# include <cstdlib>
# include <iostream>
# include <iomanip>
# include <cmath>
# include <ctime>

using namespace std;

int main ( );
void daxpy ( int n, double da, double dx[], int incx, double dy[], int incy );
double ddot ( int n, double dx[], int incx, double dy[], int incy );
double dnrm2 ( int n, double x[], int incx );
void drot ( int n, double x[], int incx, double y[], int incy, double c,
  double s );
void drotg ( double *sa, double *sb, double *c, double *s );
void dscal ( int n, double sa, double x[], int incx );
int dsvdc ( double a[], int lda, int m, int n, double s[], double e[], 
  double u[], int ldu, double v[], int ldv, double work[], int job );
void dswap ( int n, double x[], int incx, double y[], int incy );
int i4_max ( int i1, int i2 );
int i4_min ( int i1, int i2 );
double r8_abs ( double x );
double r8_max ( double x, double y );
double r8_sign ( double x );
void r8mat_print ( int m, int n, double a[], string title );
void r8mat_print_some ( int m, int n, double a[], int ilo, int jlo, int ihi,
  int jhi, string title );
double *r8mat_transpose_new ( int m, int n, double a[] );
double *r8mat_uniform_01_new ( int m, int n, int *seed );
void svd_truncated_u ( int m, int n, double a[], double un[], double sn[], 
  double v[] );
void svd_truncated_u_test ( int m, int n );
void svd_truncated_v ( int m, int n, double a[], double u[], double sm[], 
  double vm[] );
void svd_truncated_v_test ( int m, int n );
void timestamp ( );

//****************************************************************************80

int main ( )

//****************************************************************************80
//
//  Purpose:
//
//    MAIN is the main program for SVD_TRUNCATED.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    19 March 2012
//
//  Author:
//
//    John Burkardt
//
{
  int m;
  int n;

  timestamp ( );
  cout << "\n";
  cout << "SVD_TRUNCATED\n";
  cout << "  C++ version\n";
  cout << "  Demonstrate the use of the truncated or economy-size\n";
  cout << "  Singular Value Decomposition (SVD) for cases where\n";
  cout << "  the sizes of M and N are very different.\n";

  m = 4;
  n = 3;
  svd_truncated_u_test ( m, n );

  m = 3;
  n = 4;
  svd_truncated_v_test ( m, n );

  cout << "\n";
  cout << "SVD_TRUNCATED\n";
  cout << "  Normal end of execution.\n";
  cout << "\n";
  timestamp ( );

  return 0;
}
//****************************************************************************80

void daxpy ( int n, double da, double dx[], int incx, double dy[], int incy )

//****************************************************************************80
//
//  Purpose:
//
//    DAXPY computes constant times a vector plus a vector.
//
//  Discussion:
//
//    This routine uses unrolled loops for increments equal to one.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    02 May 2005
//
//  Author:
//
//    C++ version by John Burkardt
//
//  Reference:
//
//    Jack Dongarra, Jim Bunch, Cleve Moler, Pete Stewart,
//    LINPACK User's Guide,
//    SIAM, 1979,
//    ISBN13: 978-0-898711-72-1,
//    LC: QA214.L56.
//
//    Charles Lawson, Richard Hanson, David Kincaid, Fred Krogh,
//    Basic Linear Algebra Subprograms for Fortran Usage,
//    Algorithm 539,
//    ACM Transactions on Mathematical Software,
//    Volume 5, Number 3, September 1979, pages 308-323.
//
//  Parameters:
//
//    Input, int N, the number of elements in DX and DY.
//
//    Input, double DA, the multiplier of DX.
//
//    Input, double DX[*], the first vector.
//
//    Input, int INCX, the increment between successive entries of DX.
//
//    Input/output, double DY[*], the second vector.
//    On output, DY[*] has been replaced by DY[*] + DA * DX[*].
//
//    Input, int INCY, the increment between successive entries of DY.
//
{
  int i;
  int ix;
  int iy;
  int m;

  if ( n <= 0 )
  {
    return;
  }

  if ( da == 0.0 )
  {
    return;
  }
//
//  Code for unequal increments or equal increments
//  not equal to 1.
//
  if ( incx != 1 || incy != 1 )
  {
    if ( 0 <= incx )
    {
      ix = 0;
    }
    else
    {
      ix = ( - n + 1 ) * incx;
    }

    if ( 0 <= incy )
    {
      iy = 0;
    }
    else
    {
      iy = ( - n + 1 ) * incy;
    }

    for ( i = 0; i < n; i++ )
    {
      dy[iy] = dy[iy] + da * dx[ix];
      ix = ix + incx;
      iy = iy + incy;
    }
  }
//
//  Code for both increments equal to 1.
//
  else
  {
    m = n % 4;

    for ( i = 0; i < m; i++ )
    {
      dy[i] = dy[i] + da * dx[i];
    }

    for ( i = m; i < n; i = i + 4 )
    {
      dy[i  ] = dy[i  ] + da * dx[i  ];
      dy[i+1] = dy[i+1] + da * dx[i+1];
      dy[i+2] = dy[i+2] + da * dx[i+2];
      dy[i+3] = dy[i+3] + da * dx[i+3];
    }

  }

  return;
}
//****************************************************************************80

double ddot ( int n, double dx[], int incx, double dy[], int incy )

//****************************************************************************80
//
//  Purpose:
//
//    DDOT forms the dot product of two vectors.
//
//  Discussion:
//
//    This routine uses unrolled loops for increments equal to one.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    02 May 2005
//
//  Author:
//
//    C++ version by John Burkardt
//
//  Reference:
//
//    Jack Dongarra, Jim Bunch, Cleve Moler, Pete Stewart,
//    LINPACK User's Guide,
//    SIAM, 1979,
//    ISBN13: 978-0-898711-72-1,
//    LC: QA214.L56.
//
//    Charles Lawson, Richard Hanson, David Kincaid, Fred Krogh,
//    Basic Linear Algebra Subprograms for Fortran Usage,
//    Algorithm 539,
//    ACM Transactions on Mathematical Software,
//    Volume 5, Number 3, September 1979, pages 308-323.
//
//  Parameters:
//
//    Input, int N, the number of entries in the vectors.
//
//    Input, double DX[*], the first vector.
//
//    Input, int INCX, the increment between successive entries in DX.
//
//    Input, double DY[*], the second vector.
//
//    Input, int INCY, the increment between successive entries in DY.
//
//    Output, double DDOT, the sum of the product of the corresponding
//    entries of DX and DY.
//
{
  double dtemp;
  int i;
  int ix;
  int iy;
  int m;

  dtemp = 0.0;

  if ( n <= 0 )
  {
    return dtemp;
  }
//
//  Code for unequal increments or equal increments
//  not equal to 1.
//
  if ( incx != 1 || incy != 1 )
  {
    if ( 0 <= incx )
    {
      ix = 0;
    }
    else
    {
      ix = ( - n + 1 ) * incx;
    }

    if ( 0 <= incy )
    {
      iy = 0;
    }
    else
    {
      iy = ( - n + 1 ) * incy;
    }

    for ( i = 0; i < n; i++ )
    {
      dtemp = dtemp + dx[ix] * dy[iy];
      ix = ix + incx;
      iy = iy + incy;
    }
  }
//
//  Code for both increments equal to 1.
//
  else
  {
    m = n % 5;

    for ( i = 0; i < m; i++ )
    {
      dtemp = dtemp + dx[i] * dy[i];
    }

    for ( i = m; i < n; i = i + 5 )
    {
      dtemp = dtemp + dx[i  ] * dy[i  ]
                    + dx[i+1] * dy[i+1]
                    + dx[i+2] * dy[i+2]
                    + dx[i+3] * dy[i+3]
                    + dx[i+4] * dy[i+4];
    }

  }

  return dtemp;
}
//****************************************************************************80

double dnrm2 ( int n, double x[], int incx )

//****************************************************************************80
//
//  Purpose:
//
//    DNRM2 returns the euclidean norm of a vector.
//
//  Discussion:
//
//     DNRM2 ( X ) = sqrt ( X' * X )
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    02 May 2005
//
//  Author:
//
//    C++ version by John Burkardt
//
//  Reference:
//
//    Jack Dongarra, Jim Bunch, Cleve Moler, Pete Stewart,
//    LINPACK User's Guide,
//    SIAM, 1979,
//    ISBN13: 978-0-898711-72-1,
//    LC: QA214.L56.
//
//    Charles Lawson, Richard Hanson, David Kincaid, Fred Krogh,
//    Basic Linear Algebra Subprograms for Fortran Usage,
//    Algorithm 539,
//    ACM Transactions on Mathematical Software,
//    Volume 5, Number 3, September 1979, pages 308-323.
//
//  Parameters:
//
//    Input, int N, the number of entries in the vector.
//
//    Input, double X[*], the vector whose norm is to be computed.
//
//    Input, int INCX, the increment between successive entries of X.
//
//    Output, double DNRM2, the Euclidean norm of X.
//
{
  double absxi;
  int i;
  int ix;
  double norm;
  double scale;
  double ssq;
  double value;

  if ( n < 1 || incx < 1 )
  {
    norm = 0.0;
  }
  else if ( n == 1 )
  {
    norm = r8_abs ( x[0] );
  }
  else
  {
    scale = 0.0;
    ssq = 1.0;
    ix = 0;

    for ( i = 0; i < n; i++ )
    {
      if ( x[ix] != 0.0 )
      {
        absxi = r8_abs ( x[ix] );
        if ( scale < absxi )
        {
          ssq = 1.0 + ssq * ( scale / absxi ) * ( scale / absxi );
          scale = absxi;
        }
        else
        {
          ssq = ssq + ( absxi / scale ) * ( absxi / scale );
        }
      }
      ix = ix + incx;
    }

    norm  = scale * sqrt ( ssq );
  }

  return norm;
}
//****************************************************************************80

void drot ( int n, double x[], int incx, double y[], int incy, double c,
  double s )

//****************************************************************************80
//
//  Purpose:
//
//    DROT applies a plane rotation.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    02 May 2005
//
//  Author:
//
//    C++ version by John Burkardt
//
//  Reference:
//
//    Jack Dongarra, Jim Bunch, Cleve Moler, Pete Stewart,
//    LINPACK User's Guide,
//    SIAM, 1979,
//    ISBN13: 978-0-898711-72-1,
//    LC: QA214.L56.
//
//    Charles Lawson, Richard Hanson, David Kincaid, Fred Krogh,
//    Basic Linear Algebra Subprograms for Fortran Usage,
//    Algorithm 539,
//    ACM Transactions on Mathematical Software,
//    Volume 5, Number 3, September 1979, pages 308-323.
//
//  Parameters:
//
//    Input, int N, the number of entries in the vectors.
//
//    Input/output, double X[*], one of the vectors to be rotated.
//
//    Input, int INCX, the increment between successive entries of X.
//
//    Input/output, double Y[*], one of the vectors to be rotated.
//
//    Input, int INCY, the increment between successive elements of Y.
//
//    Input, double C, S, parameters (presumably the cosine and
//    sine of some angle) that define a plane rotation.
//
{
  int i;
  int ix;
  int iy;
  double stemp;

  if ( n <= 0 )
  {
  }
  else if ( incx == 1 && incy == 1 )
  {
    for ( i = 0; i < n; i++ )
    {
      stemp = c * x[i] + s * y[i];
      y[i]  = c * y[i] - s * x[i];
      x[i]  = stemp;
    }
  }
  else
  {
    if ( 0 <= incx )
    {
      ix = 0;
    }
    else
    {
      ix = ( - n + 1 ) * incx;
    }

    if ( 0 <= incy )
    {
      iy = 0;
    }
    else
    {
      iy = ( - n + 1 ) * incy;
    }

    for ( i = 0; i < n; i++ )
    {
      stemp = c * x[ix] + s * y[iy];
      y[iy] = c * y[iy] - s * x[ix];
      x[ix] = stemp;
      ix = ix + incx;
      iy = iy + incy;
    }

  }

  return;
}
//****************************************************************************80

void drotg ( double *sa, double *sb, double *c, double *s )

//****************************************************************************80
//
//  Purpose:
//
//    DROTG constructs a Givens plane rotation.
//
//  Discussion:
//
//    Given values A and B, this routine computes
//
//    SIGMA = sign ( A ) if abs ( A ) >  abs ( B )
//          = sign ( B ) if abs ( A ) <= abs ( B );
//
//    R     = SIGMA * ( A * A + B * B );
//
//    C = A / R if R is not 0
//      = 1     if R is 0;
//
//    S = B / R if R is not 0,
//        0     if R is 0.
//
//    The computed numbers then satisfy the equation
//
//    (  C  S ) ( A ) = ( R )
//    ( -S  C ) ( B ) = ( 0 )
//
//    The routine also computes
//
//    Z = S     if abs ( A ) > abs ( B ),
//      = 1 / C if abs ( A ) <= abs ( B ) and C is not 0,
//      = 1     if C is 0.
//
//    The single value Z encodes C and S, and hence the rotation:
//
//    If Z = 1, set C = 0 and S = 1;
//    If abs ( Z ) < 1, set C = sqrt ( 1 - Z * Z ) and S = Z;
//    if abs ( Z ) > 1, set C = 1/ Z and S = sqrt ( 1 - C * C );
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    15 May 2006
//
//  Author:
//
//    C++ version by John Burkardt
//
//  Reference:
//
//    Jack Dongarra, Jim Bunch, Cleve Moler, Pete Stewart,
//    LINPACK User's Guide,
//    SIAM, 1979,
//    ISBN13: 978-0-898711-72-1,
//    LC: QA214.L56.
//
//    Charles Lawson, Richard Hanson, David Kincaid, Fred Krogh,
//    Basic Linear Algebra Subprograms for Fortran Usage,
//    Algorithm 539,
//    ACM Transactions on Mathematical Software,
//    Volume 5, Number 3, September 1979, pages 308-323.
//
//  Parameters:
//
//    Input/output, double *SA, *SB,  On input, SA and SB are the values
//    A and B.  On output, SA is overwritten with R, and SB is
//    overwritten with Z.
//
//    Output, double *C, *S, the cosine and sine of the Givens rotation.
//
{
  double r;
  double roe;
  double scale;
  double z;

  if ( r8_abs ( *sb ) < r8_abs ( *sa ) )
  {
    roe = *sa;
  }
  else
  {
    roe = *sb;
  }

  scale = r8_abs ( *sa ) + r8_abs ( *sb );

  if ( scale == 0.0 )
  {
    *c = 1.0;
    *s = 0.0;
    r = 0.0;
  }
  else
  {
    r = scale * sqrt ( ( *sa / scale ) * ( *sa / scale )
                     + ( *sb / scale ) * ( *sb / scale ) );
    r = r8_sign ( roe ) * r;
    *c = *sa / r;
    *s = *sb / r;
  }

  if ( 0.0 < r8_abs ( *c ) && r8_abs ( *c ) <= *s )
  {
    z = 1.0 / *c;
  }
  else
  {
    z = *s;
  }

  *sa = r;
  *sb = z;

  return;
}
//****************************************************************************80

void dscal ( int n, double sa, double x[], int incx )

//****************************************************************************80
//
//  Purpose:
//
//    DSCAL scales a vector by a constant.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    02 May 2005
//
//  Author:
//
//    C++ version by John Burkardt
//
//  Reference:
//
//    Jack Dongarra, Jim Bunch, Cleve Moler, Pete Stewart,
//    LINPACK User's Guide,
//    SIAM, 1979,
//    ISBN13: 978-0-898711-72-1,
//    LC: QA214.L56.
//
//    Charles Lawson, Richard Hanson, David Kincaid, Fred Krogh,
//    Basic Linear Algebra Subprograms for Fortran Usage,
//    Algorithm 539,
//    ACM Transactions on Mathematical Software,
//    Volume 5, Number 3, September 1979, pages 308-323.
//
//  Parameters:
//
//    Input, int N, the number of entries in the vector.
//
//    Input, double SA, the multiplier.
//
//    Input/output, double X[*], the vector to be scaled.
//
//    Input, int INCX, the increment between successive entries of X.
//
{
  int i;
  int ix;
  int m;

  if ( n <= 0 )
  {
  }
  else if ( incx == 1 )
  {
    m = n % 5;

    for ( i = 0; i < m; i++ )
    {
      x[i] = sa * x[i];
    }

    for ( i = m; i < n; i = i + 5 )
    {
      x[i]   = sa * x[i];
      x[i+1] = sa * x[i+1];
      x[i+2] = sa * x[i+2];
      x[i+3] = sa * x[i+3];
      x[i+4] = sa * x[i+4];
    }
  }
  else
  {
    if ( 0 <= incx )
    {
      ix = 0;
    }
    else
    {
      ix = ( - n + 1 ) * incx;
    }

    for ( i = 0; i < n; i++ )
    {
      x[ix] = sa * x[ix];
      ix = ix + incx;
    }

  }

  return;
}
//****************************************************************************80

int dsvdc ( double a[], int lda, int m, int n, double s[], double e[], 
  double u[], int ldu, double v[], int ldv, double work[], int job )

//****************************************************************************80
//
//  Purpose:
//
//    DSVDC computes the singular value decomposition of a real rectangular matrix.
//
//  Discussion:
//
//    This routine reduces an M by N matrix A to diagonal form by orthogonal
//    transformations U and V.  The diagonal elements S(I) are the singular
//    values of A.  The columns of U are the corresponding left singular
//    vectors, and the columns of V the right singular vectors.
//
//    The form of the singular value decomposition is then
//
//      A(MxN) = U(MxM) * S(MxN) * V(NxN)'
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    03 May 2007
//
//  Author:
//
//    C++ version by John Burkardt.
//
//  Reference:
//
//    Jack Dongarra, Cleve Moler, Jim Bunch and Pete Stewart,
//    LINPACK User's Guide,
//    SIAM, (Society for Industrial and Applied Mathematics),
//    3600 University City Science Center,
//    Philadelphia, PA, 19104-2688.
//    ISBN 0-89871-172-X
//
//  Parameters:
//
//    Input/output, double A[LDA*N].  On input, the M by N matrix whose
//    singular value decomposition is to be computed.  On output, the matrix
//    has been destroyed.  Depending on the user's requests, the matrix may 
//    contain other useful information.
//
//    Input, int LDA, the leading dimension of the array A.
//    LDA must be at least M.
//
//    Input, int M, the number of rows of the matrix.
//
//    Input, int N, the number of columns of the matrix A.
//
//    Output, double S[MM], where MM = min(M+1,N).  The first
//    min(M,N) entries of S contain the singular values of A arranged in
//    descending order of magnitude.
//
//    Output, double E[MM], where MM = min(M+1,N), ordinarily contains zeros.
//    However see the discussion of INFO for exceptions.
//
//    Output, double U[LDU*K].  If JOBA = 1 then K = M;
//    if 2 <= JOBA, then K = min(M,N).  U contains the M by M matrix of left singular
//    vectors.  U is not referenced if JOBA = 0.  If M <= N or if JOBA = 2, then
//    U may be identified with A in the subroutine call.
//
//    Input, int LDU, the leading dimension of the array U.
//    LDU must be at least M.
//
//    Output, double V[LDV*N], the N by N matrix of right singular vectors.
//    V is not referenced if JOB is 0.  If N <= M, then V may be identified
//    with A in the subroutine call.
//
//    Input, int LDV, the leading dimension of the array V.
//    LDV must be at least N.
//
//    Workspace, double WORK[M].
//
//    Input, int JOB, controls the computation of the singular
//    vectors.  It has the decimal expansion AB with the following meaning:
//      A =  0, do not compute the left singular vectors.
//      A =  1, return the M left singular vectors in U.
//      A >= 2, return the first min(M,N) singular vectors in U.
//      B =  0, do not compute the right singular vectors.
//      B =  1, return the right singular vectors in V.
//
//    Output, int *DSVDC, status indicator INFO.
//    The singular values (and their corresponding singular vectors)
//    S(*INFO+1), S(*INFO+2),...,S(MN) are correct.  Here MN = min ( M, N ).
//    Thus if *INFO is 0, all the singular values and their vectors are
//    correct.  In any event, the matrix B = U' * A * V is the bidiagonal
//    matrix with the elements of S on its diagonal and the elements of E on
//    its superdiagonal.  Thus the singular values of A and B are the same.
//
{
  double b;
  double c;
  double cs;
  double el;
  double emm1;
  double f;
  double g;
  int i;
  int info;
  int iter;
  int j;
  int jobu;
  int k;
  int kase;
  int kk;
  int l;
  int ll;
  int lls;
  int ls;
  int lu;
  int maxit = 30;
  int mm;
  int mm1;
  int mn;
  int mp1;
  int nct;
  int nctp1;
  int ncu;
  int nrt;
  int nrtp1;
  double scale;
  double shift;
  double sl;
  double sm;
  double smm1;
  double sn;
  double t;
  double t1;
  double test;
  bool wantu;
  bool wantv;
  double ztest;
//
//  Determine what is to be computed.
//
  info = 0;
  wantu = false;
  wantv = false;
  jobu = ( job % 100 ) / 10;

  if ( 1 < jobu )
  {
    ncu = i4_min ( m, n );
  }
  else
  {
    ncu = m;
  }

  if ( jobu != 0 )
  {
    wantu = true;
  }

  if ( ( job % 10 ) != 0 )
  {
    wantv = true;
  }
//
//  Reduce A to bidiagonal form, storing the diagonal elements
//  in S and the super-diagonal elements in E.
//
  nct = i4_min ( m-1, n );
  nrt = i4_max ( 0, i4_min ( m, n-2 ) );
  lu = i4_max ( nct, nrt );

  for ( l = 1; l <= lu; l++ )
  {
//
//  Compute the transformation for the L-th column and
//  place the L-th diagonal in S(L).
//
    if ( l <= nct )
    {
      s[l-1] = dnrm2 ( m-l+1, a+l-1+(l-1)*lda, 1 );

      if ( s[l-1] != 0.0 )
      {
        if ( a[l-1+(l-1)*lda] != 0.0 )
        {
          s[l-1] = r8_sign ( a[l-1+(l-1)*lda] ) * r8_abs ( s[l-1] );
        }
        dscal ( m-l+1, 1.0 / s[l-1], a+l-1+(l-1)*lda, 1 );
        a[l-1+(l-1)*lda] = 1.0 + a[l-1+(l-1)*lda];
      }
      s[l-1] = -s[l-1];
    }

    for ( j = l+1; j <= n; j++ )
    {
//
//  Apply the transformation.
//
      if ( l <= nct && s[l-1] != 0.0 )
      {
        t = - ddot ( m-l+1, a+l-1+(l-1)*lda, 1, a+l-1+(j-1)*lda, 1 ) 
          / a[l-1+(l-1)*lda];
        daxpy ( m-l+1, t, a+l-1+(l-1)*lda, 1, a+l-1+(j-1)*lda, 1 );
      }
//
//  Place the L-th row of A into E for the
//  subsequent calculation of the row transformation.
//
      e[j-1] = a[l-1+(j-1)*lda];
    }
//
//  Place the transformation in U for subsequent back multiplication.
//
    if ( wantu && l <= nct )
    {
      for ( i = l; i <= m; i++ )
      {
        u[i-1+(l-1)*ldu] = a[i-1+(l-1)*lda];
      }
    }

    if ( l <= nrt )
    {
//
//  Compute the L-th row transformation and place the
//  L-th superdiagonal in E(L).
//
      e[l-1] = dnrm2 ( n-l, e+l, 1 );

      if ( e[l-1] != 0.0 )
      {
        if ( e[l] != 0.0 )
        {
          e[l-1] = r8_sign ( e[l] ) * r8_abs ( e[l-1] );
        }
        dscal ( n-l, 1.0 / e[l-1], e+l, 1 );
        e[l] = 1.0 + e[l];
      }

      e[l-1] = -e[l-1];
//
//  Apply the transformation.
//
      if ( l+1 <= m && e[l-1] != 0.0 )
      {
        for ( j = l+1; j <= m; j++ )
        {
          work[j-1] = 0.0;
        }

        for ( j = l+1; j <= n; j++ )
        {
          daxpy ( m-l, e[j-1], a+l+(j-1)*lda, 1, work+l, 1 );
        }

        for ( j = l+1; j <= n; j++ )
        {
          daxpy ( m-l, -e[j-1]/e[l], work+l, 1, a+l+(j-1)*lda, 1 );
        }
      }
//
//  Place the transformation in V for subsequent back multiplication.
//
      if ( wantv )
      {
        for ( j = l+1; j <= n; j++ )
        {
          v[j-1+(l-1)*ldv] = e[j-1];
        }
      }
    }
  }
//
//  Set up the final bidiagonal matrix of order MN.
//
  mn = i4_min ( m + 1, n );
  nctp1 = nct + 1;
  nrtp1 = nrt + 1;

  if ( nct < n )
  {
    s[nctp1-1] = a[nctp1-1+(nctp1-1)*lda];
  }

  if ( m < mn )
  {
    s[mn-1] = 0.0;
  }

  if ( nrtp1 < mn )
  {
    e[nrtp1-1] = a[nrtp1-1+(mn-1)*lda];
  }

  e[mn-1] = 0.0;
//
//  If required, generate U.
//
  if ( wantu )
  {
    for ( i = 1; i <= m; i++ )
    {
      for ( j = nctp1; j <= ncu; j++ )
      {
        u[(i-1)+(j-1)*ldu] = 0.0;
      }
    }

    for ( j = nctp1; j <= ncu; j++ )
    {
      u[j-1+(j-1)*ldu] = 1.0;
    }

    for ( ll = 1; ll <= nct; ll++ )
    {
      l = nct - ll + 1;

      if ( s[l-1] != 0.0 )
      {
        for ( j = l+1; j <= ncu; j++ )
        {
          t = - ddot ( m-l+1, u+(l-1)+(l-1)*ldu, 1, u+(l-1)+(j-1)*ldu, 1 ) 
            / u[l-1+(l-1)*ldu];
          daxpy ( m-l+1, t, u+(l-1)+(l-1)*ldu, 1, u+(l-1)+(j-1)*ldu, 1 );
        }

        dscal ( m-l+1, -1.0, u+(l-1)+(l-1)*ldu, 1 );
        u[l-1+(l-1)*ldu] = 1.0 + u[l-1+(l-1)*ldu];
        for ( i = 1; i <= l-1; i++ )
        {
          u[i-1+(l-1)*ldu] = 0.0;
        }
      }
      else
      {
        for ( i = 1; i <= m; i++ )
        {
          u[i-1+(l-1)*ldu] = 0.0;
        }
        u[l-1+(l-1)*ldu] = 1.0;
      }
    }
  }
//
//  If it is required, generate V.
//
  if ( wantv )
  {
    for ( ll = 1; ll <= n; ll++ )
    {
      l = n - ll + 1;

      if ( l <= nrt && e[l-1] != 0.0 )
      {
        for ( j = l+1; j <= n; j++ )
        {
          t = - ddot ( n-l, v+l+(l-1)*ldv, 1, v+l+(j-1)*ldv, 1 ) 
            / v[l+(l-1)*ldv];
          daxpy ( n-l, t, v+l+(l-1)*ldv, 1, v+l+(j-1)*ldv, 1 );
        }

      }
      for ( i = 1; i <= n; i++ )
      {
        v[i-1+(l-1)*ldv] = 0.0;
      }
      v[l-1+(l-1)*ldv] = 1.0;
    }
  }
//
//  Main iteration loop for the singular values.
//
  mm = mn;
  iter = 0;

  while ( 0 < mn )
  {
//
//  If too many iterations have been performed, set flag and return.
//
    if ( maxit <= iter )
    {
      info = mn;
      return info;
    }
//
//  This section of the program inspects for
//  negligible elements in the S and E arrays.
//
//  On completion the variables KASE and L are set as follows:
//
//  KASE = 1     if S(MN) and E(L-1) are negligible and L < MN
//  KASE = 2     if S(L) is negligible and L < MN
//  KASE = 3     if E(L-1) is negligible, L < MN, and
//               S(L), ..., S(MN) are not negligible (QR step).
//  KASE = 4     if E(MN-1) is negligible (convergence).
//
    for ( ll = 1; ll <= mn; ll++ )
    {
      l = mn - ll;

      if ( l == 0 )
      {
        break;
      }

      test = r8_abs ( s[l-1] ) + r8_abs ( s[l] );
      ztest = test + r8_abs ( e[l-1] );

      if ( ztest == test )
      {
        e[l-1] = 0.0;
        break;
      }
    }

    if ( l == mn - 1 )
    {
      kase = 4;
    }
    else
    {
      mp1 = mn + 1;

      for ( lls = l+1; lls <= mn+1; lls++ )
      {
        ls = mn - lls + l + 1;

        if ( ls == l )
        {
          break;
        }

        test = 0.0;
        if ( ls != mn )
        {
          test = test + r8_abs ( e[ls-1] );
        }

        if ( ls != l + 1 )
        {
          test = test + r8_abs ( e[ls-2] );
        }

        ztest = test + r8_abs ( s[ls-1] );

        if ( ztest == test )
        {
          s[ls-1] = 0.0;
          break;
        }

      }

      if ( ls == l )
      {
        kase = 3;
      }
      else if ( ls == mn )
      {
        kase = 1;
      }
      else
      {
        kase = 2;
        l = ls;
      }
    }

    l = l + 1;
//
//  Deflate negligible S(MN).
//
    if ( kase == 1 )
    {
      mm1 = mn - 1;
      f = e[mn-2];
      e[mn-2] = 0.0;

      for ( kk = 1; kk <= mm1; kk++ )
      {
        k = mm1 - kk + l;
        t1 = s[k-1];
        drotg ( &t1, &f, &cs, &sn );
        s[k-1] = t1;

        if ( k != l )
        {
          f = -sn * e[k-2];
          e[k-2] = cs * e[k-2];
        }

        if ( wantv )
        {
          drot ( n, v+0+(k-1)*ldv, 1, v+0+(mn-1)*ldv, 1, cs, sn );
        }
      }
    }
//
//  Split at negligible S(L).
//
    else if ( kase == 2 )
    {
      f = e[l-2];
      e[l-2] = 0.0;

      for ( k = l; k <= mn; k++ )
      {
        t1 = s[k-1];
        drotg ( &t1, &f, &cs, &sn );
        s[k-1] = t1;
        f = - sn * e[k-1];
        e[k-1] = cs * e[k-1];
        if ( wantu )
        {
          drot ( m, u+0+(k-1)*ldu, 1, u+0+(l-2)*ldu, 1, cs, sn );
        }
      }
    }
//
//  Perform one QR step.
//
    else if ( kase == 3 )
    {
//
//  Calculate the shift.
//
      scale = r8_max ( r8_abs ( s[mn-1] ), 
              r8_max ( r8_abs ( s[mn-2] ), 
              r8_max ( r8_abs ( e[mn-2] ), 
              r8_max ( r8_abs ( s[l-1] ), r8_abs ( e[l-1] ) ) ) ) );

      sm = s[mn-1] / scale;
      smm1 = s[mn-2] / scale;
      emm1 = e[mn-2] / scale;
      sl = s[l-1] / scale;
      el = e[l-1] / scale;
      b = ( ( smm1 + sm ) * ( smm1 - sm ) + emm1 * emm1 ) / 2.0;
      c = ( sm * emm1 ) * ( sm * emm1 );
      shift = 0.0;

      if ( b != 0.0 || c != 0.0 )
      {
        shift = sqrt ( b * b + c );
        if ( b < 0.0 )
        {
          shift = -shift;
        }
        shift = c / ( b + shift );
      }

      f = ( sl + sm ) * ( sl - sm ) - shift;
      g = sl * el;
//
//  Chase zeros.
//
      mm1 = mn - 1;

      for ( k = l; k <= mm1; k++ )
      {
        drotg ( &f, &g, &cs, &sn );

        if ( k != l )
        {
          e[k-2] = f;
        }

        f = cs * s[k-1] + sn * e[k-1];
        e[k-1] = cs * e[k-1] - sn * s[k-1];
        g = sn * s[k];
        s[k] = cs * s[k];

        if ( wantv )
        {
          drot ( n, v+0+(k-1)*ldv, 1, v+0+k*ldv, 1, cs, sn );
        }

        drotg ( &f, &g, &cs, &sn );
        s[k-1] = f;
        f = cs * e[k-1] + sn * s[k];
        s[k] = -sn * e[k-1] + cs * s[k];
        g = sn * e[k];
        e[k] = cs * e[k];

        if ( wantu && k < m )
        {
          drot ( m, u+0+(k-1)*ldu, 1, u+0+k*ldu, 1, cs, sn );
        }
      }
      e[mn-2] = f;
      iter = iter + 1;
    }
//
//  Convergence.
//
    else if ( kase == 4 )
    {
//
//  Make the singular value nonnegative.
//
      if ( s[l-1] < 0.0 )
      {
        s[l-1] = -s[l-1];
        if ( wantv )
        {
          dscal ( n, -1.0, v+0+(l-1)*ldv, 1 );
        }
      }
//
//  Order the singular value.
//
      for ( ; ; )
      {
        if ( l == mm )
        {
          break;
        }

        if ( s[l] <= s[l-1] )
        {
          break;
        }

        t = s[l-1];
        s[l-1] = s[l];
        s[l] = t;

        if ( wantv && l < n )
        {
          dswap ( n, v+0+(l-1)*ldv, 1, v+0+l*ldv, 1 );
        }

        if ( wantu && l < m )
        {
          dswap ( m, u+0+(l-1)*ldu, 1, u+0+l*ldu, 1 );
        }

        l = l + 1;
      }
      iter = 0;
      mn = mn - 1;
    }
  }

  return info;
}
//****************************************************************************80

void dswap ( int n, double x[], int incx, double y[], int incy )

//****************************************************************************80
//
//  Purpose:
//
//    DSWAP interchanges two vectors.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    02 May 2005
//
//  Author:
//
//    C++ version by John Burkardt
//
//  Reference:
//
//    Jack Dongarra, Jim Bunch, Cleve Moler, Pete Stewart,
//    LINPACK User's Guide,
//    SIAM, 1979,
//    ISBN13: 978-0-898711-72-1,
//    LC: QA214.L56.
//
//    Charles Lawson, Richard Hanson, David Kincaid, Fred Krogh,
//    Basic Linear Algebra Subprograms for Fortran Usage,
//    Algorithm 539,
//    ACM Transactions on Mathematical Software,
//    Volume 5, Number 3, September 1979, pages 308-323.
//
//  Parameters:
//
//    Input, int N, the number of entries in the vectors.
//
//    Input/output, double X[*], one of the vectors to swap.
//
//    Input, int INCX, the increment between successive entries of X.
//
//    Input/output, double Y[*], one of the vectors to swap.
//
//    Input, int INCY, the increment between successive elements of Y.
//
{
  int i;
  int ix;
  int iy;
  int m;
  double temp;

  if ( n <= 0 )
  {
  }
  else if ( incx == 1 && incy == 1 )
  {
    m = n % 3;

    for ( i = 0; i < m; i++ )
    {
      temp = x[i];
      x[i] = y[i];
      y[i] = temp;
    }

    for ( i = m; i < n; i = i + 3 )
    {
      temp = x[i];
      x[i] = y[i];
      y[i] = temp;

      temp = x[i+1];
      x[i+1] = y[i+1];
      y[i+1] = temp;

      temp = x[i+2];
      x[i+2] = y[i+2];
      y[i+2] = temp;
    }
  }
  else
  {
    if ( 0 <= incx )
    {
      ix = 0;
    }
    else
    {
      ix = ( - n + 1 ) * incx;
    }

    if ( 0 <= incy )
    {
      iy = 0;
    }
    else
    {
      iy = ( - n + 1 ) * incy;
    }

    for ( i = 0; i < n; i++ )
    {
      temp = x[ix];
      x[ix] = y[iy];
      y[iy] = temp;
      ix = ix + incx;
      iy = iy + incy;
    }

  }

  return;
}
//****************************************************************************80

int i4_max ( int i1, int i2 )

//****************************************************************************80
//
//  Purpose:
//
//    I4_MAX returns the maximum of two I4's.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    13 October 1998
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int I1, I2, are two integers to be compared.
//
//    Output, int I4_MAX, the larger of I1 and I2.
//
{
  int value;

  if ( i2 < i1 )
  {
    value = i1;
  }
  else
  {
    value = i2;
  }
  return value;
}
//****************************************************************************80

int i4_min ( int i1, int i2 )

//****************************************************************************80
//
//  Purpose:
//
//    I4_MIN returns the minimum of two I4's.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    13 October 1998
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int I1, I2, two integers to be compared.
//
//    Output, int I4_MIN, the smaller of I1 and I2.
//
{
  int value;

  if ( i1 < i2 )
  {
    value = i1;
  }
  else
  {
    value = i2;
  }
  return value;
}
//****************************************************************************80

double r8_abs ( double x )

//****************************************************************************80
//
//  Purpose:
//
//    R8_ABS returns the absolute value of a R8.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    02 April 2005
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, double X, the quantity whose absolute value is desired.
//
//    Output, double R8_ABS, the absolute value of X.
//
{
  double value;

  if ( 0.0 <= x )
  {
    value = x;
  }
  else
  {
    value = -x;
  }
  return value;
}
//****************************************************************************80

double r8_max ( double x, double y )

//****************************************************************************80
//
//  Purpose:
//
//    R8_MAX returns the maximum of two R8's.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    18 August 2004
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, double X, Y, the quantities to compare.
//
//    Output, double R8_MAX, the maximum of X and Y.
//
{
  double value;

  if ( y < x )
  {
    value = x;
  }
  else
  {
    value = y;
  }
  return value;
}
//****************************************************************************80

double r8_sign ( double x )

//****************************************************************************80
//
//  Purpose:
//
//    R8_SIGN returns the sign of a R8.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    18 October 2004
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, double X, the number whose sign is desired.
//
//    Output, double R8_SIGN, the sign of X.
//
{
  double value;

  if ( x < 0.0 )
  {
    value = -1.0;
  }
  else
  {
    value = 1.0;
  }
  return value;
}
//****************************************************************************80

void r8mat_print ( int m, int n, double a[], string title )

//****************************************************************************80
//
//  Purpose:
//
//    R8MAT_PRINT prints an R8MAT.
//
//  Discussion:
//
//    An R8MAT is a doubly dimensioned array of R8 values, stored as a vector
//    in column-major order.
//
//    Entry A(I,J) is stored as A[I+J*M]
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    10 September 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int M, the number of rows in A.
//
//    Input, int N, the number of columns in A.
//
//    Input, double A[M*N], the M by N matrix.
//
//    Input, string TITLE, a title.
//
{
  r8mat_print_some ( m, n, a, 1, 1, m, n, title );

  return;
}
//****************************************************************************80

void r8mat_print_some ( int m, int n, double a[], int ilo, int jlo, int ihi,
  int jhi, string title )

//****************************************************************************80
//
//  Purpose:
//
//    R8MAT_PRINT_SOME prints some of an R8MAT.
//
//  Discussion:
//
//    An R8MAT is a doubly dimensioned array of R8 values, stored as a vector
//    in column-major order.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    20 August 2010
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int M, the number of rows of the matrix.
//    M must be positive.
//
//    Input, int N, the number of columns of the matrix.
//    N must be positive.
//
//    Input, double A[M*N], the matrix.
//
//    Input, int ILO, JLO, IHI, JHI, designate the first row and
//    column, and the last row and column to be printed.
//
//    Input, string TITLE, a title.
//
{
# define INCX 5

  int i;
  int i2hi;
  int i2lo;
  int j;
  int j2hi;
  int j2lo;

  cout << "\n";
  cout << title << "\n";

  if ( m <= 0 || n <= 0 )
  {
    cout << "\n";
    cout << "  (None)\n";
    return;
  }
//
//  Print the columns of the matrix, in strips of 5.
//
  for ( j2lo = jlo; j2lo <= jhi; j2lo = j2lo + INCX )
  {
    j2hi = j2lo + INCX - 1;
    j2hi = i4_min ( j2hi, n );
    j2hi = i4_min ( j2hi, jhi );

    cout << "\n";
//
//  For each column J in the current range...
//
//  Write the header.
//
    cout << "  Col:    ";
    for ( j = j2lo; j <= j2hi; j++ )
    {
      cout << setw(7) << j - 1 << "       ";
    }
    cout << "\n";
    cout << "  Row\n";
    cout << "\n";
//
//  Determine the range of the rows in this strip.
//
    i2lo = i4_max ( ilo, 1 );
    i2hi = i4_min ( ihi, m );

    for ( i = i2lo; i <= i2hi; i++ )
    {
//
//  Print out (up to) 5 entries in row I, that lie in the current strip.
//
      cout << setw(5) << i - 1 << ": ";
      for ( j = j2lo; j <= j2hi; j++ )
      {
        cout << setw(12) << a[i-1+(j-1)*m] << "  ";
      }
      cout << "\n";
    }
  }

  return;
# undef INCX
}
//****************************************************************************80

double *r8mat_transpose_new ( int m, int n, double a[] )

//****************************************************************************80
//
//  Purpose:
//
//    R8MAT_TRANSPOSE_NEW returns the transpose of an R8MAT.
//
//  Discussion:
//
//    An R8MAT is a doubly dimensioned array of R8 values, stored as a vector
//    in column-major order.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    12 October 2005
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int M, N, the number of rows and columns of the matrix A.
//
//    Input, double A[M*N], the matrix whose transpose is desired.
//
//    Output, double R8MAT_TRANSPOSE_NEW[N*M], the transposed matrix.
//
{
  double *b;
  int i;
  int j;

  b = new double[n*m];

  for ( j = 0; j < n; j++ )
  {
    for ( i = 0; i < m; i++ )
    {
      b[j+i*n] = a[i+j*m];
    }
  }
  return b;
}
//****************************************************************************80

double *r8mat_uniform_01_new ( int m, int n, int *seed )

//****************************************************************************80
//
//  Purpose:
//
//    R8MAT_UNIFORM_01_NEW returns a unit pseudorandom R8MAT.
//
//  Discussion:
//
//    An R8MAT is a doubly dimensioned array of R8's,  stored as a vector
//    in column-major order.
//
//    This routine implements the recursion
//
//      seed = 16807 * seed mod ( 2^31 - 1 )
//      unif = seed / ( 2^31 - 1 )
//
//    The integer arithmetic never requires more than 32 bits,
//    including a sign bit.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    03 October 2005
//
//  Author:
//
//    John Burkardt
//
//  Reference:
//
//    Paul Bratley, Bennett Fox, Linus Schrage,
//    A Guide to Simulation,
//    Springer Verlag, pages 201-202, 1983.
//
//    Bennett Fox,
//    Algorithm 647:
//    Implementation and Relative Efficiency of Quasirandom
//    Sequence Generators,
//    ACM Transactions on Mathematical Software,
//    Volume 12, Number 4, pages 362-376, 1986.
//
//    Philip Lewis, Allen Goodman, James Miller,
//    A Pseudo-Random Number Generator for the System/360,
//    IBM Systems Journal,
//    Volume 8, pages 136-143, 1969.
//
//  Parameters:
//
//    Input, int M, N, the number of rows and columns.
//
//    Input/output, int *SEED, the "seed" value.  Normally, this
//    value should not be 0, otherwise the output value of SEED
//    will still be 0, and R8_UNIFORM will be 0.  On output, SEED has
//    been updated.
//
//    Output, double R8MAT_UNIFORM_01_NEW[M*N], a matrix of pseudorandom values.
//
{
  int i;
  int j;
  int k;
  double *r;

  r = new double[m*n];

  for ( j = 0; j < n; j++ )
  {
    for ( i = 0; i < m; i++ )
    {
      k = *seed / 127773;

      *seed = 16807 * ( *seed - k * 127773 ) - k * 2836;

      if ( *seed < 0 )
      {
        *seed = *seed + 2147483647;
      }
      r[i+j*m] = ( double ) ( *seed ) * 4.656612875E-10;
    }
  }

  return r;
}
//****************************************************************************80

void svd_truncated_u ( int m, int n, double a[], double un[], double sn[], 
  double v[] )

//****************************************************************************80
//
//  Purpose:
//
//    SVD_TRUNCATED_U gets the truncated SVD when N <= M
//
//  Discussion:
//
//    A(mxn) = U(mxm)  * S(mxn)  * V(nxn)'
//           = Un(mxn) * Sn(nxn) * V(nxn)'
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    14 September 2006
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int M, N, the number of rows and columns in the matrix A.
//
//    Input, double A[M*N], the matrix whose singular value
//    decomposition we are investigating.
//
//    Output, double UN[M*N], SN[N*N], V[N*N], the factors
//    that form the singular value decomposition of A.
//
{
  double *a_copy;
  double *e;
  int i;
  int info;
  int j;
  int lda;
  int ldu;
  int ldv;
  int job;
  int lwork;
  double *sdiag;
  double *work;
//
//  The correct size of E and SDIAG is min ( m+1, n).
//
  a_copy = new double[m*n];
  e = new double[m+n];
  sdiag = new double[m+n];
  work = new double[m];
//
//  Compute the eigenvalues and eigenvectors.
//
  job = 21;
  lda = m;
  ldu = m;
  ldv = n;
//
//  The input matrix is destroyed by the routine.  Since we need to keep
//  it around, we only pass a copy to the routine.
//
  for ( j = 0; j < n; j++ )
  {
    for ( i = 0; i < m; i++ )
    { 
      a_copy[i+j*m] = a[i+j*m];
    }
  }
  info = dsvdc ( a_copy, lda, m, n, sdiag, e, un, ldu, v, ldv, work, job );
 
  if ( info != 0 )
  {
    cout << "\n";
    cout << "SVD_TRUNCATED_U - Failure!\n";
    cout << "  The SVD could not be calculated.\n";
    cout << "  LINPACK routine DSVDC returned a nonzero\n";
    cout << "  value of the error flag, INFO = " << info << "\n";
    return;
  }
//
//  Make the NxN matrix S from the diagonal values in SDIAG.
//
  for ( j = 0; j < n; j++ )
  {
    for ( i = 0; i < n; i++ )
    {
      if ( i == j )
      {
        sn[i+j*n] = sdiag[i];
      }
      else
      {
        sn[i+j*n] = 0.0;
      }
    }
  }

  delete [] a_copy;
  delete [] e;
  delete [] sdiag;
  delete [] work;

  return;
}
//****************************************************************************80

void svd_truncated_u_test ( int m, int n )

//****************************************************************************80
//
//  Purpose:
//
//    SVD_TRUNCATED_U_TEST tests SVD_TRUNCATED_U.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    20 March 2012
//
//  Author:
//
//    John Burkardt
//
{
  double *a;
  double *a_save;
  double err;
  int i;
  int info;
  int j;
  int k;
  int seed;
  double *sn;
  double *un;
  double *v;

  cout << "\n";
  cout << "SVD_TRUNCATED_U_TEST\n";
  cout << "  M = " << m << "\n";
  cout << "  N = " << n << "\n";

  a = new double[m*n];
  un = new double[m*n];
  sn = new double[n*n];
  v = new double[n*n];

  seed = 123456789;

  a_save = r8mat_uniform_01_new ( m, n, &seed );

  r8mat_print ( m, n, a_save, "  A:" );

  for ( j = 0; j < n; j++ )
  {
    for ( i = 0; i < m; i++ )
    {
      a[i+j*m] = a_save[i+j*m];
    }
  }

  svd_truncated_u ( m, n, a, un, sn, v );

  r8mat_print ( m, n, un, "  UN:" );
  r8mat_print ( n, n, sn, "  SN:" );
  r8mat_print ( n, n, v, "  V:" );
//
//  Check the factorization by computing A = U * S * V'
//
  for ( j = 0; j < n; j++)
  {
    for ( i = 0; i < m; i++ )
    {
      a[i+j*m] = 0.0;
      for ( k = 0; k < n; k++ )
      {
        a[i+j*m] = a[i+j*m] + un[i+k*m] * sn[k+k*n] * v[j+k*n];
      }
    }
  }

  err = 0.0;
  for ( j = 0; j < n; j++ )
  {
    for ( i = 0; i < m; i++ )
    {
      err = r8_max ( err, r8_abs ( a[i+j*m] - a_save[i+j*m] ) );
    }
  }

  cout << "\n";
  cout << "  Maximum error |A - U*S*V'| = " << err << "\n";

  r8mat_print ( m, n, a, "  Recomputed A = U * S * V':" );

  delete [] a;
  delete [] a_save;
  delete [] sn;
  delete [] un;
  delete [] v;

  return;
}
//****************************************************************************80

void svd_truncated_v ( int m, int n, double a[], double u[], double sm[], 
  double vm[] )

//****************************************************************************80
//
//  Purpose:
//
//    SVD_TRUNCATED_V gets the truncated SVD when M <= N.
//
//  Discussion:
//
//    A(mxn) = U(mxm) * S(mxn)  * V(nxn)'
//           = U(mxm) * Sm(mxm) * Vm(mxn)'
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    20 March 2012
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int M, N, the number of rows and columns in the matrix A.
//
//    Input, double A[M*N], the matrix whose singular value
//    decomposition we are investigating.
//
//    Output, double U[M*M], SM[M*M], VM[M*N], the factors
//    that form the singular value decomposition of A.
//
{
  double *a2;
  int m2;
  int n2;
//
//  Transpose the matrix!
//
  a2 = r8mat_transpose_new ( m, n, a );
  m2 = n;
  n2 = m;

  svd_truncated_u ( m2, n2, a2, vm, sm, u );

  delete [] a2;

  return;
}
//****************************************************************************80

void svd_truncated_v_test ( int m, int n )

//****************************************************************************80
//
//  Purpose:
//
//    SVD_TRUNCATED_V_TEST tests SVD_TRUNCATED_V.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    20 March 2012
//
//  Author:
//
//    John Burkardt
//
{
  double *a;
  double *a_save;
  double err;
  int i;
  int info;
  int j;
  int k;
  int seed;
  double *sm;
  double *u;
  double *vm;

  cout << "\n";
  cout << "SVD_TRUNCATED_V_TEST\n";
  cout << "  M = " << m << "\n";
  cout << "  N = " << n << "\n";

  a = new double[m*n];
  u = new double[m*m];
  sm = new double[m*m];
  vm = new double[n*m];

  seed = 123456789;

  a_save = r8mat_uniform_01_new ( m, n, &seed );

  r8mat_print ( m, n, a_save, "  A:" );

  for ( j = 0; j < n; j++ )
  {
    for ( i = 0; i < m; i++ )
    {
      a[i+j*m] = a_save[i+j*m];
    }
  }

  svd_truncated_v ( m, n, a, u, sm, vm );

  r8mat_print ( m, m, u, "  U:" );
  r8mat_print ( m, m, sm, "  SM:" );
  r8mat_print ( n, m, vm, "  VM:" );
//
//  Check the factorization by computing A = U * S * V'
//
  for ( j = 0; j < n; j++)
  {
    for ( i = 0; i < m; i++ )
    {
      a[i+j*m] = 0.0;
      for ( k = 0; k < m; k++ )
      {
        a[i+j*m] = a[i+j*m] + u[i+k*m] * sm[k+k*m] * vm[j+k*n];
      }
    }
  }

  err = 0.0;
  for ( j = 0; j < n; j++ )
  {
    for ( i = 0; i < m; i++ )
    {
      err = r8_max ( err, r8_abs ( a[i+j*m] - a_save[i+j*m] ) );
    }
  }

  cout << "\n";
  cout << "  Maximum error |A - U*S*V'| = " << err << "\n";

  r8mat_print ( m, n, a, "  Recomputed A = U * S * V':" );

  delete [] a;
  delete [] a_save;
  delete [] sm;
  delete [] u;
  delete [] vm;

  return;
}
//****************************************************************************80

void timestamp ( )

//****************************************************************************80
//
//  Purpose:
//
//    TIMESTAMP prints the current YMDHMS date as a time stamp.
//
//  Example:
//
//    31 May 2001 09:45:54 AM
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    08 July 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    None
//
{
# define TIME_SIZE 40

  static char time_buffer[TIME_SIZE];
  const struct std::tm *tm_ptr;
  size_t len;
  std::time_t now;

  now = std::time ( NULL );
  tm_ptr = std::localtime ( &now );

  len = std::strftime ( time_buffer, TIME_SIZE, "%d %B %Y %I:%M:%S %p", tm_ptr );

  std::cout << time_buffer << "\n";

  return;
# undef TIME_SIZE
}