Max(mx, zrows);
        mx = Math.Max(mx, zcolumns);
        v = new double[mx+1];
        work = new double[mx+1];
        v = new double[mx+1];
        work = new double[mx+1];
        if( m>=n )
        {
            
            //
            // setup
            //
            if( fromtheright )
            {
                i1 = n-2;
                i2 = 0;
                istep = -1;
            }
            else
            {
                i1 = 0;
                i2 = n-2;
                istep = +1;
            }
            if( !dotranspose )
            {
                i = i1;
                i1 = i2;
                i2 = i;
                istep = -istep;
            }
            
            //
            // Process
            //
            if( n-1>0 )
            {
                i = i1;
                do
                {
                    i1_ = (i+1) - (1);
                    for(i_=1; i_<=n-1-i;i_++)
                    {
                        v[i_] = qp[i,i_+i1_];
                    }
                    v[1] = 1;
                    if( fromtheright )
                    {
                        reflections.applyreflectionfromtheright(ref z, taup[i], ref v, 0, zrows-1, i+1, n-1, ref work);
                    }
                    else
                    {
                        reflections.applyreflectionfromtheleft(ref z, taup[i], ref v, i+1, n-1, 0, zcolumns-1, ref work);
                    }
                    i = i+istep;
                }
                while( i!=i2+istep );
            }
        }
        else
        {
            
            //
            // setup
            //
            if( fromtheright )
            {
                i1 = m-1;
                i2 = 0;
                istep = -1;
            }
            else
            {
                i1 = 0;
                i2 = m-1;
                istep = +1;
            }
            if( !dotranspose )
            {
                i = i1;
                i1 = i2;
                i2 = i;
                istep = -istep;
            }
            
            //
            // Process
            //
            i = i1;
            do
            {
                i1_ = (i) - (1);
                for(i_=1; i_<=n-i;i_++)
                {
                    v[i_] = qp[i,i_+i1_];
                }
                v[1] = 1;
                if( fromtheright )
                {
                    reflections.applyreflectionfromtheright(ref z, taup[i], ref v, 0, zrows-1, i, n-1, ref work);
                }
                else
                {
                    reflections.applyreflectionfromtheleft(ref z, taup[i], ref v, i, n-1, 0, zcolumns-1, ref work);
                }
                i = i+istep;
            }
            while( i!=i2+istep );
        }
    }


    /*************************************************************************
    Unpacking of the main and secondary diagonals of bidiagonal decomposition
    of matrix A.

    Input parameters:
        B   -   output of RMatrixBD subroutine.
        M   -   number of rows in matrix B.
        N   -   number of columns in matrix B.

    Output parameters:
        IsUpper -   True, if the matrix is upper bidiagonal.
                    otherwise IsUpper is False.
        D       -   the main diagonal.
                    Array whose index ranges within [0..Min(M,N)-1].
        E       -   the secondary diagonal (upper or lower, depending on
                    the value of IsUpper).
                    Array index ranges within [0..Min(M,N)-1], the last
                    element is not used.

      -- ALGLIB --
         Copyright 2005-2007 by Bochkanov Sergey
    *************************************************************************/
    public static void rmatrixbdunpackdiagonals(ref double[,] b,
        int m,
        int n,
        ref bool isupper,
        ref double[] d,
        ref double[] e)
    {
        int i = 0;

        isupper = m>=n;
        if( m<=0 | n<=0 )
        {
            return;
        }
        if( isupper )
        {
            d = new double[n-1+1];
            e = new double[n-1+1];
            for(i=0; i<=n-2; i++)
            {
                d[i] = b[i,i];
                e[i] = b[i,i+1];
            }
            d[n-1] = b[n-1,n-1];
        }
        else
        {
            d = new double[m-1+1];
            e = new double[m-1+1];
            for(i=0; i<=m-2; i++)
            {
                d[i] = b[i,i];
                e[i] = b[i+1,i];
            }
            d[m-1] = b[m-1,m-1];
        }
    }


    /*************************************************************************
    Obsolete 1-based subroutine.
    See RMatrixBD for 0-based replacement.
    *************************************************************************/
    public static void tobidiagonal(ref double[,] a,
        int m,
        int n,
        ref double[] tauq,
        ref double[] taup)
    {
        double[] work = new double[0];
        double[] t = new double[0];
        int minmn = 0;
        int maxmn = 0;
        int i = 0;
        double ltau = 0;
        int mmip1 = 0;
        int nmi = 0;
        int ip1 = 0;
        int nmip1 = 0;
        int mmi = 0;
        int i_ = 0;
        int i1_ = 0;

        minmn = Math.Min(m, n);
        maxmn = Math.Max(m, n);
        work = new double[maxmn+1];
        t = new double[maxmn+1];
        taup = new double[minmn+1];
        tauq = new double[minmn+1];
        if( m>=n )
        {
            
            //
            // Reduce to upper bidiagonal form
            //
            for(i=1; i<=n; i++)
            {
                
                //
                // Generate elementary reflector H(i) to annihilate A(i+1:m,i)
                //
                mmip1 = m-i+1;
                i1_ = (i) - (1);
                for(i_=1; i_<=mmip1;i_++)
                {
                    t[i_] = a[i_+i1_,i];
                }
                reflections.generatereflection(ref t, mmip1, ref ltau);
                tauq[i] = ltau;
                i1_ = (1) - (i);
                for(i_=i; i_<=m;i_++)
                {
                    a[i_,i] = t[i_+i1_];
                }
                t[1] = 1;
                
                //
                // Apply H(i) to A(i:m,i+1:n) from the left
                //
                reflections.applyreflectionfromtheleft(ref a, ltau, ref t, i, m, i+1, n, ref work);
                if( i<n )
                {
                    
                    //
                    // Generate elementary reflector G(i) to annihilate
                    // A(i,i+2:n)
                    //
                    nmi = n-i;
                    ip1 = i+1;
                    i1_ = (ip1) - (1);
                    for(i_=1; i_<=nmi;i_++)
                    {
                        t[i_] = a[i,i_+i1_];
                    }
                    reflections.generatereflection(ref t, nmi, ref ltau);
                    taup[i] = ltau;
                    i1_ = (1) - (ip1);
                    for(i_=ip1; i_<=n;i_++)
                    {
                        a[i,i_] = t[i_+i1_];
                    }
                    t[1] = 1;
                    
                    //
                    // Apply G(i) to A(i+1:m,i+1:n) from the right
                    //
                    reflections.applyreflectionfromtheright(ref a, ltau, ref t, i+1, m, i+1, n, ref work);
                }
                else
                {
                    taup[i] = 0;
                }
            }
        }
        else
        {
            
            //
            // Reduce to lower bidiagonal form
            //
            for(i=1; i<=m; i++)
            {
                
                //
                // Generate elementary reflector G(i) to annihilate A(i,i+1:n)
                //
                nmip1 = n-i+1;
                i1_ = (i) - (1);
                for(i_=1; i_<=nmip1;i_++)
                {
                    t[i_] = a[i,i_+i1_];
                }
                reflections.generatereflection(ref t, nmip1, ref ltau);
                taup[i] = ltau;
                i1_ = (1) - (i);
                for(i_=i; i_<=n;i_++)
                {
                    a[i,i_] = t[i_+i1_];
                }
                t[1] = 1;
                
                //
                // Apply G(i) to A(i+1:m,i:n) from the right
                //
                reflections.applyreflectionfromtheright(ref a, ltau, ref t, i+1, m, i, n, ref work);
                if( i<m )
                {
                    
                    //
                    // Generate elementary reflector H(i) to annihilate
                    // A(i+2:m,i)
                    //
                    mmi = m-i;
                    ip1 = i+1;
                    i1_ = (ip1) - (1);
                    for(i_=1; i_<=mmi;i_++)
                    {
                        t[i_] = a[i_+i1_,i];
                    }
                    reflections.generatereflection(ref t, mmi, ref ltau);
                    tauq[i] = ltau;
                    i1_ = (1) - (ip1);
                    for(i_=ip1; i_<=m;i_++)
                    {
                        a[i_,i] = t[i_+i1_];
                    }
                    t[1] = 1;
                    
                    //
                    // Apply H(i) to A(i+1:m,i+1:n) from the left
                    //
                    reflections.applyreflectionfromtheleft(ref a, ltau, ref t, i+1, m, i+1, n, ref work);
                }
                else
                {
                    tauq[i] = 0;
                }
            }
        }
    }


    /*************************************************************************
    Obsolete 1-based subroutine.
    See RMatrixBDUnpackQ for 0-based replacement.
    *************************************************************************/
    public static void unpackqfrombidiagonal(ref double[,] qp,
        int m,
        int n,
        ref double[] tauq,
        int qcolumns,
        ref double[,] q)
    {
        int i = 0;
        int j = 0;
        int ip1 = 0;
        double[] v = new double[0];
        double[] work = new double[0];
        int vm = 0;
        int i_ = 0;
        int i1_ = 0;

        System.Diagnostics.Debug.Assert(qcolumns<=m, "UnpackQFromBidiagonal: QColumns>M!");
        if( m==0 | n==0 | qcolumns==0 )
        {
            return;
        }
        
        //
        // init
        //
        q = new double[m+1, qcolumns+1];
        v = new double[m+1];
        work = new double[qcolumns+1];
        
        //
        // prepare Q
        //
        for(i=1; i<=m; i++)
        {
            for(j=1; j<=qcolumns; j++)
            {
                if( i==j )
                {
                    q[i,j] = 1;
                }
                else
                {
                    q[i,j] = 0;
                }
            }
        }
        if( m>=n )
        {
            for(i=Math.Min(n, qcolumns); i>=1; i--)
            {
                vm = m-i+1;
                i1_ = (i) - (1);
                for(i_=1; i_<=vm;i_++)
                {
                    v[i_] = qp[i_+i1_,i];
                }
                v[1] = 1;
                reflections.applyreflectionfromtheleft(ref q, tauq[i], ref v, i, m, 1, qcolumns, ref work);
            }
        }
        else
        {
            for(i=Math.Min(m-1, qcolumns-1); i>=1; i--)
            {
                vm = m-i;
                ip1 = i+1;
                i1_ = (ip1) - (1);
                for(i_=1; i_<=vm;i_++)
                {
                    v[i_] = qp[i_+i1_,i];
                }
                v[1] = 1;
                reflections.applyreflectionfromtheleft(ref q, tauq[i], ref v, i+1, m, 1, qcolumns, ref work);
            }
        }
    }


    /*************************************************************************
    Obsolete 1-based subroutine.
    See RMatrixBDMultiplyByQ for 0-based replacement.
    *************************************************************************/
    public static void multiplybyqfrombidiagonal(ref double[,] qp,
        int m,
        int n,
        ref double[] tauq,
        ref double[,] z,
        int zrows,
        int zcolumns,
        bool fromtheright,
        bool dotranspose)
    {
        int i = 0;
        int ip1 = 0;
        int i1 = 0;
        int i2 = 0;
        int istep = 0;
        double[] v = new double[0];
        double[] work = new double[0];
        int vm = 0;
        int mx = 0;
        int i_ = 0;
        int i1_ = 0;

        if( m<=0 | n<=0 | zrows<=0 | zcolumns<=0 )
        {
            return;
        }
        System.Diagnostics.Debug.Assert(fromtheright & zcolumns==m | !fromtheright & zrows==m, "MultiplyByQFromBidiagonal: incorrect Z size!");
        
        //
        // init
        //
        mx = Math.Max(m, n);
        mx = Math.Max(mx, zrows);
        mx = Math.Max(mx, zcolumns);
        v = new double[mx+1];
        work = new double[mx+1];
        if( m>=n )
        {
            
            //
            // setup
            //
            if( fromtheright )
            {
                i1 = 1;
                i2 = n;
                istep = +1;
            }
            else
            {
                i1 = n;
                i2 = 1;
                istep = -1;
            }
            if( dotranspose )
            {
                i = i1;
                i1 = i2;
                i2 = i;
                istep = -istep;
            }
            
            //
            // Process
            //
            i = i1;
            do
            {
                vm = m-i+1;
                i1_ = (i) - (1);
                for(i_=1; i_<=vm;i_++)
                {
                    v[i_] = qp[i_+i1_,i];
                }
                v[1] = 1;
                if( fromtheright )
                {
                    reflections.applyreflectionfromtheright(ref z, tauq[i], ref v, 1, zrows, i, m, ref work);
                }
                else
                {
                    reflections.applyreflectionfromtheleft(ref z, tauq[i], ref v, i, m, 1, zcolumns, ref work);
                }
                i = i+istep;
            }
            while( i!=i2+istep );
        }
        else
        {
            
            //
            // setup
            //
            if( fromtheright )
            {
                i1 = 1;
                i2 = m-1;
                istep = +1;
            }
            else
            {
                i1 = m-1;
                i2 = 1;
                istep = -1;
            }
            if( dotranspose )
            {
                i = i1;
                i1 = i2;
                i2 = i;
                istep = -istep;
            }
            
            //
            // Process
            //
            if( m-1>0 )
            {
                i = i1;
                do
                {
                    vm = m-i;
                    ip1 = i+1;
                    i1_ = (ip1) - (1);
                    for(i_=1; i_<=vm;i_++)
                    {
                        v[i_] = qp[i_+i1_,i];
                    }
                    v[1] = 1;
                    if( fromtheright )
                    {
                        reflections.applyreflectionfromtheright(ref z, tauq[i], ref v, 1, zrows, i+1, m, ref work);
                    }
                    else
                    {
                        reflections.applyreflectionfromtheleft(ref z, tauq[i], ref v, i+1, m, 1, zcolumns, ref work);
                    }
                    i = i+istep;
                }
                while( i!=i2+istep );
            }
        }
    }


    /*************************************************************************
    Obsolete 1-based subroutine.
    See RMatrixBDUnpackPT for 0-based replacement.
    *************************************************************************/
    public static void unpackptfrombidiagonal(ref double[,] qp,
        int m,
        int n,
        ref double[] taup,
        int ptrows,
        ref double[,] pt)
    {
        int i = 0;
        int j = 0;
        int ip1 = 0;
        double[] v = new double[0];
        double[] work = new double[0];
        int vm = 0;
        int i_ = 0;
        int i1_ = 0;

        System.Diagnostics.Debug.Assert(ptrows<=n, "UnpackPTFromBidiagonal: PTRows>N!");
        if( m==0 | n==0 | ptrows==0 )
        {
            return;
        }
        
        //
        // init
        //
        pt = new double[ptrows+1, n+1];
        v = new double[n+1];
        work = new double[ptrows+1];
        
        //
        // prepare PT
        //
        for(i=1; i<=ptrows; i++)
        {
            for(j=1; j<=n; j++)
            {
                if( i==j )
                {
                    pt[i,j] = 1;
                }
                else
                {
                    pt[i,j] = 0;
                }
            }
        }
        if( m>=n )
        {
            for(i=Math.Min(n-1, ptrows-1); i>=1; i--)
            {
                vm = n-i;
                ip1 = i+1;
                i1_ = (ip1) - (1);
                for(i_=1; i_<=vm;i_++)
                {
                    v[i_] = qp[i,i_+i1_];
                }
                v[1] = 1;
                reflections.applyreflectionfromtheright(ref pt, taup[i], ref v, 1, ptrows, i+1, n, ref work);
            }
        }
        else
        {
            for(i=Math.Min(m, ptrows); i>=1; i--)
            {
                vm = n-i+1;
                i1_ = (i) - (1);
                for(i_=1; i_<=vm;i_++)
                {
                    v[i_] = qp[i,i_+i1_];
                }
                v[1] = 1;
                reflections.applyreflectionfromtheright(ref pt, taup[i], ref v, 1, ptrows, i, n, ref work);
            }
        }
    }


    /*************************************************************************
    Obsolete 1-based subroutine.
    See RMatrixBDMultiplyByP for 0-based replacement.
    *************************************************************************/
    public static void multiplybypfrombidiagonal(ref double[,] qp,
        int m,
        int n,
        ref double[] taup,
        ref double[,] z,
        int zrows,
        int zcolumns,
        bool fromtheright,
        bool dotranspose)
    {
        int i = 0;
        int ip1 = 0;
        double[] v = new double[0];
        double[] work = new double[0];
        int vm = 0;
        int mx = 0;
        int i1 = 0;
        int i2 = 0;
        int istep = 0;
        int i_ = 0;
        int i1_ = 0;

        if( m<=0 | n<=0 | zrows<=0 | zcolumns<=0 )
        {
            return;
        }
        System.Diagnostics.Debug.Assert(fromtheright & zcolumns==n | !fromtheright & zrows==n, "MultiplyByQFromBidiagonal: incorrect Z size!");
        
        //
        // init
        //
        mx = Math.Max(m, n);
        mx = Math.Max(mx, zrows);
        mx = Math.Max(mx, zcolumns);
        v = new double[mx+1];
        work = new double[mx+1];
        v = new double[mx+1];
        work = new double[mx+1];
        if( m>=n )
        {
            
            //
            // setup
            //
            if( fromtheright )
            {
                i1 = n-1;
                i2 = 1;
                istep = -1;
            }
            else
            {
                i1 = 1;
                i2 = n-1;
                istep = +1;
            }
            if( !dotranspose )
            {
                i = i1;
                i1 = i2;
                i2 = i;
                istep = -istep;
            }
            
            //
            // Process
            //
            if( n-1>0 )
            {
                i = i1;
                do
                {
                    vm = n-i;
                    ip1 = i+1;
                    i1_ = (ip1) - (1);
                    for(i_=1; i_<=vm;i_++)
                    {
                        v[i_] = qp[i,i_+i1_];
                    }
                    v[1] = 1;
                    if( fromtheright )
                    {
                        reflections.applyreflectionfromtheright(ref z, taup[i], ref v, 1, zrows, i+1, n, ref work);
                    }
                    else
                    {
                        reflections.applyreflectionfromtheleft(ref z, taup[i], ref v, i+1, n, 1, zcolumns, ref work);
                    }
                    i = i+istep;
                }
                while( i!=i2+istep );
            }
        }
        else
        {
            
            //
            // setup
            //
            if( fromtheright )
            {
                i1 = m;
                i2 = 1;
                istep = -1;
            }
            else
            {
                i1 = 1;
                i2 = m;
                istep = +1;
            }
            if( !dotranspose )
            {
                i = i1;
                i1 = i2;
                i2 = i;
                istep = -istep;
            }
            
            //
            // Process
            //
            i = i1;
            do
            {
                vm = n-i+1;
                i1_ = (i) - (1);
                for(i_=1; i_<=vm;i_++)
                {
                    v[i_] = qp[i,i_+i1_];
                }
                v[1] = 1;
                if( fromtheright )
                {
                    reflections.applyreflectionfromtheright(ref z, taup[i], ref v, 1, zrows, i, n, ref work);
                }
                else
                {
                    reflections.applyreflectionfromtheleft(ref z, taup[i], ref v, i, n, 1, zcolumns, ref work);
                }
                i = i+istep;
            }
            while( i!=i2+istep );
        }
    }


    /*************************************************************************
    Obsolete 1-based subroutine.
    See RMatrixBDUnpackDiagonals for 0-based replacement.
    *************************************************************************/
    public static void unpackdiagonalsfrombidiagonal(ref double[,] b,
        int m,
        int n,
        ref bool isupper,
        ref double[] d,
        ref double[] e)
    {
        int i = 0;

        isupper = m>=n;
        if( m==0 | n==0 )
        {
            return;
        }
        if( isupper )
        {
            d = new double[n+1];
            e = new double[n+1];
            for(i=1; i<=n-1; i++)
            {
                d[i] = b[i,i];
                e[i] = b[i,i+1];
            }
            d[n] = b[n,n];
        }
        else
        {
            d = new double[m+1];
            e = new double[m+1];
            for(i=1; i<=m-1; i++)
            {
                d[i] = b[i,i];
                e[i] = b[i+1,i];
            }
            d[m] = b[m,m];
        }
    }
}