#! /usr/bin/env python
#
def legendre_ek_compute ( n ):

#*****************************************************************************80
#
## LEGENDRE_EK_COMPUTE: Gauss-Legendre, Elhay-Kautsky method.
#
#  Discussion:
#
#    The integral:
#
#      integral ( -1 < x < +1 ) f(x) dx
#
#    The quadrature rule:
#
#      sum ( 1 <= i <= n ) w(i) * f ( x(i) )
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    16 June 2015
#
#  Author:
#
#    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, integer N, the number of abscissas.
#
#    Output, real X(N), the abscissas.
#
#    Output, real W(N), the weights.
#
  import numpy as np
  from imtqlx import imtqlx
#
#  Define the zero-th moment.
#
  zemu = 2.0
#
#  Define the Jacobi matrix.
#
  bj = np.zeros ( n )
  for i in range ( 0, n ):
    ip1_r8 = float ( i + 1 )
    bj[i] = ip1_r8 * ip1_r8 / ( 4.0 * ip1_r8 * ip1_r8 - 1.0 )
    bj[i] = np.sqrt ( bj[i] )

  x = np.zeros ( n )

  w = np.zeros ( n )
  w[0] = np.sqrt ( zemu )
#
#  Diagonalize the Jacobi matrix.
#
  x, w = imtqlx ( n, x, bj, w )

  for i in range ( 0, n ):
    w[i] = w[i] ** 2

  return x, w

def legendre_ek_compute_test ( ):

#*****************************************************************************80
#
## LEGENDRE_EK_COMPUTE_TEST tests LEGENDRE_EK_COMPUTE.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license. 
#
#  Modified:
#
#    16 June 2015
#
#  Author:
#
#    John Burkardt
#
  import platform

  print ( '' )
  print ( 'LEGENDRE_EK_COMPUTE_TEST' )
  print ( '  Python version: %s' % ( platform.python_version ( ) ) )
  print ( '  LEGENDRE_EK_COMPUTE computes' )
  print ( '  a Legendre quadrature rule' )
  print ( '  using the Elhay-Kautsky algorithm.' )
  print ( '' )
  print ( '  Index       X             W' )

  for n in range ( 1, 11 ):

    x, w = legendre_ek_compute ( n )

    print ( '' )

    for i in range ( 0, n ):
      print ( '  %2d  %24.16g  %24.16g' % ( i, x[i], w[i] ) )
#
#  Terminate.
#
  print ( '' )
  print ( 'LEGENDRE_EK_COMPUTE_TEST:' )
  print ( '  Normal end of execution.' )
  return

if ( __name__ == '__main__' ):
  from timestamp import timestamp
  timestamp ( )
  legendre_ek_compute_test ( )
  timestamp ( )