pvfmm-static-build/src/legendre_rule.cpp

# include <cstdlib>
# include <cmath>
# include <iostream>
# include <fstream>
# include <iomanip>
# include <ctime>
# include <cstring>
# include <cassert>
# include <legendre_rule.hpp>

//using namespace std;

//****************************************************************************80
/*
int main ( int argc, char *argv[] )

// ***************************************************************************80
//
//  Purpose:
//
//    MAIN is the main program for LEGENDRE_RULE.
//
//  Discussion:
//
//    This program computes a standard Gauss-Legendre quadrature rule
//    and writes it to a file.
//
//    The user specifies:
//    * the ORDER (number of points) in the rule
//    * A, the left endpoint;
//    * B, the right endpoint;
//    * FILENAME, the root name of the output files.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    23 February 2010
//
//  Author:
//
//    John Burkardt
//
{
  double a;
  double alpha;
  double b;
  double beta;
  string filename;
  int kind;
  int order;
  double *r;
  double *w;
  double *x;

  timestamp ( );
  std::cout << "\n";
  std::cout << "LEGENDRE_RULE\n";
  std::cout << "  C++ version\n";
  std::cout << "\n";
  std::cout << "  Compiled on " << __DATE__ << " at " << __TIME__ << ".\n";
  std::cout << "\n";
  std::cout << "  Compute a Gauss-Legendre rule for approximating\n";
  std::cout << "\n";
  std::cout << "    Integral ( A <= x <= B ) f(x) dx\n";
  std::cout << "\n";
  std::cout << "  of order ORDER.\n";
  std::cout << "\n";
  std::cout << "  The user specifies ORDER, A, B and FILENAME.\n";
  std::cout << "\n";
  std::cout << "  Order is the number of points.\n";
  std::cout << "\n";
  std::cout << "  A is the left endpoint.\n";
  std::cout << "\n";
  std::cout << "  B is the right endpoint.\n";
  std::cout << "\n";
  std::cout << "  FILENAME is used to generate 3 files:\n";
  std::cout << "\n";
  std::cout << "    filename_w.txt - the weight file\n";
  std::cout << "    filename_x.txt - the abscissa file.\n";
  std::cout << "    filename_r.txt - the region file.\n";
//
//  Initialize parameters;
//
  alpha = 0.0;
  beta = 0.0;
//
//  Get ORDER.
//
  if ( 1 < argc )
  {
    order = atoi ( argv[1] );
  }
  else
  {
    std::cout << "\n";
    std::cout << "  Enter the value of ORDER (1 or greater)\n";
    cin >> order;
  }
//
//  Get A.
//
  if ( 2 < argc )
  {
    a = atof ( argv[2] );
  }
  else
  {
    std::cout << "\n";
    std::cout << "  Enter the value of A:\n";
    cin >> a;
  }
//
//  Get B.
//
  if ( 3 < argc )
  {
    b = atof ( argv[3] );
  }
  else
  {
    std::cout << "\n";
    std::cout << "  Enter the value of B:\n";
    cin >> b;
  }
//
//  Get FILENAME:
//
  if ( 4 < argc )
  {
    filename = argv[4];
  }
  else
  {
    std::cout << "\n";
    std::cout << "  Enter FILENAME, the \"root name\" of the quadrature files).\n";
    cin >> filename;
  }
//
//  Input summary.
//
  std::cout << "\n";
  std::cout << "  ORDER = " << order << "\n";
  std::cout << "  A = " << a << "\n";
  std::cout << "  B = " << b << "\n";
  std::cout << "  FILENAME = \"" << filename << "\".\n";
//
//  Construct the rule.
//
  w = new double[order];
  x = new double[order];
  
  kind = 1;
  cgqf ( order, kind, alpha, beta, a, b, x, w );
//
//  Write the rule.
//
  r = new double[2];
  r[0] = a;
  r[1] = b;

  rule_write ( order, filename, x, w, r );
//
//  Terminate.
//
  delete [] r;
  delete [] w;
  delete [] x;
  std::cout << "\n";
  std::cout << "LEGENDRE_RULE:\n";
  std::cout << "  Normal end of execution.\n";

  std::cout << "\n";
  timestamp ( );

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

void cdgqf ( int nt, int kind, double alpha, double beta, double t[], 
  double wts[] )

//****************************************************************************80
//
//  Purpose:
//
//    CDGQF computes a Gauss quadrature formula with default A, B and simple knots.
//
//  Discussion:
//
//    This routine computes all the knots and weights of a Gauss quadrature
//    formula with a classical weight function with default values for A and B,
//    and only simple knots.
//
//    There are no moments checks and no printing is done.
//
//    Use routine EIQFS to evaluate a quadrature computed by CGQFS.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    08 January 2010
//
//  Author:
//
//    Original FORTRAN77 version by Sylvan Elhay, Jaroslav Kautsky.
//    C++ version by John Burkardt.
//
//  Reference:
//
//    Sylvan Elhay, Jaroslav Kautsky,
//    Algorithm 655: IQPACK, FORTRAN Subroutines for the Weights of 
//    Interpolatory Quadrature,
//    ACM Transactions on Mathematical Software,
//    Volume 13, Number 4, December 1987, pages 399-415.
//
//  Parameters:
//
//    Input, int NT, the number of knots.
//
//    Input, int KIND, the rule.
//    1, Legendre,             (a,b)       1.0
//    2, Chebyshev Type 1,     (a,b)       ((b-x)*(x-a))^(-0.5)
//    3, Gegenbauer,           (a,b)       ((b-x)*(x-a))^alpha
//    4, Jacobi,               (a,b)       (b-x)^alpha*(x-a)^beta
//    5, Generalized Laguerre, (a,inf)     (x-a)^alpha*exp(-b*(x-a))
//    6, Generalized Hermite,  (-inf,inf)  |x-a|^alpha*exp(-b*(x-a)^2)
//    7, Exponential,          (a,b)       |x-(a+b)/2.0|^alpha
//    8, Rational,             (a,inf)     (x-a)^alpha*(x+b)^beta
//
//    Input, double ALPHA, the value of Alpha, if needed.
//
//    Input, double BETA, the value of Beta, if needed.
//
//    Output, double T[NT], the knots.
//
//    Output, double WTS[NT], the weights.
//
{
  double *aj;
  double *bj;
  double zemu;

  parchk ( kind, 2 * nt, alpha, beta );
//
//  Get the Jacobi matrix and zero-th moment.
//
  aj = new double[nt];
  bj = new double[nt];

  zemu = class_matrix ( kind, nt, alpha, beta, aj, bj );
//
//  Compute the knots and weights.
//
  sgqf ( nt, aj, bj, zemu, t, wts );

  delete [] aj;
  delete [] bj;

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

void cgqf ( int nt, int kind, double alpha, double beta, double a, double b, 
  double t[], double wts[] )

//****************************************************************************80
//
//  Purpose:
//
//    CGQF computes knots and weights of a Gauss quadrature formula.
//
//  Discussion:
//
//    The user may specify the interval (A,B).
//
//    Only simple knots are produced.
//
//    Use routine EIQFS to evaluate this quadrature formula.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    16 February 2010
//
//  Author:
//
//    Original FORTRAN77 version by Sylvan Elhay, Jaroslav Kautsky.
//    C++ version by John Burkardt.
//
//  Reference:
//
//    Sylvan Elhay, Jaroslav Kautsky,
//    Algorithm 655: IQPACK, FORTRAN Subroutines for the Weights of 
//    Interpolatory Quadrature,
//    ACM Transactions on Mathematical Software,
//    Volume 13, Number 4, December 1987, pages 399-415.
//
//  Parameters:
//
//    Input, int NT, the number of knots.
//
//    Input, int KIND, the rule.
//    1, Legendre,             (a,b)       1.0
//    2, Chebyshev Type 1,     (a,b)       ((b-x)*(x-a))^-0.5)
//    3, Gegenbauer,           (a,b)       ((b-x)*(x-a))^alpha
//    4, Jacobi,               (a,b)       (b-x)^alpha*(x-a)^beta
//    5, Generalized Laguerre, (a,+oo)     (x-a)^alpha*exp(-b*(x-a))
//    6, Generalized Hermite,  (-oo,+oo)   |x-a|^alpha*exp(-b*(x-a)^2)
//    7, Exponential,          (a,b)       |x-(a+b)/2.0|^alpha
//    8, Rational,             (a,+oo)     (x-a)^alpha*(x+b)^beta
//    9, Chebyshev Type 2,     (a,b)       ((b-x)*(x-a))^(+0.5)
//
//    Input, double ALPHA, the value of Alpha, if needed.
//
//    Input, double BETA, the value of Beta, if needed.
//
//    Input, double A, B, the interval endpoints, or
//    other parameters.
//
//    Output, double T[NT], the knots.
//
//    Output, double WTS[NT], the weights.
//
{
  int i;
  int *mlt;
  int *ndx;
//
//  Compute the Gauss quadrature formula for default values of A and B.
//
  cdgqf ( nt, kind, alpha, beta, t, wts );
//
//  Prepare to scale the quadrature formula to other weight function with 
//  valid A and B.
//
  mlt = new int[nt];
  for ( i = 0; i < nt; i++ )
  {
    mlt[i] = 1;
  }
  ndx = new int[nt];
  for ( i = 0; i < nt; i++ )
  {
    ndx[i] = i + 1;
  }
  scqf ( nt, t, mlt, wts, nt, ndx, wts, t, kind, alpha, beta, a, b );

  delete [] mlt;
  delete [] ndx;

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

double class_matrix ( int kind, int m, double alpha, double beta, double aj[], 
  double bj[] )

//****************************************************************************80
//
//  Purpose:
//
//    CLASS_MATRIX computes the Jacobi matrix for a quadrature rule.
//
//  Discussion:
//
//    This routine computes the diagonal AJ and sub-diagonal BJ
//    elements of the order M tridiagonal symmetric Jacobi matrix
//    associated with the polynomials orthogonal with respect to
//    the weight function specified by KIND.
//
//    For weight functions 1-7, M elements are defined in BJ even
//    though only M-1 are needed.  For weight function 8, BJ(M) is
//    set to zero.
//
//    The zero-th moment of the weight function is returned in ZEMU.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    08 January 2010
//
//  Author:
//
//    Original FORTRAN77 version by Sylvan Elhay, Jaroslav Kautsky.
//    C++ version by John Burkardt.
//
//  Reference:
//
//    Sylvan Elhay, Jaroslav Kautsky,
//    Algorithm 655: IQPACK, FORTRAN Subroutines for the Weights of 
//    Interpolatory Quadrature,
//    ACM Transactions on Mathematical Software,
//    Volume 13, Number 4, December 1987, pages 399-415.
//
//  Parameters:
//
//    Input, int KIND, the rule.
//    1, Legendre,             (a,b)       1.0
//    2, Chebyshev Type 1,     (a,b)       ((b-x)*(x-a))^(-0.5)
//    3, Gegenbauer,           (a,b)       ((b-x)*(x-a))^alpha
//    4, Jacobi,               (a,b)       (b-x)^alpha*(x-a)^beta
//    5, Generalized Laguerre, (a,inf)     (x-a)^alpha*exp(-b*(x-a))
//    6, Generalized Hermite,  (-inf,inf)  |x-a|^alpha*exp(-b*(x-a)^2)
//    7, Exponential,          (a,b)       |x-(a+b)/2.0|^alpha
//    8, Rational,             (a,inf)     (x-a)^alpha*(x+b)^beta
//
//    Input, int M, the order of the Jacobi matrix.
//
//    Input, double ALPHA, the value of Alpha, if needed.
//
//    Input, double BETA, the value of Beta, if needed.
//
//    Output, double AJ[M], BJ[M], the diagonal and subdiagonal
//    of the Jacobi matrix.
//
//    Output, double CLASS_MATRIX, the zero-th moment.
//
{
  double a2b2;
  double ab;
  double aba;
  double abi;
  double abj;
  double abti;
  double apone;
  int i;
  double pi = 3.14159265358979323846264338327950;
  double zemu=0;

  parchk ( kind, 2 * m - 1, alpha, beta );

/*
  double temp = r8_epsilon ( );
  double temp2 = 0.5;
  if ( 500.0 * temp < r8_abs ( pow ( tgamma ( temp2 ), 2 ) - pi ) )
  {
    std::cout << "\n";
    std::cout << "CLASS_MATRIX - Fatal error!\n";
    std::cout << "  Gamma function does not match machine parameters.\n";
    exit ( 1 );
  }
*/
  if ( kind == 1 )
  {
    ab = 0.0;

    zemu = 2.0 / ( ab + 1.0 );

    for ( i = 0; i < m; i++ )
    {
      aj[i] = 0.0;
    }

    for ( i = 1; i <= m; i++ )
    {
      abi = i + ab * ( i % 2 );
      abj = 2 * i + ab;
      bj[i-1] = sqrt ( abi * abi / ( abj * abj - 1.0 ) );
    }
  }
  else if ( kind == 2 )
  {
    zemu = pi;

    for ( i = 0; i < m; i++ )
    {
      aj[i] = 0.0;
    }

    bj[0] =  sqrt ( 0.5 );
    for ( i = 1; i < m; i++ )
    {
      bj[i] = 0.5;
    }
  }
  else if ( kind == 3 )
  {
    ab = alpha * 2.0;
    zemu = pow ( 2.0, ab + 1.0 ) * pow ( tgamma ( alpha + 1.0 ), 2 )
      / tgamma ( ab + 2.0 );

    for ( i = 0; i < m; i++ )
    {
      aj[i] = 0.0;
    }

    bj[0] = sqrt ( 1.0 / ( 2.0 * alpha + 3.0 ) );
    for ( i = 2; i <= m; i++ )
    {
      bj[i-1] = sqrt ( i * ( i + ab ) / ( 4.0 * pow ( i + alpha, 2 ) - 1.0 ) );
    }
  }
  else if ( kind == 4 )
  {
    ab = alpha + beta;
    abi = 2.0 + ab;
    zemu = pow ( 2.0, ab + 1.0 ) * tgamma ( alpha + 1.0 ) 
      * tgamma ( beta + 1.0 ) / tgamma ( abi );
    aj[0] = ( beta - alpha ) / abi;
    bj[0] = sqrt ( 4.0 * ( 1.0 + alpha ) * ( 1.0 + beta ) 
      / ( ( abi + 1.0 ) * abi * abi ) );
    a2b2 = beta * beta - alpha * alpha;

    for ( i = 2; i <= m; i++ )
    {
      abi = 2.0 * i + ab;
      aj[i-1] = a2b2 / ( ( abi - 2.0 ) * abi );
      abi = abi * abi;
      bj[i-1] = sqrt ( 4.0 * i * ( i + alpha ) * ( i + beta ) * ( i + ab ) 
        / ( ( abi - 1.0 ) * abi ) );
    }
  }
  else if ( kind == 5 )
  {
    zemu = tgamma ( alpha + 1.0 );

    for ( i = 1; i <= m; i++ )
    {
      aj[i-1] = 2.0 * i - 1.0 + alpha;
      bj[i-1] = sqrt ( i * ( i + alpha ) );
    }
  }
  else if ( kind == 6 )
  {
    zemu = tgamma ( ( alpha + 1.0 ) / 2.0 );

    for ( i = 0; i < m; i++ )
    {
      aj[i] = 0.0;
    }

    for ( i = 1; i <= m; i++ )
    {
      bj[i-1] = sqrt ( ( i + alpha * ( i % 2 ) ) / 2.0 );
    }
  }
  else if ( kind == 7 )
  {
    ab = alpha;
    zemu = 2.0 / ( ab + 1.0 );

    for ( i = 0; i < m; i++ )
    {
      aj[i] = 0.0;
    }

    for ( i = 1; i <= m; i++ )
    {
      abi = i + ab * ( i % 2 );
      abj = 2 * i + ab;
      bj[i-1] = sqrt ( abi * abi / ( abj * abj - 1.0 ) );
    }
  }
  else if ( kind == 8 )
  {
    ab = alpha + beta;
    zemu = tgamma ( alpha + 1.0 ) * tgamma ( - ( ab + 1.0 ) ) 
      / tgamma ( - beta );
    apone = alpha + 1.0;
    aba = ab * apone;
    aj[0] = - apone / ( ab + 2.0 );
    bj[0] = - aj[0] * ( beta + 1.0 ) / ( ab + 2.0 ) / ( ab + 3.0 );
    for ( i = 2; i <= m; i++ )
    {
      abti = ab + 2.0 * i;
      aj[i-1] = aba + 2.0 * ( ab + i ) * ( i - 1 );
      aj[i-1] = - aj[i-1] / abti / ( abti - 2.0 );
    }

    for ( i = 2; i <= m - 1; i++ )
    {
      abti = ab + 2.0 * i;
      bj[i-1] = i * ( alpha + i ) / ( abti - 1.0 ) * ( beta + i ) 
        / ( abti * abti ) * ( ab + i ) / ( abti + 1.0 );
    }
    bj[m-1] = 0.0;
    for ( i = 0; i < m; i++ )
    {
      bj[i] =  sqrt ( bj[i] );
    }
  }

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

void imtqlx ( int n, double d[], double e[], double z[] )

//****************************************************************************80
//
//  Purpose:
//
//    IMTQLX diagonalizes a symmetric tridiagonal matrix.
//
//  Discussion:
//
//    This routine is a slightly modified version of the EISPACK routine to 
//    perform the implicit QL algorithm on a symmetric tridiagonal matrix. 
//
//    The authors thank the authors of EISPACK for permission to use this
//    routine. 
//
//    It has been modified to produce the product Q' * Z, where Z is an input 
//    vector and Q is the orthogonal matrix diagonalizing the input matrix.  
//    The changes consist (essentialy) of applying the orthogonal transformations
//    directly to Z as they are generated.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    08 January 2010
//
//  Author:
//
//    Original FORTRAN77 version by Sylvan Elhay, Jaroslav Kautsky.
//    C++ version by John Burkardt.
//
//  Reference:
//
//    Sylvan Elhay, Jaroslav Kautsky,
//    Algorithm 655: IQPACK, FORTRAN Subroutines for the Weights of 
//    Interpolatory Quadrature,
//    ACM Transactions on Mathematical Software,
//    Volume 13, Number 4, December 1987, pages 399-415.
//
//    Roger Martin, James Wilkinson,
//    The Implicit QL Algorithm,
//    Numerische Mathematik,
//    Volume 12, Number 5, December 1968, pages 377-383.
//
//  Parameters:
//
//    Input, int N, the order of the matrix.
//
//    Input/output, double D(N), the diagonal entries of the matrix.
//    On output, the information in D has been overwritten.
//
//    Input/output, double E(N), the subdiagonal entries of the 
//    matrix, in entries E(1) through E(N-1).  On output, the information in
//    E has been overwritten.
//
//    Input/output, double Z(N).  On input, a vector.  On output,
//    the value of Q' * Z, where Q is the matrix that diagonalizes the
//    input symmetric tridiagonal matrix.
//
{
  double b;
  double c;
  double f;
  double g;
  int i;
  int ii;
  int itn = 30;
  int j;
  int k;
  int l;
  int m=0;
  int mml;
  double p;
  double prec;
  double r;
  double s;

  prec = r8_epsilon ( );

  if ( n == 1 )
  {
    return;
  }

  e[n-1] = 0.0;

  for ( l = 1; l <= n; l++ )
  {
    j = 0;
    for ( ; ; )
    {
      for ( m = l; m <= n; m++ )
      {
        if ( m == n )
        {
          break;
        }

        if ( r8_abs ( e[m-1] ) <= prec * ( r8_abs ( d[m-1] ) + r8_abs ( d[m] ) ) )
        {
          break;
        }
      }
      p = d[l-1];
      if ( m == l )
      {
        break;
      }
      if ( itn <= j )
      {
        std::cout << "\n";
        std::cout << "IMTQLX - Fatal error!\n";
        std::cout << "  Iteration limit exceeded\n";
        exit ( 1 );
      }
      j = j + 1;
      g = ( d[l] - p ) / ( 2.0 * e[l-1] );
      r =  sqrt ( g * g + 1.0 );
      g = d[m-1] - p + e[l-1] / ( g + r8_abs ( r ) * r8_sign ( g ) );
      s = 1.0;
      c = 1.0;
      p = 0.0;
      mml = m - l;

      for ( ii = 1; ii <= mml; ii++ )
      {
        i = m - ii;
        f = s * e[i-1];
        b = c * e[i-1];

        if ( r8_abs ( g ) <= r8_abs ( f ) )
        {
          c = g / f;
          r =  sqrt ( c * c + 1.0 );
          e[i] = f * r;
          s = 1.0 / r;
          c = c * s;
        }
        else
        {
          s = f / g;
          r =  sqrt ( s * s + 1.0 );
          e[i] = g * r;
          c = 1.0 / r;
          s = s * c;
        }
        g = d[i] - p;
        r = ( d[i-1] - g ) * s + 2.0 * c * b;
        p = s * r;
        d[i] = g + p;
        g = c * r - b;
        f = z[i];
        z[i] = s * z[i-1] + c * f;
        z[i-1] = c * z[i-1] - s * f;
      }
      d[l-1] = d[l-1] - p;
      e[l-1] = g;
      e[m-1] = 0.0;
    }
  }
//
//  Sorting.
//
  for ( ii = 2; ii <= m; ii++ )
  {
    i = ii - 1;
    k = i;
    p = d[i-1];

    for ( j = ii; j <= n; j++ )
    {
      if ( d[j-1] < p )
      {
         k = j;
         p = d[j-1];
      }
    }

    if ( k != i )
    {
      d[k-1] = d[i-1];
      d[i-1] = p;
      p = z[i-1];
      z[i-1] = z[k-1];
      z[k-1] = p;
    }
  }
  return;
}
//****************************************************************************80

void parchk ( int kind, int m, double alpha, double beta )

//****************************************************************************80
//
//  Purpose:
//
//    PARCHK checks parameters ALPHA and BETA for classical weight functions. 
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    07 January 2010
//
//  Author:
//
//    Original FORTRAN77 version by Sylvan Elhay, Jaroslav Kautsky.
//    C++ version by John Burkardt.
//
//  Reference:
//
//    Sylvan Elhay, Jaroslav Kautsky,
//    Algorithm 655: IQPACK, FORTRAN Subroutines for the Weights of 
//    Interpolatory Quadrature,
//    ACM Transactions on Mathematical Software,
//    Volume 13, Number 4, December 1987, pages 399-415.
//
//  Parameters:
//
//    Input, int KIND, the rule.
//    1, Legendre,             (a,b)       1.0
//    2, Chebyshev Type 1,     (a,b)       ((b-x)*(x-a))^(-0.5)
//    3, Gegenbauer,           (a,b)       ((b-x)*(x-a))^alpha
//    4, Jacobi,               (a,b)       (b-x)^alpha*(x-a)^beta
//    5, Generalized Laguerre, (a,inf)     (x-a)^alpha*exp(-b*(x-a))
//    6, Generalized Hermite,  (-inf,inf)  |x-a|^alpha*exp(-b*(x-a)^2)
//    7, Exponential,          (a,b)       |x-(a+b)/2.0|^alpha
//    8, Rational,             (a,inf)     (x-a)^alpha*(x+b)^beta
//
//    Input, int M, the order of the highest moment to
//    be calculated.  This value is only needed when KIND = 8.
//
//    Input, double ALPHA, BETA, the parameters, if required
//    by the value of KIND.
//
{
  double tmp;

  if ( kind <= 0 )
  {
    std::cout << "\n";
    std::cout << "PARCHK - Fatal error!\n";
    std::cout << "  KIND <= 0.\n";
    exit ( 1 );
  }
//
//  Check ALPHA for Gegenbauer, Jacobi, Laguerre, Hermite, Exponential.
//
  if ( 3 <= kind && alpha <= -1.0 )
  {
    std::cout << "\n";
    std::cout << "PARCHK - Fatal error!\n";
    std::cout << "  3 <= KIND and ALPHA <= -1.\n";
    exit ( 1 );
  }
//
//  Check BETA for Jacobi.
//
  if ( kind == 4 && beta <= -1.0 )
  {
    std::cout << "\n";
    std::cout << "PARCHK - Fatal error!\n";
    std::cout << "  KIND == 4 and BETA <= -1.0.\n";
    exit ( 1 );
  }
//
//  Check ALPHA and BETA for rational.
//
  if ( kind == 8 )
  {
    tmp = alpha + beta + m + 1.0;
    if ( 0.0 <= tmp || tmp <= beta )
    {
      std::cout << "\n";
      std::cout << "PARCHK - Fatal error!\n";
      std::cout << "  KIND == 8 but condition on ALPHA and BETA fails.\n";
      exit ( 1 );
    }
  }
  return;
}
//****************************************************************************80

double r8_abs ( double x )

//****************************************************************************80
//
//  Purpose:
//
//    R8_ABS returns the absolute value of an R8.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    14 November 2006
//
//  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_epsilon ( )

//****************************************************************************80
//
//  Purpose:
//
//    R8_EPSILON returns the R8 roundoff unit.
//
//  Discussion:
//
//    The roundoff unit is a number R which is a power of 2 with the 
//    property that, to the precision of the computer's arithmetic,
//      1 < 1 + R
//    but 
//      1 = ( 1 + R / 2 )
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    01 July 2004
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Output, double R8_EPSILON, the R8 round-off unit.
//
{
  double value;

  value = 1.0;

  while ( 1.0 < ( double ) ( 1.0 + value )  )
  {
    value = value / 2.0;
  }

  value = 2.0 * value;

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

double r8_sign ( double x )

//****************************************************************************80
//
//  Purpose:
//
//    R8_SIGN returns the sign of an 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_write ( std::string output_filename, int m, int n, double table[] )

//****************************************************************************80
//
//  Purpose:
//
//    R8MAT_WRITE writes an R8MAT file with no header.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    29 June 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, string OUTPUT_FILENAME, the output filename.
//
//    Input, int M, the spatial dimension.
//
//    Input, int N, the number of points.
//
//    Input, double TABLE[M*N], the table data.
//
{
  int i;
  int j;
  std::ofstream output;
//
//  Open the file.
//
  output.open ( output_filename.c_str ( ) );

  if ( !output )
  {
    std::cerr << "\n";
    std::cerr << "R8MAT_WRITE - Fatal error!\n";
    std::cerr << "  Could not open the output file.\n";
    return;
  }
//
//  Write the data.
//
  for ( j = 0; j < n; j++ )
  {
    for ( i = 0; i < m; i++ )
    {
      output << "  " << std::setw(24) << std::setprecision(16) << table[i+j*m];
    }
    output << "\n";
  }
//
//  Close the file.
//
  output.close ( );

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

void rule_write ( int order, std::string filename, double x[], double w[], 
  double r[] )

//****************************************************************************80
//
//  Purpose:
//
//    RULE_WRITE writes a quadrature rule to three files.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    18 February 2010
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int ORDER, the order of the rule.
//
//    Input, double A, the left endpoint.
//
//    Input, double B, the right endpoint.
//
//    Input, string FILENAME, specifies the output filenames.
//    "filename_w.txt", "filename_x.txt", "filename_r.txt" 
//    defining weights, abscissas, and region.
// 
{
  std::string filename_r;
  std::string filename_w;
  std::string filename_x;

  filename_w = filename + "_w.txt";
  filename_x = filename + "_x.txt";
  filename_r = filename + "_r.txt";

  std::cout << "\n";
  std::cout << "  Creating quadrature files.\n";
  std::cout << "\n";
  std::cout << "  Root file name is     \"" << filename   << "\".\n";
  std::cout << "\n";
  std::cout << "  Weight file will be   \"" << filename_w << "\".\n";
  std::cout << "  Abscissa file will be \"" << filename_x << "\".\n";
  std::cout << "  Region file will be   \"" << filename_r << "\".\n";
            
  r8mat_write ( filename_w, 1, order, w );
  r8mat_write ( filename_x, 1, order, x );
  r8mat_write ( filename_r, 1, 2,     r );
  
  return;
}
//****************************************************************************80

void scqf ( int nt, double t[], int mlt[], double wts[], int nwts, int ndx[], 
  double swts[], double st[], int kind, double alpha, double beta, double a, 
  double b )

//****************************************************************************80
//
//  Purpose:
//
//    SCQF scales a quadrature formula to a nonstandard interval.
//
//  Discussion:
//
//    The arrays WTS and SWTS may coincide.
//
//    The arrays T and ST may coincide.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    16 February 2010
//
//  Author:
//
//    Original FORTRAN77 version by Sylvan Elhay, Jaroslav Kautsky.
//    C++ version by John Burkardt.
//
//  Reference:
//
//    Sylvan Elhay, Jaroslav Kautsky,
//    Algorithm 655: IQPACK, FORTRAN Subroutines for the Weights of 
//    Interpolatory Quadrature,
//    ACM Transactions on Mathematical Software,
//    Volume 13, Number 4, December 1987, pages 399-415.
//
//  Parameters:
//
//    Input, int NT, the number of knots.
//
//    Input, double T[NT], the original knots.
//
//    Input, int MLT[NT], the multiplicity of the knots.
//
//    Input, double WTS[NWTS], the weights.
//
//    Input, int NWTS, the number of weights.
//
//    Input, int NDX[NT], used to index the array WTS.  
//    For more details see the comments in CAWIQ.
//
//    Output, double SWTS[NWTS], the scaled weights.
//
//    Output, double ST[NT], the scaled knots.
//
//    Input, int KIND, the rule.
//    1, Legendre,             (a,b)       1.0
//    2, Chebyshev Type 1,     (a,b)       ((b-x)*(x-a))^(-0.5)
//    3, Gegenbauer,           (a,b)       ((b-x)*(x-a))^alpha
//    4, Jacobi,               (a,b)       (b-x)^alpha*(x-a)^beta
//    5, Generalized Laguerre, (a,+oo)     (x-a)^alpha*exp(-b*(x-a))
//    6, Generalized Hermite,  (-oo,+oo)   |x-a|^alpha*exp(-b*(x-a)^2)
//    7, Exponential,          (a,b)       |x-(a+b)/2.0|^alpha
//    8, Rational,             (a,+oo)     (x-a)^alpha*(x+b)^beta
//    9, Chebyshev Type 2,     (a,b)       ((b-x)*(x-a))^(+0.5)
//
//    Input, double ALPHA, the value of Alpha, if needed.
//
//    Input, double BETA, the value of Beta, if needed.
//
//    Input, double A, B, the interval endpoints.
//
{
  double al=0;
  double be=0;
  int i;
  int k;
  int l;
  double p;
  double shft=0;
  double slp=0;
  double temp;
  double tmp;

  temp = r8_epsilon ( );

  parchk ( kind, 1, alpha, beta );

  if ( kind == 1 )
  {
    al = 0.0;
    be = 0.0;
    if ( r8_abs ( b - a ) <= temp )
    {
      std::cout << "\n";
      std::cout << "SCQF - Fatal error!\n";
      std::cout << "  |B - A| too small.\n";
      exit ( 1 );
    }
    shft = ( a + b ) / 2.0;
    slp = ( b - a ) / 2.0;
  }
  else if ( kind == 2 )
  {
    al = -0.5;
    be = -0.5;
    if ( r8_abs ( b - a ) <= temp )
    {
      std::cout << "\n";
      std::cout << "SCQF - Fatal error!\n";
      std::cout << "  |B - A| too small.\n";
      exit ( 1 );
    }
    shft = ( a + b ) / 2.0;
    slp = ( b - a ) / 2.0;
  }
  else if ( kind == 3 )
  {
    al = alpha;
    be = alpha;
    if ( r8_abs ( b - a ) <= temp )
    {
      std::cout << "\n";
      std::cout << "SCQF - Fatal error!\n";
      std::cout << "  |B - A| too small.\n";
      exit ( 1 );
    }
    shft = ( a + b ) / 2.0;
    slp = ( b - a ) / 2.0;
  }
  else if ( kind == 4 )
  {
    al = alpha;
    be = beta;

    if ( r8_abs ( b - a ) <= temp )
    {
      std::cout << "\n";
      std::cout << "SCQF - Fatal error!\n";
      std::cout << "  |B - A| too small.\n";
      exit ( 1 );
    }
    shft = ( a + b ) / 2.0;
    slp = ( b - a ) / 2.0;
  }
  else if ( kind == 5 )
  {
    if ( b <= 0.0 )
    {
      std::cout << "\n";
      std::cout << "SCQF - Fatal error!\n";
      std::cout << "  B <= 0\n";
      exit ( 1 );
    }
    shft = a;
    slp = 1.0 / b;
    al = alpha;
    be = 0.0;
  }
  else if ( kind == 6 )
  {
    if ( b <= 0.0 )
    {
      std::cout << "\n";
      std::cout << "SCQF - Fatal error!\n";
      std::cout << "  B <= 0.\n";
      exit ( 1 );
    }
    shft = a;
    slp = 1.0 / sqrt ( b );
    al = alpha;
    be = 0.0;
  }
  else if ( kind == 7 )
  {
    al = alpha;
    be = 0.0;
    if ( r8_abs ( b - a ) <= temp )
    {
      std::cout << "\n";
      std::cout << "SCQF - Fatal error!\n";
      std::cout << "  |B - A| too small.\n";
      exit ( 1 );
    }
    shft = ( a + b ) / 2.0;
    slp = ( b - a ) / 2.0;
  }
  else if ( kind == 8 )
  {
    if ( a + b <= 0.0 )
    {
      std::cout << "\n";
      std::cout << "SCQF - Fatal error!\n";
      std::cout << "  A + B <= 0.\n";
      exit ( 1 );
    }
    shft = a;
    slp = a + b;
    al = alpha;
    be = beta;
  }
  else if ( kind == 9 )
  {
    al = 0.5;
    be = 0.5;
    if ( r8_abs ( b - a ) <= temp )
    {
      std::cout << "\n";
      std::cout << "SCQF - Fatal error!\n";
      std::cout << "  |B - A| too small.\n";
      exit ( 1 );
    }
    shft = ( a + b ) / 2.0;
    slp = ( b - a ) / 2.0;
  }

  p = pow ( slp, al + be + 1.0 );

  for ( k = 0; k < nt; k++ )
  {
    st[k] = shft + slp * t[k];
    l = abs ( ndx[k] );

    if ( l != 0 )
    {
      tmp = p;
      for ( i = l - 1; i <= l - 1 + mlt[k] - 1; i++ )
      {
        swts[i] = wts[i] * tmp;
        tmp = tmp * slp;
      }
    }
  }
  return;
}
//****************************************************************************80

void sgqf ( int nt, double aj[], double bj[], double zemu, double t[], 
  double wts[] )

//****************************************************************************80
//
//  Purpose:
//
//    SGQF computes knots and weights of a Gauss Quadrature formula.
//
//  Discussion:
//
//    This routine computes all the knots and weights of a Gauss quadrature
//    formula with simple knots from the Jacobi matrix and the zero-th
//    moment of the weight function, using the Golub-Welsch technique.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    08 January 2010
//
//  Author:
//
//    Original FORTRAN77 version by Sylvan Elhay, Jaroslav Kautsky.
//    C++ version by John Burkardt.
//
//  Reference:
//
//    Sylvan Elhay, Jaroslav Kautsky,
//    Algorithm 655: IQPACK, FORTRAN Subroutines for the Weights of 
//    Interpolatory Quadrature,
//    ACM Transactions on Mathematical Software,
//    Volume 13, Number 4, December 1987, pages 399-415.
//
//  Parameters:
//
//    Input, int NT, the number of knots.
//
//    Input, double AJ[NT], the diagonal of the Jacobi matrix.
//
//    Input/output, double BJ[NT], the subdiagonal of the Jacobi 
//    matrix, in entries 1 through NT-1.  On output, BJ has been overwritten.
//
//    Input, double ZEMU, the zero-th moment of the weight function.
//
//    Output, double T[NT], the knots.
//
//    Output, double WTS[NT], the weights.
//
{
  int i;
//
//  Exit if the zero-th moment is not positive.
//
  if ( zemu <= 0.0 )
  {
    std::cout << "\n";
    std::cout << "SGQF - Fatal error!\n";
    std::cout << "  ZEMU <= 0.\n";
    exit ( 1 );
  }
//
//  Set up vectors for IMTQLX.
//
  for ( i = 0; i < nt; i++ )
  {
    t[i] = aj[i];
  }
  wts[0] = sqrt ( zemu );
  for ( i = 1; i < nt; i++ )
  {
    wts[i] = 0.0;
  }
//
//  Diagonalize the Jacobi matrix.
//
  imtqlx ( nt, t, bj, wts );

  for ( i = 0; i < nt; i++ )
  {
    wts[i] = wts[i] * wts[i];
  }

  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 );
  assert(len>0);
  std::cout << time_buffer << "\n";

  return;
# undef TIME_SIZE
}