# include <math.h>
# include <stdlib.h>
# include <stdio.h>
# include <time.h>
# include <string.h>

# include "differ.h"

/******************************************************************************/

void differ_backward ( double h, int o, int p, double c[], double x[] )

/******************************************************************************/
/*
  Purpose:

    DIFFER_BACKWARD computes backward difference coefficients.

  Discussion:

    We determine coefficients C to approximate the derivative at X0
    of order O and precision P, using equally spaced backward
    differences, so that 

      d^o f(x)/dx^o = sum ( 0 <= i <= o+p-1 ) c(i) f(x-ih) + O(h^(p))

  Licensing:

    This code is distributed under the MIT license.

  Modified:

    10 November 2013

  Author:

    John Burkardt

  Parameters:

    Input, double H, the spacing.  0 < H.

    Input, int O, the order of the derivative to be 
    approximated.  1 <= O.

    Input, int P, the order of the error, as a power of H.

    Output, double C[O+P], the coefficients.

    Output, double X[O+P], the evaluation points.
*/
{
  double *b;
  int i;
  int info;
  int job;
  int n;
  double t;

  n = o + p;

  for ( i = 0; i < n; i++ )
  {
    x[i] = ( double ) ( i + 1 - n ) * h;
  }

  b = ( double * ) malloc ( n * sizeof ( double ) );

  for ( i = 0; i < n; i++ )
  {
    b[i] = 0.0;
  }
  b[o] = 1.0;

  job = 0;
  r8vm_sl ( n, x, b, job, c, &info );

  if ( info != 0 )
  {
    fprintf ( stderr, "\n" );
    fprintf ( stderr, "DIFFER_BACKWARD - Fatal error!\n" );
    fprintf ( stderr, "  Vandermonde linear system is singular.\n" );
    exit ( 1 );
  }

  t = r8_factorial ( o );
  for ( i = 0; i < n; i++ )
  {
    c[i] = c[i] * t;
  }

  free ( b );

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

void differ_central ( double h, int  o, int p, double c[], double x[] )

/******************************************************************************/
/*
  Purpose:

    DIFFER_CENTRAL computes central difference coefficients.

  Discussion:

    We determine coefficients C to approximate the derivative at X0
    of order O and precision P, using equally spaced central
    differences, so that 

      d^o f(x)/dx^o = sum ( 0 <= i <= o+p-1 ) c(i) f(x+(2*i-o-p+1)*h/2) 
        + O(h^(p))

  Licensing:

    This code is distributed under the MIT license.

  Modified:

    10 November 2013

  Author:

    John Burkardt

  Parameters:

    Input, double H, the spacing.  0 < H.

    Input, int O, the order of the derivative to 
    be approximated.  1 <= O.

    Input, int P, the order of the error, as a power of H.

    Output, double C[O+P], the coefficients.

    Output, double X[O+P], the evaluation points.
*/
{
  double *b;
  int i;
  int info;
  int job;
  int n;
  double t;

  n = o + p;

  for ( i = 0; i < n; i++ )
  {
    x[i] = ( double ) ( - n + 1 + 2 * i ) * h / 2.0;
  }

  b = ( double * ) malloc ( n * sizeof ( double ) );
  for ( i = 0; i < n; i++ )
  {
    b[i] = 0.0;
  }
  b[o] = 1.0;

  job = 0;
  r8vm_sl ( n, x, b, job, c, &info );

  if ( info != 0 )
  {
    fprintf ( stderr, "\n" );
    fprintf ( stderr, "DIFFER_CENTRAL - Fatal error!\n" );
    fprintf ( stderr, "  Vandermonde linear system is singular.\n" );
    exit ( 1 );
  }

  t = r8_factorial ( o );
  for ( i = 0; i < n; i++ )
  {
    c[i] = c[i] * t;
  }

  free ( b );

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

void differ_forward ( double h, int o, int p, double c[], double x[] )

/******************************************************************************/
/*
  Purpose:

    DIFFER_FORWARD computes forward difference coefficients.

  Discussion:

    We determine coefficients C to approximate the derivative at X0
    of order O and precision P, using equally spaced forward
    differences, so that 

      d^o f(x)/dx^o = sum ( 0 <= i <= o+p-1 ) c(i) f(x+ih) + O(h^(p))

  Licensing:

    This code is distributed under the MIT license.

  Modified:

    10 November 2013

  Author:

    John Burkardt

  Parameters:

    Input, real H, the spacing.  0 < H.

    Input, integer O, the order of the derivative to be approximated.
    1 <= O.

    Input, integer P, the order of the error, as a power of H.

    Output, real C[O+P], the coefficients.

    Output, real X[O+P], the evaluation points.
*/
{
  double *b;
  int i;
  int info;
  int job;
  int n;
  double t;

  n = o + p;

  for ( i = 0; i < n; i++ )
  {
    x[i] = ( double ) ( i ) * h;
  }

  b = ( double * ) malloc ( n * sizeof ( double ) );
  for ( i = 0; i < n; i++ )
  {
    b[i] = 0.0;
  }
  b[o] = 1.0;

  job = 0;
  r8vm_sl ( n, x, b, job, c, &info );

  if ( info != 0 )
  {
    fprintf ( stderr, "\n" );
    fprintf ( stderr, "DIFFER_FORWARD - Fatal error!\n" );
    fprintf ( stderr, "  Vandermonde linear system is singular.\n" );
    exit ( 1 );
  }

  t = r8_factorial ( o );
  for ( i = 0; i < n; i++ )
  {
    c[i] = c[i] * t;
  }

  free ( b );

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

double *differ_inverse ( int n, double stencil[] )

/******************************************************************************/
/*
  Purpose:

    DIFFER_INVERSE returns the inverse of the DIFFER matrix.

  Licensing:

    This code is distributed under the MIT license.

  Modified:

    03 November 2013

  Author:

    John Burkardt

  Parameters:

    Input, int N, the order of the matrix.

    Input, double STENCIL[N], the values that define A.

    Output, double DIFFER_INVERSE[N*N], the matrix.
*/
{
  double *a;
  int i;
  int indx;
  int j;
  int k;

  a = ( double * ) malloc ( n * n * sizeof ( double ) );

  for ( j = 0; j < n; j++ )
  {
    for ( i = 0; i < n; i++ )
    {
      if ( j == 0 )
      {
        a[i+j*n] = 1.0;
      }
      else
      {
        a[i+j*n] = 0.0;
      }
    }
  }

  for ( i = 0; i < n; i++ )
  {
    indx = 0;

    for ( k = 0; k < n; k++ )
    {
      if ( k != i )
      {
        for ( j = indx + 1; 0 <= j; j-- )
        {
          a[i+j*n] = - stencil[k] * a[i+j*n] / ( stencil[i] - stencil[k] );

          if ( 0 < j )
          {
            a[i+j*n] = a[i+j*n] + a[i+(j-1)*n] / ( stencil[i] - stencil[k] );
          }
        }
        indx = indx + 1;
      }
    }
  }

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

  return a;
}
/******************************************************************************/

double *differ_matrix ( int n, double stencil[] )

/******************************************************************************/
/*
  Purpose:

    DIFFER_MATRIX computes the stencil matrix from the stencil vector.

  Discussion:

    If N = 4, and STENCIL = ( -3, -2, -1, 1 ), then A will be

    -3  -2  -1  1
     9   4   1  1
   -27  -8  -1  1
    81  16   1  1

    This matrix is a generalized form of a Vandermonde matrix A2:

     1   1   1  1
    -3  -2  -1  1
     9   4   1  1
   -27  -8  -1  1    

    and if A * x = b, the A2 * x2 = b, where x2(i) = x(i) * stencil(i)

  Licensing:

    This code is distributed under the MIT license.

  Modified:

    03 November 2013

  Author:

    John Burkardt

  Parameters:

    Input, int N, the number of stencil points.

    Input, double STENCIL[N], the stencil vector.
    The entries in this vector must be distinct.
    No entry of STENCIL may be 0.

    Output, double DIFFER_MATRIX[N*N], the stencil matrix.
*/
{
  double *a;
  int i;
  int j;

  a = ( double * ) malloc ( n * n * sizeof ( double ) );

  for ( j = 0; j < n; j++ )
  {
    a[0+j*n] = stencil[j];
    for ( i = 1; i < n; i++ )
    {
      a[i+j*n] = a[i-1+j*n] * stencil[j];
    }
  }

  return a;
}
/******************************************************************************/

double *differ_solve ( int n, double stencil[], int order )

/******************************************************************************/
/*
  Purpose:

    DIFFER_SOLVE solves for finite difference coefficients.

  Licensing:

    This code is distributed under the MIT license.

  Modified:

    03 November 2013

  Author:

    John Burkardt

  Parameters:

    Input, int N, the number of stencil points.

    Input, double STENCIL[N], the stencil vector.
    The entries in this vector must be distinct.
    No entry of STENCIL may be 0.

    Input, int ORDER, the order of the derivative to
    be approximated.  1 <= ORDER <= N.

    Output, double DIFFER_SOLVE[N], the coefficients to be used
    to multiply U(STENCIL(I))-U(0), so that the sum forms an
    approximation to the derivative of order ORDER, with error 
    of order H^(N+1-ORDER).
*/
{ 
  double *a;
  double *b;
  double *c;
  int i;

  a = differ_matrix ( n, stencil );

  b = ( double * ) malloc ( n * sizeof ( double ) );
  for ( i = 0; i < n; i++ )
  {
    b[i] = 0.0;
  }
  b[order-1] = 1.0;
/*
  Solve A * C = B.
*/
  c = r8mat_fs_new ( n, a, b );

  free ( a );
  free ( b );

  return c;
}
/******************************************************************************/

void differ_stencil ( double x0, int o, int p, double x[], double c[] )

/******************************************************************************/
/*
  Purpose:

    DIFFER_STENCIL computes finite difference coefficients.

  Discussion:

    We determine coefficients C to approximate the derivative at X0
    of order O and precision P, using finite differences, so that 

      d^o f(x)/dx^o (x0) = sum ( 0 <= i <= o+p-1 ) c(i) f(x(i)) 
        + O(h^(p))

    where H is the maximum spacing between X0 and any X(I).

  Licensing:

    This code is distributed under the MIT license.

  Modified:

    10 November 2013

  Author:

    John Burkardt

  Parameters:

    Input, double X0, the point where the derivative is to 
    be approximated.

    Input, int O, the order of the derivative to be 
    approximated.  1 <= O.

    Input, int P, the order of the error, as a power of H.

    Input, double X[O+P], the evaluation points.

    Output, double C[O+P], the coefficients.
*/
{
  double *b;
  double *dx;
  int i;
  int info;
  int job;
  int n;
  double t;

  n = o + p;

  dx = ( double * ) malloc ( n * sizeof ( double ) );

  for ( i = 0; i < n; i++ )
  {
    dx[i] = x[i] - x0;
  }

  b = ( double * ) malloc ( n * sizeof ( double ) );

  for ( i = 0; i < n; i++ )
  {
    b[i] = 0.0;
  }
  b[o] = 1.0;

  job = 0;
  r8vm_sl ( n, dx, b, job, c, &info );

  if ( info != 0 )
  {
    fprintf ( stderr, "\n" );
    fprintf ( stderr, "DIFFER_STENCIL - Fatal error!\n" );
    fprintf ( stderr, "  Vandermonde linear system is singular.\n" );
    exit ( 1 );
  }

  t = r8_factorial ( o );
  for ( i = 0; i < n; i++ )
  {
    c[i] = c[i] * t;
  }

  free ( b );
  free ( dx );

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

int i4_max ( int i1, int i2 )

/******************************************************************************/
/*
  Purpose:

    I4_MAX returns the maximum of two I4's.

  Licensing:

    This code is distributed under the MIT license.

  Modified:

    29 August 2006

  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;
}
/******************************************************************************/

int i4_min ( int i1, int i2 )

/******************************************************************************/
/*
  Purpose:

    I4_MIN returns the smaller of two I4's.

  Licensing:

    This code is distributed under the MIT license.

  Modified:

    29 August 2006

  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;
}
/******************************************************************************/

double inverse_error ( int n, double a[], double b[] )

/******************************************************************************/
/*
  Purpose:

    INVERSE_ERROR determines the error in an inverse matrix.

  Licensing:

    This code is distributed under the MIT license.

  Modified:

    30 October 2013

  Author:

    John Burkardt

  Parameters:

    Input, int N, the order of the matrix.

    Input, double A[N*N], the matrix.

    Input, double B[N*N], the inverse.

    Output, double ERROR_FROBENIUS, the Frobenius norm
    of (A*B-I) + (B*A-I).
*/
{
  double *c;
  int j;
  double value;

  c = r8mat_mm_new ( n, n, n, a, b );

  for ( j = 0; j < n; j++ )
  {
    c[j+j*n] = c[j+j*n] - 1.0;
  }

  value = r8mat_norm_fro ( n, n, c );

  free ( c );

  c = r8mat_mm_new ( n, n, n, b, a );

  for ( j = 0; j < n; j++ )
  {
    c[j+j*n] = c[j+j*n] - 1.0;
  }

  value = value + r8mat_norm_fro ( n, n, c );

  free ( c );

  return value;
}
/******************************************************************************/

double r8_factorial ( int n )

/******************************************************************************/
/*
  Purpose:

    R8_FACTORIAL computes the factorial of N.

  Discussion:

    factorial ( N ) = product ( 1 <= I <= N ) I

  Licensing:

    This code is distributed under the MIT license.

  Modified:

    26 June 2008

  Author:

    John Burkardt

  Parameters:

    Input, int N, the argument of the factorial function.
    If N is less than 1, the function value is returned as 1.

    Output, double R8_FACTORIAL, the factorial of N.
*/
{
  int i;
  double value;

  value = 1.0;

  for ( i = 1; i <= n; i++ )
  {
    value = value * ( double ) ( i );
  }

  return value;
}
/******************************************************************************/

double *r8mat_fs_new ( int n, double a[], double b[] )

/******************************************************************************/
/*
  Purpose:

    R8MAT_FS_NEW factors and solves a system with one right hand side.

  Discussion:

    This routine uses partial pivoting, but no pivot vector is required.

    This routine differs from R8MAT_FSS_NEW in two ways:
    * only one right hand side is allowed;
    * the input matrix A is not modified.

    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 MIT license. 

  Modified:

    21 January 2013

  Author:

    John Burkardt

  Parameters:

    Input, int N, the order of the matrix.
    N must be positive.

    Input, double A[N*N], the coefficient matrix of the linear system.

    Input, double B[N], on input, the right hand side of the
    linear system.

    Output, double R8MAT_FS[N], the solution of the linear system.
*/
{
  double *a2;
  int i;
  int ipiv;
  int j;
  int jcol;
  double piv;
  double t;
  double *x;

  a2 = ( double * ) malloc ( n * n * sizeof ( double ) );
  for ( j = 0; j < n; j++ )
  {
    for ( i = 0; i < n; i++ )
    {
      a2[i+j*n] = a[i+j*n];
    }
  }

  x = ( double * ) malloc ( n * sizeof ( double ) );
  for ( i = 0; i < n; i++ )
  {
    x[i] = b[i];
  }

  for ( jcol = 1; jcol <= n; jcol++ )
  {
/*
  Find the maximum element in column I.
*/
    piv = fabs ( a2[jcol-1+(jcol-1)*n] );
    ipiv = jcol;
    for ( i = jcol+1; i <= n; i++ )
    {
      if ( piv < fabs ( a2[i-1+(jcol-1)*n] ) )
      {
        piv = fabs ( a2[i-1+(jcol-1)*n] );
        ipiv = i;
      }
    }

    if ( piv == 0.0 )
    {
      fprintf ( stderr, "\n" );
      fprintf ( stderr, "R8MAT_FS_NEW - Fatal error!\n" );
      fprintf ( stderr, "  Zero pivot on step %d\n", jcol );
      exit ( 1 );
    }
/*
  Switch rows JCOL and IPIV, and X.
*/
    if ( jcol != ipiv )
    {
      for ( j = 1; j <= n; j++ )
      {
        t                  = a2[jcol-1+(j-1)*n];
        a2[jcol-1+(j-1)*n] = a2[ipiv-1+(j-1)*n];
        a2[ipiv-1+(j-1)*n] = t;
      }
      t         = x[jcol-1];
      x[jcol-1] = x[ipiv-1];
      x[ipiv-1] = t;
    }
/*
  Scale the pivot row.
*/
    t = a2[jcol-1+(jcol-1)*n];
    a2[jcol-1+(jcol-1)*n] = 1.0;
    for ( j = jcol+1; j <= n; j++ )
    {
      a2[jcol-1+(j-1)*n] = a2[jcol-1+(j-1)*n] / t;
    }
    x[jcol-1] = x[jcol-1] / t;
/*
  Use the pivot row to eliminate lower entries in that column.
*/
    for ( i = jcol+1; i <= n; i++ )
    {
      if ( a2[i-1+(jcol-1)*n] != 0.0 )
      {
        t = - a2[i-1+(jcol-1)*n];
        a2[i-1+(jcol-1)*n] = 0.0;
        for ( j = jcol+1; j <= n; j++ )
        {
          a2[i-1+(j-1)*n] = a2[i-1+(j-1)*n] + t * a2[jcol-1+(j-1)*n];
        }
        x[i-1] = x[i-1] + t * x[jcol-1];
      }
    }
  }
/*
  Back solve.
*/
  for ( jcol = n; 2 <= jcol; jcol-- )
  {
    for ( i = 1; i < jcol; i++ )
    {
      x[i-1] = x[i-1] - a2[i-1+(jcol-1)*n] * x[jcol-1];
    }
  }

  free ( a2 );

  return x;
}
/******************************************************************************/

double *r8mat_mm_new ( int n1, int n2, int n3, double a[], double b[] )

/******************************************************************************/
/*
  Purpose:

    R8MAT_MM_NEW multiplies two matrices.

  Discussion:

    An R8MAT is a doubly dimensioned array of R8 values, stored as a vector
    in column-major order.

    For this routine, the result is returned as the function value.

  Licensing:

    This code is distributed under the MIT license.

  Modified:

    08 April 2009

  Author:

    John Burkardt

  Parameters:

    Input, int N1, N2, N3, the order of the matrices.

    Input, double A[N1*N2], double B[N2*N3], the matrices to multiply.

    Output, double R8MAT_MM_NEW[N1*N3], the product matrix C = A * B.
*/
{
  double *c;
  int i;
  int j;
  int k;

  c = ( double * ) malloc ( n1 * n3 * sizeof ( double ) );

  for ( i = 0; i < n1; i++ )
  {
    for ( j = 0; j < n3; j++ )
    {
      c[i+j*n1] = 0.0;
      for ( k = 0; k < n2; k++ )
      {
        c[i+j*n1] = c[i+j*n1] + a[i+k*n1] * b[k+j*n2];
      }
    }
  }

  return c;
}
/******************************************************************************/

double *r8mat_mv_new ( int m, int n, double a[], double x[] )

/******************************************************************************/
/*
  Purpose:

    R8MAT_MV_NEW multiplies a matrix times a vector.

  Discussion:

    An R8MAT is a doubly dimensioned array of R8 values, stored as a vector
    in column-major order.

    For this routine, the result is returned as the function value.

  Licensing:

    This code is distributed under the MIT license.

  Modified:

    11 April 2007

  Author:

    John Burkardt

  Parameters:

    Input, int M, N, the number of rows and columns of the matrix.

    Input, double A[M,N], the M by N matrix.

    Input, double X[N], the vector to be multiplied by A.

    Output, double R8MAT_MV_NEW[M], the product A*X.
*/
{
  int i;
  int j;
  double *y;

  y = ( double * ) malloc ( m * sizeof ( double ) );

  for ( i = 0; i < m; i++ )
  {
    y[i] = 0.0;
    for ( j = 0; j < n; j++ )
    {
      y[i] = y[i] + a[i+j*m] * x[j];
    }
  }

  return y;
}
/******************************************************************************/

double r8mat_norm_fro ( int m, int n, double a[] )

/******************************************************************************/
/*
  Purpose:

    R8MAT_NORM_FRO returns the Frobenius norm of an R8MAT.

  Discussion:

    An R8MAT is a doubly dimensioned array of R8 values, stored as a vector
    in column-major order.

    The Frobenius norm is defined as

      R8MAT_NORM_FRO = sqrt (
        sum ( 1 <= I <= M ) sum ( 1 <= j <= N ) A(I,J)^2 )
    The matrix Frobenius norm is not derived from a vector norm, but
    is compatible with the vector L2 norm, so that:

      r8vec_norm_l2 ( A * x ) <= r8mat_norm_fro ( A ) * r8vec_norm_l2 ( x ).

  Licensing:

    This code is distributed under the MIT license.

  Modified:

    26 July 2008

  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 matrix whose Frobenius
    norm is desired.

    Output, double R8MAT_NORM_FRO, the Frobenius norm of A.
*/
{
  int i;
  int j;
  double value;

  value = 0.0;
  for ( j = 0; j < n; j++ )
  {
    for ( i = 0; i < m; i++ )
    {
      value = value + pow ( a[i+j*m], 2 );
    }
  }
  value = sqrt ( value );

  return value;
}
/******************************************************************************/

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

/******************************************************************************/
/*
  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 MIT license.

  Modified:

    28 May 2008

  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, char *TITLE, a title.
*/
{
  r8mat_print_some ( m, n, a, 1, 1, m, n, title );

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

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

/******************************************************************************/
/*
  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 MIT license.

  Modified:

    26 June 2013

  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, char *TITLE, a title.
*/
{
# define INCX 5

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

  fprintf ( stdout, "\n" );
  fprintf ( stdout, "%s\n", title );

  if ( m <= 0 || n <= 0 )
  {
    fprintf ( stdout, "\n" );
    fprintf ( stdout, "  (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;
    if ( n < j2hi )
    {
      j2hi = n;
    }
    if ( jhi < j2hi )
    {
      j2hi = jhi;
    }

    fprintf ( stdout, "\n" );
/*
  For each column J in the current range...

  Write the header.
*/
    fprintf ( stdout, "  Col:  ");
    for ( j = j2lo; j <= j2hi; j++ )
    {
      fprintf ( stdout, "  %7d     ", j - 1 );
    }
    fprintf ( stdout, "\n" );
    fprintf ( stdout, "  Row\n" );
    fprintf ( stdout, "\n" );
/*
  Determine the range of the rows in this strip.
*/
    if ( 1 < ilo )
    {
      i2lo = ilo;
    }
    else
    {
      i2lo = 1;
    }
    if ( m < ihi )
    {
      i2hi = m;
    }
    else
    {
      i2hi = ihi;
    }

    for ( i = i2lo; i <= i2hi; i++ )
    {
/*
  Print out (up to) 5 entries in row I, that lie in the current strip.
*/
      fprintf ( stdout, "%5d:", i - 1 );
      for ( j = j2lo; j <= j2hi; j++ )
      {
        fprintf ( stdout, "  %14g", a[i-1+(j-1)*m] );
      }
      fprintf ( stdout, "\n" );
    }
  }

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

double *r8mat_sub_new ( int m, int n, double a[], double b[] )

/******************************************************************************/
/*
  Purpose:

    R8MAT_SUB_NEW computes C = A - B.

  Discussion:

    An R8MAT is a doubly dimensioned array of R8 values, stored as a vector
    in column-major order.

    For this routine, the result is returned as the function value.

  Licensing:

    This code is distributed under the MIT license.

  Modified:

    30 October 2013

  Author:

    John Burkardt

  Parameters:

    Input, int M, N, the order of the matrices.

    Input, double A[M*N], double B[M*N], the matrices.

    Output, double R8MAT_SUB_NEW[M*N], the value of A-B.
*/
{
  double *c;
  int i;
  int j;

  c = ( double * ) malloc ( m * n * sizeof ( double ) );

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

  return c;
}
/******************************************************************************/

void r8vec_print ( int n, double a[], char *title )

/******************************************************************************/
/*
  Purpose:

    R8VEC_PRINT prints an R8VEC.

  Discussion:

    An R8VEC is a vector of R8's.

  Licensing:

    This code is distributed under the MIT license.

  Modified:

    08 April 2009

  Author:

    John Burkardt

  Parameters:

    Input, int N, the number of components of the vector.

    Input, double A[N], the vector to be printed.

    Input, char *TITLE, a title.
*/
{
  int i;

  fprintf ( stdout, "\n" );
  fprintf ( stdout, "%s\n", title );
  fprintf ( stdout, "\n" );
  for ( i = 0; i < n; i++ )
  {
    fprintf ( stdout, "  %8d: %14g\n", i, a[i] );
  }

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

double *r8vec_uniform_01_new ( int n, int *seed )

/******************************************************************************/
/*
  Purpose:

    R8VEC_UNIFORM_01_NEW returns a unit pseudorandom R8VEC.

  Discussion:

    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 MIT license.

  Modified:

    19 August 2004

  Author:

    John Burkardt

  Reference:

    Paul Bratley, Bennett Fox, Linus Schrage,
    A Guide to Simulation,
    Second Edition,
    Springer, 1987,
    ISBN: 0387964673,
    LC: QA76.9.C65.B73.

    Bennett Fox,
    Algorithm 647:
    Implementation and Relative Efficiency of Quasirandom
    Sequence Generators,
    ACM Transactions on Mathematical Software,
    Volume 12, Number 4, December 1986, pages 362-376.

    Pierre L'Ecuyer,
    Random Number Generation,
    in Handbook of Simulation,
    edited by Jerry Banks,
    Wiley, 1998,
    ISBN: 0471134031,
    LC: T57.62.H37.

    Peter Lewis, Allen Goodman, James Miller,
    A Pseudo-Random Number Generator for the System/360,
    IBM Systems Journal,
    Volume 8, Number 2, 1969, pages 136-143.

  Parameters:

    Input, int N, the number of entries in the vector.

    Input/output, int *SEED, a seed for the random number generator.

    Output, double R8VEC_UNIFORM_01_NEW[N], the vector of pseudorandom values.
*/
{
  int i;
  int i4_huge = 2147483647;
  int k;
  double *r;

  if ( *seed == 0 )
  {
    fprintf ( stderr, "\n" );
    fprintf ( stderr, "R8VEC_UNIFORM_01_NEW - Fatal error!\n" );
    fprintf ( stderr, "  Input value of SEED = 0.\n" );
    exit ( 1 );
  }

  r = ( double * ) malloc ( n * sizeof ( double ) );

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

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

    if ( *seed < 0 )
    {
      *seed = *seed + i4_huge;
    }

    r[i] = ( double ) ( *seed ) * 4.656612875E-10;
  }

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

void r8vec2_print ( int n, double a1[], double a2[], char *title )

/******************************************************************************/
/*
  Purpose:

    R8VEC2_PRINT prints an R8VEC2.

  Discussion:

    An R8VEC2 is a dataset consisting of N pairs of real values, stored
    as two separate vectors A1 and A2.

  Licensing:

    This code is distributed under the MIT license.

  Modified:

    26 March 2009

  Author:

    John Burkardt

  Parameters:

    Input, int N, the number of components of the vector.

    Input, double A1[N], double A2[N], the vectors to be printed.

    Input, char *TITLE, a title.
*/
{
  int i;

  fprintf ( stdout, "\n" );
  fprintf ( stdout, "%s\n", title );
  fprintf ( stdout, "\n" );
  for ( i = 0; i < n; i++ )
  {
    fprintf ( stdout, "  %4d: %14g  %14g\n", i, a1[i], a2[i] );
  }

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

void r8vm_sl ( int n, double a[], double b[], int job, double x[], int *info )

/******************************************************************************/
/*
  Purpose:

    R8VM_SL solves a R8VM system.

  Discussion:

    The R8VM storage format is used for an M by N Vandermonde matrix.
    An M by N Vandermonde matrix is defined by the values in its second
    row, which will be written here as X(1:N).  The matrix has a first 
    row of 1's, a second row equal to X(1:N), a third row whose entries
    are the squares of the X values, up to the M-th row whose entries
    are the (M-1)th powers of the X values.  The matrix can be stored
    compactly by listing just the values X(1:N).

    Vandermonde systems are very close to singularity.  The singularity
    gets worse as N increases, and as any pair of values defining
    the matrix get close.  Even a system as small as N = 10 will
    involve the 9th power of the defining values.

  Licensing:

    This code is distributed under the MIT license. 

  Modified:

    27 January 2013

  Author:

    Original FORTRAN77 version by Golub, VanLoan.
    C version by John Burkardt.

  Reference:

    Gene Golub, Charles Van Loan,
    Matrix Computations,
    Third Edition,
    Johns Hopkins, 1996.

  Parameters:

    Input, int N, the number of rows and columns of the matrix.

    Input, double A[N], the R8VM matrix.

    Input, double B[N], the right hand side.

    Input, int JOB, specifies the system to solve.
    0, solve A * x = b.
    nonzero, solve A' * x = b.

    Output, double X[N], the solution of the linear system.

    Output, int *INFO.
    0, no error.
    nonzero, at least two of the values in A are equal.
*/
{
  int i;
  int j;
/*
  Check for explicit singularity.
*/
  *info = 0;

  for ( j = 0; j < n; j++ )
  {
    for ( i = j+1; i < n; i++ )
    {
      if ( a[i] == a[j] )
      {
        *info = 1;
        return;
      }
    }
  }

  for ( i = 0; i < n; i++ )
  {
    x[i] = b[i];
  }

  if ( job == 0 )
  {
    for ( j = 1; j <= n-1; j++ )
    {
      for ( i = n; j+1 <= i; i-- )
      {
        x[i-1] = x[i-1] - a[j-1] * x[i-2];
      }
    }

    for ( j = n-1; 1 <= j; j-- )
    {
      for ( i = j+1; i <= n; i++ )
      {
        x[i-1] = x[i-1] / ( a[i-1] - a[i-j-1] );
      }

      for ( i = j; i <= n-1; i++ )
      {
        x[i-1] = x[i-1] - x[i];
      }
    }
  }
  else
  {
    for ( j = 1; j <= n-1; j++ )
    {
      for ( i = n; j+1 <= i; i-- )
      {
        x[i-1] = ( x[i-1] - x[i-2] ) / ( a[i-1] - a[i-j-1] );
      }
    }

    for ( j = n-1; 1 <= j; j-- )
    {
      for ( i = j; i <= n-1; i++ )
      {
        x[i-1] = x[i-1] - x[i] * a[j-1];
      }
    }

  }

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

double *r8vm_sl_new ( int n, double a[], double b[], int job, int *info )

/******************************************************************************/
/*
  Purpose:

    R8VM_SL_NEW solves a R8VM system.

  Discussion:

    The R8VM storage format is used for an M by N Vandermonde matrix.
    An M by N Vandermonde matrix is defined by the values in its second
    row, which will be written here as X(1:N).  The matrix has a first 
    row of 1's, a second row equal to X(1:N), a third row whose entries
    are the squares of the X values, up to the M-th row whose entries
    are the (M-1)th powers of the X values.  The matrix can be stored
    compactly by listing just the values X(1:N).

    Vandermonde systems are very close to singularity.  The singularity
    gets worse as N increases, and as any pair of values defining
    the matrix get close.  Even a system as small as N = 10 will
    involve the 9th power of the defining values.

  Licensing:

    This code is distributed under the MIT license. 

  Modified:

    27 January 2013

  Author:

    Original FORTRAN77 version by Golub, VanLoan.
    C version by John Burkardt.

  Reference:

    Gene Golub, Charles Van Loan,
    Matrix Computations,
    Third Edition,
    Johns Hopkins, 1996.

  Parameters:

    Input, int N, the number of rows and columns of the matrix.

    Input, double A[N], the R8VM matrix.

    Input, double B[N], the right hand side.

    Input, int JOB, specifies the system to solve.
    0, solve A * x = b.
    nonzero, solve A' * x = b.

    Output, int *INFO.
    0, no error.
    nonzero, at least two of the values in A are equal.

    Output, double R8VM_SL[N], the solution of the linear system.
*/
{
  int i;
  int j;
  double *x;
/*
  Check for explicit singularity.
*/
  *info = 0;

  for ( j = 0; j < n; j++ )
  {
    for ( i = j+1; i < n; i++ )
    {
      if ( a[i] == a[j] )
      {
        *info = 1;
        return NULL;
      }
    }
  }

  x = ( double * ) malloc ( n * sizeof ( double ) );

  for ( i = 0; i < n; i++ )
  {
    x[i] = b[i];
  }

  if ( job == 0 )
  {
    for ( j = 1; j <= n-1; j++ )
    {
      for ( i = n; j+1 <= i; i-- )
      {
        x[i-1] = x[i-1] - a[j-1] * x[i-2];
      }
    }

    for ( j = n-1; 1 <= j; j-- )
    {
      for ( i = j+1; i <= n; i++ )
      {
        x[i-1] = x[i-1] / ( a[i-1] - a[i-j-1] );
      }

      for ( i = j; i <= n-1; i++ )
      {
        x[i-1] = x[i-1] - x[i];
      }
    }
  }
  else
  {
    for ( j = 1; j <= n-1; j++ )
    {
      for ( i = n; j+1 <= i; i-- )
      {
        x[i-1] = ( x[i-1] - x[i-2] ) / ( a[i-1] - a[i-j-1] );
      }
    }

    for ( j = n-1; 1 <= j; j-- )
    {
      for ( i = j; i <= n-1; i++ )
      {
        x[i-1] = x[i-1] - x[i] * a[j-1];
      }
    }

  }

  return x;
}
/******************************************************************************/

void timestamp ( )

/******************************************************************************/
/*
  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 MIT license.

  Modified:

    24 September 2003

  Author:

    John Burkardt

  Parameters:

    None
*/
{
# define TIME_SIZE 40

  static char time_buffer[TIME_SIZE];
  const struct tm *tm;
  time_t now;

  now = time ( NULL );
  tm = localtime ( &now );

  strftime ( time_buffer, TIME_SIZE, "%d %B %Y %I:%M:%S %p", tm );

  fprintf ( stdout, "%s\n", time_buffer );

  return;
# undef TIME_SIZE
}