#! /usr/bin/env python3
#
def cauchy_principal_value ( f, a, b, n ):

#*****************************************************************************80
#
## cauchy_principal_value() estimates the Cauchy Principal Value of an integral.
#
#  Discussion:
#
#    This function can be used to estimate the Cauchy Principal Value of
#    a singular integral of the form 
#      Integral f(t)/(t-x) dt 
#    over an interval which includes the singularity point t=x.
#
#    Isolate the singularity at x in a symmetric interval of finite size delta:
#
#        cauchy_principal_value ( Integral ( a         <= t <= b         ) p(t) / ( t - x ) dt )
#      =       Integral ( a         <= t <= x - delta ) p(t) / ( t - x ) dt
#      + cauchy_principal_value ( Integral ( x - delta <= t <= x + delta ) p(t) / ( t - x ) dt )
#      +       Integral ( x + delta <= t <= b         ) p(t) / ( t - x ) dt.
#
#    We assume the first and third integrals can be handled in the usual way.
#    The second integral can be rewritten as
#      Integral ( -1 <= s <= +1 ) ( p(s*delta+x) - p(x) ) / s ds
#    and approximated by
#      Sum ( 1 <= i <= N ) w(i) * ( p(xi*delta+x) - p(x) ) / xi(i)
#    = Sum ( 1 <= i <= N ) w(i) * ( p(xi*delta+x) ) / xi(i)
#    if we assume that N is even, so that coefficients of p(x) sum to zero.
#
#  Licensing:
#
#    This code is distributed under the MIT license.
#
#  Modified:
#
#    02 September 2021
#
#  Author:
#
#    John Burkardt
#
#  Reference:
#
#    Julian Noble,
#    Gauss-Legendre Principal Value Integration,
#    Computing in Science and Engineering,
#    Volume 2, Number 1, January-February 2000, pages 92-95.
#
#  Input:
#
#    real F ( X ), the function that evaluates the
#    integrand.
#
#    real A, B, the endpoints of the symmetric interval, 
#    which contains a singularity of the form 1/(X-(A+B)/2).
#
#    integer N, the number of Gauss points to use.
#    N must be even.
#
#  Output:
#
#    real VALUE, the estimate for the Cauchy Principal Value.
#

#
#  N must be even.
#
  if ( ( n % 2 ) != 0 ):
    print ( '' )
    print ( 'cauchy_principal_value - Fatal error!' )
    print ( '  N must be even.' )
    raise Exception ( 'cauchy_principal_value - Fatal error.' )
#
#  Get the Gauss-Legendre rule.
#
  [ x, w ] = legendre_set ( n );
#
#  Estimate the integral.
#
  value = 0.0;
  for i in range ( 0, n ):
    x2 = ( ( 1.0 - x[i] ) * a   \
         + ( 1.0 + x[i] ) * b ) \
         /   2.0
    value = value + w[i] * ( f ( x2 ) ) / x[i]

  return value

def cauchy_principal_value_test01 ( ):

#*****************************************************************************80
#
## cauchy_principal_value_test01() seeks CPV of Integral ( -1 <= t <= 1 ) exp(t) / t dt
#
#  Licensing:
#
#    This code is distributed under the MIT license.
#
#  Modified:
#
#    02 April 2015
#
#  Author:
#
#    John Burkardt
#
  import platform

  print ( '' )
  print ( 'cauchy_principal_value_test01():' )
  print ( '  Python version: %s' % ( platform.python_version ( ) ) )
  print ( '  cauchy_principal_value() for Integral ( -1 <= t <= 1 ) exp(t) / t dt' )

  print ( '' )
  print ( '   N           Estimate             Error' )
  print ( '' )

  exact = 2.11450175075
  a = -1.0
  b = +1.0
  for n in range ( 2, 10, 2 ):
    value = cauchy_principal_value ( f01, a, b, n )
    print ( '  %2d  %24.16g  %14.6g' % ( n, value, abs ( value - exact ) ) )
#
#  Terminate.
#
  print ( '' )
  print ( 'cauchy_principal_value_test01():' )
  print ( '  Normal end of execution.' )
  return

def f01 ( t ):

#*****************************************************************************80
#
## f01() evaluates the integrand of Integral ( -1 <= t <= 1 ) exp(t) / t dt
#
#  Licensing:
#
#    This code is distributed under the MIT license.
#
#  Modified:
#
#    01 April 2015
#
#  Author:
#
#    John Burkardt
#
#  Input:
#
#    real T, the argument.
#
#  Output:
#
#    real VALUE, the value of the integrand.
#
  import numpy as np

  value = np.exp ( t )

  return value

def cauchy_principal_value_test02 ( ):

#*****************************************************************************80
#
## cauchy_principal_value_test02() is another test.
#
#  Discussion:
#
#    We seek
#      cauchy_principal_value ( Integral ( 1-delta <= t <= 1+delta ) 1/(1-t)^3 dt )
#    which we must rewrite as
#      cauchy_principal_value ( Integral ( 1-delta <= t <= 1+delta ) 1/(1+t+t^2) 1/(1-t) dt )
#    so that our "integrand" is 1/(1+t+t^2).
#
#  Licensing:
#
#    This code is distributed under the MIT license.
#
#  Modified:
#
#    01 April 2015
#
#  Author:
#
#    John Burkardt
#
  import numpy as np
  import platform

  print ( '' )
  print ( 'cauchy_principal_value_test02():' )
  print ( '  Python version: %s' % ( platform.python_version ( ) ) )
  print ( '  cauchy_principal_value ( Integral ( 1-delta <= t <= 1+delta ) 1/(1-t)^3 dt )' )
  print ( '  Try this for delta = 1, 1/2, 1/4.' )
  print ( '' )
  print ( '   N          Estimate                  Exact                  Error' )

  delta = 1.0

  for k in range ( 0, 3 ):
    print ( '' )
    r1 = (   delta + 1.5 ) ** 2 + 0.75
    r2 = ( - delta + 1.5 ) ** 2 + 0.75
    r3 = np.arctan ( np.sqrt ( 0.75 ) / (   delta + 1.5 ) )
    r4 = np.arctan ( np.sqrt ( 0.75 ) / ( - delta + 1.5 ) )
    exact = - np.log ( r1 / r2 ) / 6.0 + ( r3 - r4 ) / np.sqrt ( 3.0 )
    for n in range ( 2, 10, 2 ):
      a = 1.0 - delta
      b = 1.0 + delta
      value = cauchy_principal_value ( f02, a, b, n )
      print ( '  %2d  %24.16g  %24.16g %14.6g' \
        % ( n, value, exact, abs ( value - exact ) ) )

    delta = delta / 2.0
#
#  Terminate.
#
  print ( '' )
  print ( 'cauchy_principal_value_test02():' )
  print ( '  Normal end of execution.' )
  return

def f02 ( t ):

#*****************************************************************************80
#
## f02(): integrand of Integral ( 1-delta <= t <= 1+delta ) 1/(1-t^3) dt
#
#  Discussion:
#
#    1/(1-t^3) = 1/(1+t+t^2) * 1/(1-t)
#
#  Licensing:
#
#    This code is distributed under the MIT license.
#
#  Modified:
#
#    01 April 2015
#
#  Author:
#
#    John Burkardt
#
#  Input:
#
#    real T, the argument.
#
#  Output:
#
#    real VALUE, the value of the integrand at T.
#
  value = 1.0 / ( 1.0 + t + t * t )

  return value

def cauchy_principal_value_test ( ):

#*****************************************************************************80
#
## cauchy_principal_value_test() tests cauchy_principal_value().
#
#  Licensing:
#
#    This code is distributed under the MIT license. 
#
#  Modified:
#
#    02 April 2015
#
#  Author:
#
#    John Burkardt
#
  import platform

  print ( '' )
  print ( 'cauchy_principal_value_test():' )
  print ( '  Python version: %s' % ( platform.python_version ( ) ) )
  print ( '  Test cauchy_principal_value().' )

  cauchy_principal_value_test01 ( )
  cauchy_principal_value_test02 ( )

  legendre_set_test01 ( )
#
#  Terminate.
#
  print ( '' )
  print ( 'cauchy_principal_value_test():' )
  print ( '  Normal end of execution.' )
  return

def legendre_set ( n ):

#*****************************************************************************80
#
## legendre_set() sets abscissas and weights for Gauss-Legendre quadrature.
#
#  Discussion:
#
#    The integral:
#
#      integral ( -1 <= x <= 1 ) f(x) dx
#
#    The quadrature rule:
#
#      sum ( 1 <= i <= n ) w(i) * f ( x(i) )
#
#    The quadrature rule is exact for polynomials through degree 2*N-1.
#
#    The abscissas are the zeroes of the Legendre polynomial P(N)(X).
#
#    Mathematica can compute the abscissas and weights of a Gauss-Legendre
#    rule of order N for the interval [A,B] with P digits of precision
#    by the commands:
#
#    Needs["NumericalDifferentialEquationAnalysis`"]
#    GaussianQuadratureWeights [ n, a, b, p ]
#
#  Licensing:
#
#    This code is distributed under the MIT license.
#
#  Modified:
#
#    01 April 2015
#
#  Author:
#
#    John Burkardt
#
#  Reference:
#
#    Milton Abramowitz, Irene Stegun,
#    Handbook of Mathematical Functions,
#    National Bureau of Standards, 1964,
#    ISBN: 0-486-61272-4,
#    LC: QA47.A34.
#
#    Vladimir Krylov,
#    Approximate Calculation of Integrals,
#    Dover, 2006,
#    ISBN: 0486445798,
#    LC: QA311.K713.
#
#    Arthur Stroud, Don Secrest,
#    Gaussian Quadrature Formulas,
#    Prentice Hall, 1966,
#    LC: QA299.4G3S7.
#
#    Stephen Wolfram,
#    The Mathematica Book,
#    Fourth Edition,
#    Cambridge University Press, 1999,
#    ISBN: 0-521-64314-7,
#    LC: QA76.95.W65.
#
#    Daniel Zwillinger, editor,
#    CRC Standard Mathematical Tables and Formulae,
#    30th Edition,
#    CRC Press, 1996,
#    ISBN: 0-8493-2479-3,
#    LC: QA47.M315.
#
#  Input:
#
#    integer N, the order.
#    N must be between 1 and 33 or 63/64/65, 127/128/129,
#    255/256/257.
#
#  Output:
#
#    real X(N), the abscissas.
#
#    real W(N), the weights.
#
  import numpy as np

  if ( n == 1 ):

    x = np.array ( [ \
      0.000000000000000000000000000000 \
    ] )

    w = np.array ( [ \
      2.000000000000000000000000000000 \
    ] )

  elif ( n == 2 ):

    x = np.array ( [ \
     -0.577350269189625764509148780502, \
      0.577350269189625764509148780502 \
    ] )

    w = np.array ( [ \
      1.000000000000000000000000000000, \
      1.000000000000000000000000000000 \
    ] )

  elif ( n == 3 ):

    x = np.array ( [ \
     -0.774596669241483377035853079956, \
      0.000000000000000000000000000000, \
      0.774596669241483377035853079956 \
    ] )

    w = np.array ( [ \
      0.555555555555555555555555555556, \
      0.888888888888888888888888888889, \
      0.555555555555555555555555555556 \
    ] )

  elif ( n == 4 ):

    x = np.array ( [ \
     -0.861136311594052575223946488893, \
     -0.339981043584856264802665759103, \
     0.339981043584856264802665759103, \
     0.861136311594052575223946488893 \
    ] )

    w = np.array ( [ \
     0.347854845137453857373063949222, \
     0.652145154862546142626936050778, \
     0.652145154862546142626936050778, \
     0.347854845137453857373063949222 \
    ] )

  elif ( n == 5 ):

    x = np.array ( [ \
     -0.906179845938663992797626878299, \
     -0.538469310105683091036314420700, \
     0.000000000000000000000000000000, \
     0.538469310105683091036314420700, \
     0.906179845938663992797626878299 \
    ] )

    w = np.array ( [ \
     0.236926885056189087514264040720, \
     0.478628670499366468041291514836, \
     0.568888888888888888888888888889, \
     0.478628670499366468041291514836, \
     0.236926885056189087514264040720 \
    ] )

  elif ( n == 6 ):

    x = np.array ( [ \
     -0.932469514203152027812301554494, \
     -0.661209386466264513661399595020, \
     -0.238619186083196908630501721681, \
     0.238619186083196908630501721681, \
     0.661209386466264513661399595020, \
     0.932469514203152027812301554494 \
    ] )

    w = np.array ( [ \
     0.171324492379170345040296142173, \
     0.360761573048138607569833513838, \
     0.467913934572691047389870343990, \
     0.467913934572691047389870343990, \
     0.360761573048138607569833513838, \
     0.171324492379170345040296142173 \
    ] )

  elif ( n == 7 ):

    x = np.array ( [ \
     -0.949107912342758524526189684048, \
     -0.741531185599394439863864773281, \
     -0.405845151377397166906606412077, \
     0.000000000000000000000000000000, \
     0.405845151377397166906606412077, \
     0.741531185599394439863864773281, \
     0.949107912342758524526189684048 \
    ] )

    w = np.array ( [ \
     0.129484966168869693270611432679, \
     0.279705391489276667901467771424, \
     0.381830050505118944950369775489, \
     0.417959183673469387755102040816, \
     0.381830050505118944950369775489, \
     0.279705391489276667901467771424, \
     0.129484966168869693270611432679 \
    ] )

  elif ( n == 8 ):

    x = np.array ( [ \
     -0.960289856497536231683560868569, \
     -0.796666477413626739591553936476, \
     -0.525532409916328985817739049189, \
     -0.183434642495649804939476142360, \
     0.183434642495649804939476142360, \
     0.525532409916328985817739049189, \
     0.796666477413626739591553936476, \
     0.960289856497536231683560868569 \
    ] )

    w = np.array ( [ \
     0.101228536290376259152531354310, \
     0.222381034453374470544355994426, \
     0.313706645877887287337962201987, \
     0.362683783378361982965150449277, \
     0.362683783378361982965150449277, \
     0.313706645877887287337962201987, \
     0.222381034453374470544355994426, \
     0.101228536290376259152531354310 \
    ] )

  elif ( n == 9 ):

    x = np.array ( [ \
     -0.968160239507626089835576203, \
     -0.836031107326635794299429788, \
     -0.613371432700590397308702039, \
     -0.324253423403808929038538015, \
     0.000000000000000000000000000, \
     0.324253423403808929038538015, \
     0.613371432700590397308702039, \
     0.836031107326635794299429788, \
     0.968160239507626089835576203 \
    ] )

    w = np.array ( [ \
     0.081274388361574411971892158111, \
     0.18064816069485740405847203124, \
     0.26061069640293546231874286942, \
     0.31234707704000284006863040658, \
     0.33023935500125976316452506929, \
     0.31234707704000284006863040658, \
     0.26061069640293546231874286942, \
     0.18064816069485740405847203124, \
     0.081274388361574411971892158111 \
    ] )

  elif ( n == 10 ):

    x = np.array ( [ \
     -0.973906528517171720077964012, \
     -0.865063366688984510732096688, \
     -0.679409568299024406234327365, \
     -0.433395394129247190799265943, \
     -0.148874338981631210884826001, \
     0.148874338981631210884826001, \
     0.433395394129247190799265943, \
     0.679409568299024406234327365, \
     0.865063366688984510732096688, \
     0.973906528517171720077964012 \
    ] )

    w = np.array ( [ \
     0.066671344308688137593568809893, \
     0.14945134915058059314577633966, \
     0.21908636251598204399553493423, \
     0.26926671930999635509122692157, \
     0.29552422471475287017389299465, \
     0.29552422471475287017389299465, \
     0.26926671930999635509122692157, \
     0.21908636251598204399553493423, \
     0.14945134915058059314577633966, \
     0.066671344308688137593568809893 \
    ] )

  elif ( n == 11 ):

    x = np.array ( [ \
     -0.978228658146056992803938001, \
     -0.887062599768095299075157769, \
     -0.730152005574049324093416252, \
     -0.519096129206811815925725669, \
     -0.269543155952344972331531985, \
     0.000000000000000000000000000, \
     0.269543155952344972331531985, \
     0.519096129206811815925725669, \
     0.730152005574049324093416252, \
     0.887062599768095299075157769, \
     0.978228658146056992803938001 \
    ] )

    w = np.array ( [ \
     0.055668567116173666482753720443, \
     0.12558036946490462463469429922, \
     0.18629021092773425142609764143, \
     0.23319376459199047991852370484, \
     0.26280454451024666218068886989, \
     0.27292508677790063071448352834, \
     0.26280454451024666218068886989, \
     0.23319376459199047991852370484, \
     0.18629021092773425142609764143, \
     0.12558036946490462463469429922, \
     0.055668567116173666482753720443 \
    ] )

  elif ( n == 12 ):

    x = np.array ( [ \
     -0.981560634246719250690549090, \
     -0.904117256370474856678465866, \
     -0.769902674194304687036893833, \
     -0.587317954286617447296702419, \
     -0.367831498998180193752691537, \
     -0.125233408511468915472441369, \
     0.125233408511468915472441369, \
     0.367831498998180193752691537, \
     0.587317954286617447296702419, \
     0.769902674194304687036893833, \
     0.904117256370474856678465866, \
     0.981560634246719250690549090 \
    ] )

    w = np.array ( [ \
     0.047175336386511827194615961485, \
     0.10693932599531843096025471819, \
     0.16007832854334622633465252954, \
     0.20316742672306592174906445581, \
     0.23349253653835480876084989892, \
     0.24914704581340278500056243604, \
     0.24914704581340278500056243604, \
     0.23349253653835480876084989892, \
     0.20316742672306592174906445581, \
     0.16007832854334622633465252954, \
     0.10693932599531843096025471819, \
     0.047175336386511827194615961485 \
    ] )

  elif ( n == 13 ):

    x = np.array ( [ \
     -0.984183054718588149472829449, \
     -0.917598399222977965206547837, \
     -0.801578090733309912794206490, \
     -0.642349339440340220643984607, \
     -0.448492751036446852877912852, \
     -0.230458315955134794065528121, \
     0.000000000000000000000000000, \
     0.230458315955134794065528121, \
     0.448492751036446852877912852, \
     0.642349339440340220643984607, \
     0.80157809073330991279420649, \
     0.91759839922297796520654784, \
     0.98418305471858814947282945 \
    ] )

    w = np.array ( [ \
     0.040484004765315879520021592201, \
     0.092121499837728447914421775954, \
     0.13887351021978723846360177687, \
     0.17814598076194573828004669200, \
     0.20781604753688850231252321931, \
     0.22628318026289723841209018604, \
     0.23255155323087391019458951527, \
     0.22628318026289723841209018604, \
     0.20781604753688850231252321931, \
     0.17814598076194573828004669200, \
     0.13887351021978723846360177687, \
     0.092121499837728447914421775954, \
     0.040484004765315879520021592201 \
    ] )

  elif ( n == 14 ):

    x = np.array ( [ \
     -0.986283808696812338841597267, \
     -0.928434883663573517336391139, \
     -0.827201315069764993189794743, \
     -0.687292904811685470148019803, \
     -0.515248636358154091965290719, \
     -0.319112368927889760435671824, \
     -0.108054948707343662066244650, \
     0.108054948707343662066244650, \
     0.31911236892788976043567182, \
     0.51524863635815409196529072, \
     0.68729290481168547014801980, \
     0.82720131506976499318979474, \
     0.92843488366357351733639114, \
     0.98628380869681233884159727 \
    ] )

    w = np.array ( [ \
     0.035119460331751863031832876138, \
     0.08015808715976020980563327706, \
     0.12151857068790318468941480907, \
     0.15720316715819353456960193862, \
     0.18553839747793781374171659013, \
     0.20519846372129560396592406566, \
     0.21526385346315779019587644332, \
     0.21526385346315779019587644332, \
     0.20519846372129560396592406566, \
     0.18553839747793781374171659013, \
     0.15720316715819353456960193862, \
     0.12151857068790318468941480907, \
     0.08015808715976020980563327706, \
     0.035119460331751863031832876138 \
    ] )

  elif ( n == 15 ):

    x = np.array ( [ \
     -0.987992518020485428489565719, \
     -0.937273392400705904307758948, \
     -0.848206583410427216200648321, \
     -0.724417731360170047416186055, \
     -0.570972172608538847537226737, \
     -0.394151347077563369897207371, \
     -0.201194093997434522300628303, \
     0.00000000000000000000000000, \
     0.20119409399743452230062830, \
     0.39415134707756336989720737, \
     0.57097217260853884753722674, \
     0.72441773136017004741618605, \
     0.84820658341042721620064832, \
     0.93727339240070590430775895, \
     0.98799251802048542848956572 \
    ] )

    w = np.array ( [ \
     0.030753241996117268354628393577, \
     0.070366047488108124709267416451, \
     0.107159220467171935011869546686, \
     0.13957067792615431444780479451, \
     0.16626920581699393355320086048, \
     0.18616100001556221102680056187, \
     0.19843148532711157645611832644, \
     0.20257824192556127288062019997, \
     0.19843148532711157645611832644, \
     0.18616100001556221102680056187, \
     0.16626920581699393355320086048, \
     0.13957067792615431444780479451, \
     0.107159220467171935011869546686, \
     0.070366047488108124709267416451, \
     0.030753241996117268354628393577 \
    ] )

  elif ( n == 16 ):

    x = np.array ( [ \
     -0.989400934991649932596154173, \
     -0.944575023073232576077988416, \
     -0.865631202387831743880467898, \
     -0.755404408355003033895101195, \
     -0.617876244402643748446671764, \
     -0.458016777657227386342419443, \
     -0.281603550779258913230460501, \
     -0.09501250983763744018531934, \
     0.09501250983763744018531934, \
     0.28160355077925891323046050, \
     0.45801677765722738634241944, \
     0.61787624440264374844667176, \
     0.75540440835500303389510119, \
     0.86563120238783174388046790, \
     0.94457502307323257607798842, \
     0.98940093499164993259615417 \
    ] )

    w = np.array ( [ \
     0.027152459411754094851780572456, \
     0.062253523938647892862843836994, \
     0.09515851168249278480992510760, \
     0.12462897125553387205247628219, \
     0.14959598881657673208150173055, \
     0.16915651939500253818931207903, \
     0.18260341504492358886676366797, \
     0.18945061045506849628539672321, \
     0.18945061045506849628539672321, \
     0.18260341504492358886676366797, \
     0.16915651939500253818931207903, \
     0.14959598881657673208150173055, \
     0.12462897125553387205247628219, \
     0.09515851168249278480992510760, \
     0.062253523938647892862843836994, \
     0.027152459411754094851780572456 \
    ] )

  elif ( n == 17 ):

    x = np.array ( [ \
     -0.990575475314417335675434020, \
     -0.950675521768767761222716958, \
     -0.880239153726985902122955694, \
     -0.781514003896801406925230056, \
     -0.657671159216690765850302217, \
     -0.512690537086476967886246569, \
     -0.35123176345387631529718552, \
     -0.17848418149584785585067749, \
     0.00000000000000000000000000, \
     0.17848418149584785585067749, \
     0.35123176345387631529718552, \
     0.51269053708647696788624657, \
     0.65767115921669076585030222, \
     0.78151400389680140692523006, \
     0.88023915372698590212295569, \
     0.95067552176876776122271696, \
     0.99057547531441733567543402 \
    ] )

    w = np.array ( [ \
     0.024148302868547931960110026288, \
     0.055459529373987201129440165359, \
     0.085036148317179180883535370191, \
     0.111883847193403971094788385626, \
     0.13513636846852547328631998170, \
     0.15404576107681028808143159480, \
     0.16800410215645004450997066379, \
     0.17656270536699264632527099011, \
     0.17944647035620652545826564426, \
     0.17656270536699264632527099011, \
     0.16800410215645004450997066379, \
     0.15404576107681028808143159480, \
     0.13513636846852547328631998170, \
     0.111883847193403971094788385626, \
     0.085036148317179180883535370191, \
     0.055459529373987201129440165359, \
     0.024148302868547931960110026288 \
    ] )

  elif ( n == 18 ):

    x = np.array ( [ \
     -0.991565168420930946730016005, \
     -0.955823949571397755181195893, \
     -0.892602466497555739206060591, \
     -0.803704958972523115682417455, \
     -0.691687043060353207874891081, \
     -0.55977083107394753460787155, \
     -0.41175116146284264603593179, \
     -0.25188622569150550958897285, \
     -0.08477501304173530124226185, \
     0.08477501304173530124226185, \
     0.25188622569150550958897285, \
     0.41175116146284264603593179, \
     0.55977083107394753460787155, \
     0.69168704306035320787489108, \
     0.80370495897252311568241746, \
     0.89260246649755573920606059, \
     0.95582394957139775518119589, \
     0.99156516842093094673001600 \
    ] )

    w = np.array ( [ \
     0.021616013526483310313342710266, \
     0.049714548894969796453334946203, \
     0.07642573025488905652912967762, \
     0.10094204410628716556281398492, \
     0.12255520671147846018451912680, \
     0.14064291467065065120473130375, \
     0.15468467512626524492541800384, \
     0.16427648374583272298605377647, \
     0.16914238296314359184065647013, \
     0.16914238296314359184065647013, \
     0.16427648374583272298605377647, \
     0.15468467512626524492541800384, \
     0.14064291467065065120473130375, \
     0.12255520671147846018451912680, \
     0.10094204410628716556281398492, \
     0.07642573025488905652912967762, \
     0.049714548894969796453334946203, \
     0.021616013526483310313342710266 \
    ] )

  elif ( n == 19 ):

    x = np.array ( [ \
     -0.992406843843584403189017670, \
     -0.960208152134830030852778841, \
     -0.903155903614817901642660929, \
     -0.822714656537142824978922487, \
     -0.72096617733522937861709586, \
     -0.60054530466168102346963816, \
     -0.46457074137596094571726715, \
     -0.31656409996362983199011733, \
     -0.16035864564022537586809612, \
     0.00000000000000000000000000, \
     0.16035864564022537586809612, \
     0.31656409996362983199011733, \
     0.46457074137596094571726715, \
     0.60054530466168102346963816, \
     0.72096617733522937861709586, \
     0.82271465653714282497892249, \
     0.90315590361481790164266093, \
     0.96020815213483003085277884, \
     0.99240684384358440318901767 \
    ] )

    w = np.array ( [ \
     0.019461788229726477036312041464, \
     0.044814226765699600332838157402, \
     0.069044542737641226580708258006, \
     0.091490021622449999464462094124, \
     0.111566645547333994716023901682, \
     0.12875396253933622767551578486, \
     0.14260670217360661177574610944, \
     0.15276604206585966677885540090, \
     0.15896884339395434764995643946, \
     0.16105444984878369597916362532, \
     0.15896884339395434764995643946, \
     0.15276604206585966677885540090, \
     0.14260670217360661177574610944, \
     0.12875396253933622767551578486, \
     0.111566645547333994716023901682, \
     0.091490021622449999464462094124, \
     0.069044542737641226580708258006, \
     0.044814226765699600332838157402, \
     0.019461788229726477036312041464 \
    ] )

  elif ( n == 20 ):

    x = np.array ( [ \
     -0.993128599185094924786122388, \
     -0.963971927277913791267666131, \
     -0.912234428251325905867752441, \
     -0.83911697182221882339452906, \
     -0.74633190646015079261430507, \
     -0.63605368072651502545283670, \
     -0.51086700195082709800436405, \
     -0.37370608871541956067254818, \
     -0.22778585114164507808049620, \
     -0.07652652113349733375464041, \
     0.07652652113349733375464041, \
     0.22778585114164507808049620, \
     0.37370608871541956067254818, \
     0.51086700195082709800436405, \
     0.63605368072651502545283670, \
     0.74633190646015079261430507, \
     0.83911697182221882339452906, \
     0.91223442825132590586775244, \
     0.96397192727791379126766613, \
     0.99312859918509492478612239 \
    ] )

    w = np.array ( [ \
     0.017614007139152118311861962352, \
     0.040601429800386941331039952275, \
     0.062672048334109063569506535187, \
     0.08327674157670474872475814322, \
     0.10193011981724043503675013548, \
     0.11819453196151841731237737771, \
     0.13168863844917662689849449975, \
     0.14209610931838205132929832507, \
     0.14917298647260374678782873700, \
     0.15275338713072585069808433195, \
     0.15275338713072585069808433195, \
     0.14917298647260374678782873700, \
     0.14209610931838205132929832507, \
     0.13168863844917662689849449975, \
     0.11819453196151841731237737771, \
     0.10193011981724043503675013548, \
     0.08327674157670474872475814322, \
     0.062672048334109063569506535187, \
     0.040601429800386941331039952275, \
     0.017614007139152118311861962352 \
    ] )

  elif ( n == 21 ):

    x = np.array ( [ \
      -0.99375217062038950026024204, \
      -0.96722683856630629431662221, \
      -0.92009933415040082879018713, \
      -0.85336336458331728364725064, \
      -0.76843996347567790861587785, \
      -0.66713880419741231930596667, \
      -0.55161883588721980705901880, \
      -0.42434212020743878357366889, \
      -0.28802131680240109660079252, \
      -0.14556185416089509093703098, \
       0.00000000000000000000000000, \
      +0.14556185416089509093703098, \
      +0.28802131680240109660079252, \
      +0.42434212020743878357366889, \
      +0.55161883588721980705901880, \
      +0.66713880419741231930596667, \
      +0.76843996347567790861587785, \
      +0.85336336458331728364725064, \
      +0.92009933415040082879018713, \
      +0.96722683856630629431662221, \
      +0.99375217062038950026024204 \
    ] )

    w = np.array ( [ \
       0.016017228257774333324224616858, \
       0.036953789770852493799950668299, \
       0.057134425426857208283635826472, \
       0.076100113628379302017051653300, \
       0.093444423456033861553289741114, \
       0.108797299167148377663474578070, \
       0.12183141605372853419536717713, \
       0.13226893863333746178105257450, \
       0.13988739479107315472213342387, \
       0.14452440398997005906382716655, \
       0.14608113364969042719198514768, \
       0.14452440398997005906382716655, \
       0.13988739479107315472213342387, \
       0.13226893863333746178105257450, \
       0.12183141605372853419536717713, \
       0.108797299167148377663474578070, \
       0.093444423456033861553289741114, \
       0.076100113628379302017051653300, \
       0.057134425426857208283635826472, \
       0.036953789770852493799950668299, \
       0.016017228257774333324224616858 \
    ] )

  elif ( n == 22 ):

    x = np.array ( [ \
     -0.99429458548239929207303142, \
     -0.97006049783542872712395099, \
     -0.92695677218717400052069294, \
     -0.86581257772030013653642564, \
     -0.78781680597920816200427796, \
     -0.69448726318668278005068984, \
     -0.58764040350691159295887693, \
     -0.46935583798675702640633071, \
     -0.34193582089208422515814742, \
     -0.20786042668822128547884653, \
     -0.06973927331972222121384180, \
     0.06973927331972222121384180, \
     0.20786042668822128547884653, \
     0.34193582089208422515814742, \
     0.46935583798675702640633071, \
     0.58764040350691159295887693, \
     0.69448726318668278005068984, \
     0.78781680597920816200427796, \
     0.86581257772030013653642564, \
     0.92695677218717400052069294, \
     0.97006049783542872712395099, \
     0.99429458548239929207303142 \
    ] )

    w = np.array ( [ \
     0.014627995298272200684991098047, \
     0.033774901584814154793302246866, \
     0.052293335152683285940312051273, \
     0.06979646842452048809496141893, \
     0.08594160621706772741444368137, \
     0.10041414444288096493207883783, \
     0.11293229608053921839340060742, \
     0.12325237681051242428556098615, \
     0.13117350478706237073296499253, \
     0.13654149834601517135257383123, \
     0.13925187285563199337541024834, \
     0.13925187285563199337541024834, \
     0.13654149834601517135257383123, \
     0.13117350478706237073296499253, \
     0.12325237681051242428556098615, \
     0.11293229608053921839340060742, \
     0.10041414444288096493207883783, \
     0.08594160621706772741444368137, \
     0.06979646842452048809496141893, \
     0.052293335152683285940312051273, \
     0.033774901584814154793302246866, \
     0.014627995298272200684991098047 \
    ] )

  elif ( n == 23 ):

    x = np.array ( [ \
     -0.99476933499755212352392572, \
     -0.97254247121811523195602408, \
     -0.93297108682601610234919699, \
     -0.87675235827044166737815689, \
     -0.80488840161883989215111841, \
     -0.71866136313195019446162448, \
     -0.61960987576364615638509731, \
     -0.50950147784600754968979305, \
     -0.39030103803029083142148887, \
     -0.26413568097034493053386954, \
     -0.13325682429846611093174268, \
     0.00000000000000000000000000, \
     0.13325682429846611093174268, \
     0.26413568097034493053386954, \
     0.39030103803029083142148887, \
     0.50950147784600754968979305, \
     0.61960987576364615638509731, \
     0.71866136313195019446162448, \
     0.80488840161883989215111841, \
     0.87675235827044166737815689, \
     0.93297108682601610234919699, \
     0.97254247121811523195602408, \
     0.99476933499755212352392572 \
    ] )

    w = np.array ( [ \
     0.013411859487141772081309493459, \
     0.030988005856979444310694219642, \
     0.048037671731084668571641071632, \
     0.064232421408525852127169615159, \
     0.079281411776718954922892524742, \
     0.092915766060035147477018617370, \
     0.104892091464541410074086185015, \
     0.11499664022241136494164351293, \
     0.12304908430672953046757840067, \
     0.12890572218808214997859533940, \
     0.13246203940469661737164246470, \
     0.13365457218610617535145711055, \
     0.13246203940469661737164246470, \
     0.12890572218808214997859533940, \
     0.12304908430672953046757840067, \
     0.11499664022241136494164351293, \
     0.104892091464541410074086185015, \
     0.092915766060035147477018617370, \
     0.079281411776718954922892524742, \
     0.064232421408525852127169615159, \
     0.048037671731084668571641071632, \
     0.030988005856979444310694219642, \
     0.013411859487141772081309493459 \
    ] )

  elif ( n == 24 ):

    x = np.array ( [ \
     -0.99518721999702136017999741, \
     -0.97472855597130949819839199, \
     -0.93827455200273275852364900, \
     -0.88641552700440103421315434, \
     -0.82000198597390292195394987, \
     -0.74012419157855436424382810, \
     -0.64809365193697556925249579, \
     -0.54542147138883953565837562, \
     -0.43379350762604513848708423, \
     -0.31504267969616337438679329, \
     -0.19111886747361630915863982, \
     -0.06405689286260562608504308, \
     0.06405689286260562608504308, \
     0.19111886747361630915863982, \
     0.31504267969616337438679329, \
     0.43379350762604513848708423, \
     0.54542147138883953565837562, \
     0.64809365193697556925249579, \
     0.74012419157855436424382810, \
     0.82000198597390292195394987, \
     0.88641552700440103421315434, \
     0.93827455200273275852364900, \
     0.97472855597130949819839199, \
     0.99518721999702136017999741 \
    ] )

    w = np.array ( [ \
     0.012341229799987199546805667070, \
     0.028531388628933663181307815952, \
     0.044277438817419806168602748211, \
     0.059298584915436780746367758500, \
     0.07334648141108030573403361525, \
     0.08619016153195327591718520298, \
     0.09761865210411388826988066446, \
     0.10744427011596563478257734245, \
     0.11550566805372560135334448391, \
     0.12167047292780339120446315348, \
     0.12583745634682829612137538251, \
     0.12793819534675215697405616522, \
     0.12793819534675215697405616522, \
     0.12583745634682829612137538251, \
     0.12167047292780339120446315348, \
     0.11550566805372560135334448391, \
     0.10744427011596563478257734245, \
     0.09761865210411388826988066446, \
     0.08619016153195327591718520298, \
     0.07334648141108030573403361525, \
     0.059298584915436780746367758500, \
     0.044277438817419806168602748211, \
     0.028531388628933663181307815952, \
     0.012341229799987199546805667070 \
    ] )

  elif ( n == 25 ):

    x = np.array ( [ \
     -0.99555696979049809790878495, \
     -0.97666392145951751149831539, \
     -0.94297457122897433941401117, \
     -0.89499199787827536885104201, \
     -0.83344262876083400142102111, \
     -0.75925926303735763057728287, \
     -0.67356636847346836448512063, \
     -0.57766293024122296772368984, \
     -0.47300273144571496052218212, \
     -0.36117230580938783773582173, \
     -0.24386688372098843204519036, \
     -0.12286469261071039638735982, \
     0.00000000000000000000000000, \
     0.12286469261071039638735982, \
     0.24386688372098843204519036, \
     0.36117230580938783773582173, \
     0.47300273144571496052218212, \
     0.57766293024122296772368984, \
     0.67356636847346836448512063, \
     0.75925926303735763057728287, \
     0.83344262876083400142102111, \
     0.89499199787827536885104201, \
     0.94297457122897433941401117, \
     0.97666392145951751149831539, \
     0.99555696979049809790878495 \
    ] )

    w = np.array ( [ \
     0.0113937985010262879479029641132, \
     0.026354986615032137261901815295, \
     0.040939156701306312655623487712, \
     0.054904695975835191925936891541, \
     0.068038333812356917207187185657, \
     0.080140700335001018013234959669, \
     0.091028261982963649811497220703, \
     0.100535949067050644202206890393, \
     0.108519624474263653116093957050, \
     0.11485825914571164833932554587, \
     0.11945576353578477222817812651, \
     0.12224244299031004168895951895, \
     0.12317605372671545120390287308, \
     0.12224244299031004168895951895, \
     0.11945576353578477222817812651, \
     0.11485825914571164833932554587, \
     0.108519624474263653116093957050, \
     0.100535949067050644202206890393, \
     0.091028261982963649811497220703, \
     0.080140700335001018013234959669, \
     0.068038333812356917207187185657, \
     0.054904695975835191925936891541, \
     0.040939156701306312655623487712, \
     0.026354986615032137261901815295, \
     0.0113937985010262879479029641132 \
    ] )

  elif ( n == 26 ):

    x = np.array ( [ \
     -0.99588570114561692900321696, \
     -0.97838544595647099110058035, \
     -0.94715906666171425013591528, \
     -0.90263786198430707421766560, \
     -0.84544594278849801879750706, \
     -0.77638594882067885619296725, \
     -0.69642726041995726486381391, \
     -0.60669229301761806323197875, \
     -0.50844071482450571769570306, \
     -0.40305175512348630648107738, \
     -0.29200483948595689514283538, \
     -0.17685882035689018396905775, \
     -0.05923009342931320709371858, \
     0.05923009342931320709371858, \
     0.17685882035689018396905775, \
     0.29200483948595689514283538, \
     0.40305175512348630648107738, \
     0.50844071482450571769570306, \
     0.60669229301761806323197875, \
     0.69642726041995726486381391, \
     0.77638594882067885619296725, \
     0.84544594278849801879750706, \
     0.90263786198430707421766560, \
     0.94715906666171425013591528, \
     0.97838544595647099110058035, \
     0.99588570114561692900321696 \
    ] )

    w = np.array ( [ \
     0.010551372617343007155651187685, \
     0.024417851092631908789615827520, \
     0.037962383294362763950303141249, \
     0.050975825297147811998319900724, \
     0.063274046329574835539453689907, \
     0.07468414976565974588707579610, \
     0.08504589431348523921044776508, \
     0.09421380035591414846366488307, \
     0.10205916109442542323841407025, \
     0.10847184052857659065657942673, \
     0.11336181654631966654944071844, \
     0.11666044348529658204466250754, \
     0.11832141527926227651637108570, \
     0.11832141527926227651637108570, \
     0.11666044348529658204466250754, \
     0.11336181654631966654944071844, \
     0.10847184052857659065657942673, \
     0.10205916109442542323841407025, \
     0.09421380035591414846366488307, \
     0.08504589431348523921044776508, \
     0.07468414976565974588707579610, \
     0.063274046329574835539453689907, \
     0.050975825297147811998319900724, \
     0.037962383294362763950303141249, \
     0.024417851092631908789615827520, \
     0.010551372617343007155651187685 \
    ] )

  elif ( n == 27 ):

    x = np.array ( [ \
     -0.99617926288898856693888721, \
     -0.97992347596150122285587336, \
     -0.95090055781470500685190803, \
     -0.90948232067749110430064502, \
     -0.85620790801829449030273722, \
     -0.79177163907050822714439734, \
     -0.71701347373942369929481621, \
     -0.63290797194649514092773464, \
     -0.54055156457945689490030094, \
     -0.44114825175002688058597416, \
     -0.33599390363850889973031903, \
     -0.22645936543953685885723911, \
     -0.11397258560952996693289498, \
     0.00000000000000000000000000, \
     0.11397258560952996693289498, \
     0.22645936543953685885723911, \
     0.33599390363850889973031903, \
     0.44114825175002688058597416, \
     0.54055156457945689490030094, \
     0.63290797194649514092773464, \
     0.71701347373942369929481621, \
     0.79177163907050822714439734, \
     0.85620790801829449030273722, \
     0.90948232067749110430064502, \
     0.95090055781470500685190803, \
     0.97992347596150122285587336, \
     0.99617926288898856693888721 \
    ] )

    w = np.array ( [ \
     0.0097989960512943602611500550912, \
     0.022686231596180623196034206447, \
     0.035297053757419711022578289305, \
     0.047449412520615062704096710114, \
     0.058983536859833599110300833720, \
     0.069748823766245592984322888357, \
     0.079604867773057771263074959010, \
     0.088423158543756950194322802854, \
     0.096088727370028507565652646558, \
     0.102501637817745798671247711533, \
     0.107578285788533187212162984427, \
     0.111252488356845192672163096043, \
     0.113476346108965148620369948092, \
     0.11422086737895698904504573690, \
     0.113476346108965148620369948092, \
     0.111252488356845192672163096043, \
     0.107578285788533187212162984427, \
     0.102501637817745798671247711533, \
     0.096088727370028507565652646558, \
     0.088423158543756950194322802854, \
     0.079604867773057771263074959010, \
     0.069748823766245592984322888357, \
     0.058983536859833599110300833720, \
     0.047449412520615062704096710114, \
     0.035297053757419711022578289305, \
     0.022686231596180623196034206447, \
     0.0097989960512943602611500550912 \
    ] )

  elif ( n == 28 ):

    x = np.array ( [ \
     -0.99644249757395444995043639, \
     -0.98130316537087275369455995, \
     -0.95425928062893819725410184, \
     -0.91563302639213207386968942, \
     -0.86589252257439504894225457, \
     -0.80564137091717917144788596, \
     -0.73561087801363177202814451, \
     -0.65665109403886496121989818, \
     -0.56972047181140171930800328, \
     -0.47587422495511826103441185, \
     -0.37625151608907871022135721, \
     -0.27206162763517807767682636, \
     -0.16456928213338077128147178, \
     -0.05507928988403427042651653, \
     0.05507928988403427042651653, \
     0.16456928213338077128147178, \
     0.27206162763517807767682636, \
     0.37625151608907871022135721, \
     0.47587422495511826103441185, \
     0.56972047181140171930800328, \
     0.65665109403886496121989818, \
     0.73561087801363177202814451, \
     0.80564137091717917144788596, \
     0.86589252257439504894225457, \
     0.91563302639213207386968942, \
     0.95425928062893819725410184, \
     0.98130316537087275369455995, \
     0.99644249757395444995043639 \
    ] )

    w = np.array ( [ \
     0.009124282593094517738816153923, \
     0.021132112592771259751500380993, \
     0.032901427782304379977630819171, \
     0.044272934759004227839587877653, \
     0.055107345675716745431482918227, \
     0.06527292396699959579339756678, \
     0.07464621423456877902393188717, \
     0.08311341722890121839039649824, \
     0.09057174439303284094218603134, \
     0.09693065799792991585048900610, \
     0.10211296757806076981421663851, \
     0.10605576592284641791041643700, \
     0.10871119225829413525357151930, \
     0.11004701301647519628237626560, \
     0.11004701301647519628237626560, \
     0.10871119225829413525357151930, \
     0.10605576592284641791041643700, \
     0.10211296757806076981421663851, \
     0.09693065799792991585048900610, \
     0.09057174439303284094218603134, \
     0.08311341722890121839039649824, \
     0.07464621423456877902393188717, \
     0.06527292396699959579339756678, \
     0.055107345675716745431482918227, \
     0.044272934759004227839587877653, \
     0.032901427782304379977630819171, \
     0.021132112592771259751500380993, \
     0.009124282593094517738816153923 \
    ] )

  elif ( n == 29 ):

    x = np.array ( [ \
     -0.99667944226059658616319153, \
     -0.98254550526141317487092602, \
     -0.95728559577808772579820804, \
     -0.92118023295305878509375344, \
     -0.87463780492010279041779342, \
     -0.81818548761525244498957221, \
     -0.75246285173447713391261008, \
     -0.67821453760268651515618501, \
     -0.59628179713822782037958621, \
     -0.50759295512422764210262792, \
     -0.41315288817400866389070659, \
     -0.31403163786763993494819592, \
     -0.21135228616600107450637573, \
     -0.10627823013267923017098239, \
     0.00000000000000000000000000, \
     0.10627823013267923017098239, \
     0.21135228616600107450637573, \
     0.31403163786763993494819592, \
     0.41315288817400866389070659, \
     0.50759295512422764210262792, \
     0.59628179713822782037958621, \
     0.67821453760268651515618501, \
     0.75246285173447713391261008, \
     0.81818548761525244498957221, \
     0.87463780492010279041779342, \
     0.92118023295305878509375344, \
     0.95728559577808772579820804, \
     0.98254550526141317487092602, \
     0.99667944226059658616319153 \
    ] )

    w = np.array ( [ \
     0.0085169038787464096542638133022, \
     0.019732085056122705983859801640, \
     0.030740492202093622644408525375, \
     0.041402062518682836104830010114, \
     0.051594826902497923912594381180, \
     0.061203090657079138542109848024, \
     0.070117933255051278569581486949, \
     0.078238327135763783828144888660, \
     0.085472257366172527545344849297, \
     0.091737757139258763347966411077, \
     0.096963834094408606301900074883, \
     0.101091273759914966121820546907, \
     0.104073310077729373913328471285, \
     0.105876155097320941406591327852, \
     0.10647938171831424424651112691, \
     0.105876155097320941406591327852, \
     0.104073310077729373913328471285, \
     0.101091273759914966121820546907, \
     0.096963834094408606301900074883, \
     0.091737757139258763347966411077, \
     0.085472257366172527545344849297, \
     0.078238327135763783828144888660, \
     0.070117933255051278569581486949, \
     0.061203090657079138542109848024, \
     0.051594826902497923912594381180, \
     0.041402062518682836104830010114, \
     0.030740492202093622644408525375, \
     0.019732085056122705983859801640, \
     0.0085169038787464096542638133022 \
    ] )

  elif ( n == 30 ):

    x = np.array ( [ \
     -0.99689348407464954027163005, \
     -0.98366812327974720997003258, \
     -0.96002186496830751221687103, \
     -0.92620004742927432587932428, \
     -0.88256053579205268154311646, \
     -0.82956576238276839744289812, \
     -0.76777743210482619491797734, \
     -0.69785049479331579693229239, \
     -0.62052618298924286114047756, \
     -0.53662414814201989926416979, \
     -0.44703376953808917678060990, \
     -0.35270472553087811347103721, \
     -0.25463692616788984643980513, \
     -0.15386991360858354696379467, \
     -0.05147184255531769583302521, \
     0.05147184255531769583302521, \
     0.15386991360858354696379467, \
     0.25463692616788984643980513, \
     0.35270472553087811347103721, \
     0.44703376953808917678060990, \
     0.53662414814201989926416979, \
     0.62052618298924286114047756, \
     0.69785049479331579693229239, \
     0.76777743210482619491797734, \
     0.82956576238276839744289812, \
     0.88256053579205268154311646, \
     0.92620004742927432587932428, \
     0.96002186496830751221687103, \
     0.98366812327974720997003258, \
     0.99689348407464954027163005 \
    ] )

    w = np.array ( [ \
     0.007968192496166605615465883475, \
     0.018466468311090959142302131912, \
     0.028784707883323369349719179611, \
     0.038799192569627049596801936446, \
     0.048402672830594052902938140423, \
     0.057493156217619066481721689402, \
     0.06597422988218049512812851512, \
     0.07375597473770520626824385002, \
     0.08075589522942021535469493846, \
     0.08689978720108297980238753072, \
     0.09212252223778612871763270709, \
     0.09636873717464425963946862635, \
     0.09959342058679526706278028210, \
     0.10176238974840550459642895217, \
     0.10285265289355884034128563671, \
     0.10285265289355884034128563671, \
     0.10176238974840550459642895217, \
     0.09959342058679526706278028210, \
     0.09636873717464425963946862635, \
     0.09212252223778612871763270709, \
     0.08689978720108297980238753072, \
     0.08075589522942021535469493846, \
     0.07375597473770520626824385002, \
     0.06597422988218049512812851512, \
     0.057493156217619066481721689402, \
     0.048402672830594052902938140423, \
     0.038799192569627049596801936446, \
     0.028784707883323369349719179611, \
     0.018466468311090959142302131912, \
     0.007968192496166605615465883475 \
    ] )



  elif ( n == 31 ):

    x = np.array ( [ \
     -0.99708748181947707405562655, \
     -0.98468590966515248400246517, \
     -0.96250392509294966178905240, \
     -0.93075699789664816495694576, \
     -0.88976002994827104337419201, \
     -0.83992032014626734008690454, \
     -0.78173314841662494040636002, \
     -0.71577678458685328390597087, \
     -0.64270672292426034618441820, \
     -0.56324916140714926272094492, \
     -0.47819378204490248044059404, \
     -0.38838590160823294306135146, \
     -0.29471806998170161661790390, \
     -0.19812119933557062877241300, \
     -0.09955531215234152032517479, \
     0.00000000000000000000000000, \
     0.09955531215234152032517479, \
     0.19812119933557062877241300, \
     0.29471806998170161661790390, \
     0.38838590160823294306135146, \
     0.47819378204490248044059404, \
     0.56324916140714926272094492, \
     0.64270672292426034618441820, \
     0.71577678458685328390597087, \
     0.78173314841662494040636002, \
     0.83992032014626734008690454, \
     0.88976002994827104337419201, \
     0.93075699789664816495694576, \
     0.96250392509294966178905240, \
     0.98468590966515248400246517, \
     0.99708748181947707405562655 \
    ] )

    w = np.array ( [ \
     0.0074708315792487758586968750322, \
     0.017318620790310582463157996087, \
     0.027009019184979421800608708092, \
     0.036432273912385464024392010468, \
     0.045493707527201102902315857895, \
     0.054103082424916853711666259087, \
     0.062174786561028426910343543687, \
     0.069628583235410366167756126255, \
     0.076390386598776616426357674901, \
     0.082392991761589263903823367432, \
     0.087576740608477876126198069695, \
     0.091890113893641478215362871607, \
     0.095290242912319512807204197488, \
     0.097743335386328725093474010979, \
     0.099225011226672307874875514429, \
     0.09972054479342645142753383373, \
     0.099225011226672307874875514429, \
     0.097743335386328725093474010979, \
     0.095290242912319512807204197488, \
     0.091890113893641478215362871607, \
     0.087576740608477876126198069695, \
     0.082392991761589263903823367432, \
     0.076390386598776616426357674901, \
     0.069628583235410366167756126255, \
     0.062174786561028426910343543687, \
     0.054103082424916853711666259087, \
     0.045493707527201102902315857895, \
     0.036432273912385464024392010468, \
     0.027009019184979421800608708092, \
     0.017318620790310582463157996087, \
     0.0074708315792487758586968750322 \
    ] )

  elif ( n == 32 ):

    x = np.array ( [ \
     -0.99726386184948156354498113, \
     -0.98561151154526833540017504, \
     -0.96476225558750643077381193, \
     -0.93490607593773968917091913, \
     -0.89632115576605212396530724, \
     -0.84936761373256997013369300, \
     -0.79448379596794240696309730, \
     -0.73218211874028968038742667, \
     -0.66304426693021520097511517, \
     -0.58771575724076232904074548, \
     -0.50689990893222939002374747, \
     -0.42135127613063534536411944, \
     -0.33186860228212764977991681, \
     -0.23928736225213707454460321, \
     -0.14447196158279649348518637, \
     -0.04830766568773831623481257, \
     0.04830766568773831623481257, \
     0.14447196158279649348518637, \
     0.23928736225213707454460321, \
     0.33186860228212764977991681, \
     0.42135127613063534536411944, \
     0.50689990893222939002374747, \
     0.58771575724076232904074548, \
     0.66304426693021520097511517, \
     0.73218211874028968038742667, \
     0.79448379596794240696309730, \
     0.84936761373256997013369300, \
     0.89632115576605212396530724, \
     0.93490607593773968917091913, \
     0.96476225558750643077381193, \
     0.98561151154526833540017504, \
     0.99726386184948156354498113 \
    ] )

    w = np.array ( [ \
     0.007018610009470096600407063739, \
     0.016274394730905670605170562206, \
     0.025392065309262059455752589789, \
     0.034273862913021433102687732252, \
     0.042835898022226680656878646606, \
     0.050998059262376176196163244690, \
     0.058684093478535547145283637300, \
     0.06582222277636184683765006371, \
     0.07234579410884850622539935648, \
     0.07819389578707030647174091883, \
     0.08331192422694675522219907460, \
     0.08765209300440381114277146275, \
     0.09117387869576388471286857711, \
     0.09384439908080456563918023767, \
     0.09563872007927485941908200220, \
     0.09654008851472780056676483006, \
     0.09654008851472780056676483006, \
     0.09563872007927485941908200220, \
     0.09384439908080456563918023767, \
     0.09117387869576388471286857711, \
     0.08765209300440381114277146275, \
     0.08331192422694675522219907460, \
     0.07819389578707030647174091883, \
     0.07234579410884850622539935648, \
     0.06582222277636184683765006371, \
     0.058684093478535547145283637300, \
     0.050998059262376176196163244690, \
     0.042835898022226680656878646606, \
     0.034273862913021433102687732252, \
     0.025392065309262059455752589789, \
     0.016274394730905670605170562206, \
     0.007018610009470096600407063739 \
    ] )

  elif ( n == 33 ):

    x = np.array ( [ \
     -0.99742469424645521726616802, \
     -0.98645572623064248811037570, \
     -0.96682290968999276892837771, \
     -0.93869437261116835035583512, \
     -0.90231676774343358304053133, \
     -0.85800965267650406464306148, \
     -0.80616235627416658979620087, \
     -0.74723049644956215785905512, \
     -0.68173195996974278626821595, \
     -0.61024234583637902730728751, \
     -0.53338990478634764354889426, \
     -0.45185001727245069572599328, \
     -0.36633925774807334107022062, \
     -0.27760909715249702940324807, \
     -0.18643929882799157233579876, \
     -0.09363106585473338567074292, \
     0.00000000000000000000000000, \
     0.09363106585473338567074292, \
     0.18643929882799157233579876, \
     0.27760909715249702940324807, \
     0.36633925774807334107022062, \
     0.45185001727245069572599328, \
     0.53338990478634764354889426, \
     0.61024234583637902730728751, \
     0.68173195996974278626821595, \
     0.74723049644956215785905512, \
     0.80616235627416658979620087, \
     0.85800965267650406464306148, \
     0.90231676774343358304053133, \
     0.93869437261116835035583512, \
     0.96682290968999276892837771, \
     0.98645572623064248811037570, \
     0.99742469424645521726616802 \
    ] )

    w = np.array ( [ \
     0.0066062278475873780586492352085, \
     0.015321701512934676127945768534, \
     0.023915548101749480350533257529, \
     0.032300358632328953281561447250, \
     0.040401541331669591563409790527, \
     0.048147742818711695670146880138, \
     0.055470846631663561284944495439, \
     0.062306482530317480031627725771, \
     0.068594572818656712805955073015, \
     0.074279854843954149342472175919, \
     0.079312364794886738363908384942, \
     0.083647876067038707613928014518, \
     0.087248287618844337607281670945, \
     0.090081958660638577239743705500, \
     0.092123986643316846213240977717, \
     0.093356426065596116160999126274, \
     0.09376844616020999656730454155, \
     0.093356426065596116160999126274, \
     0.092123986643316846213240977717, \
     0.090081958660638577239743705500, \
     0.087248287618844337607281670945, \
     0.083647876067038707613928014518, \
     0.079312364794886738363908384942, \
     0.074279854843954149342472175919, \
     0.068594572818656712805955073015, \
     0.062306482530317480031627725771, \
     0.055470846631663561284944495439, \
     0.048147742818711695670146880138, \
     0.040401541331669591563409790527, \
     0.032300358632328953281561447250, \
     0.023915548101749480350533257529, \
     0.015321701512934676127945768534, \
     0.0066062278475873780586492352085 \
    ] )

  elif ( n == 63 ):

    x = np.array ( [ \
     -0.99928298402912378037893614, \
     -0.99622401277797010860219336, \
     -0.99072854689218946681089467, \
     -0.98280881059372723486251141, \
     -0.97248403469757002280196068, \
     -0.95977944975894192707035417, \
     -0.94472613404100980296637532, \
     -0.92736092062184320544703138, \
     -0.90772630277853155803695313, \
     -0.88587032850785342629029846, \
     -0.86184648236412371953961184, \
     -0.83571355431950284347180777, \
     -0.80753549577345676005146599, \
     -0.7773812629903723355633302, \
     -0.7453246483178474178293217, \
     -0.7114440995848458078514315, \
     -0.6758225281149860901311033, \
     -0.6385471058213653850003070, \
     -0.5997090518776252357390089, \
     -0.5594034094862850132676978, \
     -0.5177288132900332481244776, \
     -0.4747872479948043999222123, \
     -0.4306837987951116006620889, \
     -0.3855263942122478924776150, \
     -0.3394255419745844024688344, \
     -0.2924940585862514400361572, \
     -0.2448467932459533627484046, \
     -0.1966003467915066845576275, \
     -0.1478727863578719685698391, \
     -0.0987833564469452795297037, \
     -0.0494521871161596272342338, \
     0.0000000000000000000000000, \
     0.0494521871161596272342338, \
     0.0987833564469452795297037, \
     0.1478727863578719685698391, \
     0.1966003467915066845576275, \
     0.2448467932459533627484046, \
     0.2924940585862514400361572, \
     0.3394255419745844024688344, \
     0.3855263942122478924776150, \
     0.4306837987951116006620889, \
     0.4747872479948043999222123, \
     0.5177288132900332481244776, \
     0.5594034094862850132676978, \
     0.5997090518776252357390089, \
     0.6385471058213653850003070, \
     0.6758225281149860901311033, \
     0.7114440995848458078514315, \
     0.7453246483178474178293217, \
     0.7773812629903723355633302, \
     0.8075354957734567600514660, \
     0.8357135543195028434718078, \
     0.8618464823641237195396118, \
     0.8858703285078534262902985, \
     0.9077263027785315580369531, \
     0.9273609206218432054470314, \
     0.9447261340410098029663753, \
     0.9597794497589419270703542, \
     0.9724840346975700228019607, \
     0.9828088105937272348625114, \
     0.9907285468921894668108947, \
     0.9962240127779701086021934, \
     0.9992829840291237803789361 \
    ] )

    w = np.array ( [ \
     0.0018398745955770841170924455540, \
     0.0042785083468637618660784110826, \
     0.0067102917659601362519069307298, \
     0.0091259686763266563540586454218, \
     0.011519376076880041750750606149, \
     0.013884612616115610824866086368, \
     0.016215878410338338882283672975, \
     0.018507464460161270409260545805, \
     0.020753761258039090775341953421, \
     0.022949271004889933148942319562, \
     0.025088620553344986618630138068, \
     0.027166574359097933225189839439, \
     0.029178047208280526945551502154, \
     0.031118116622219817508215988557, \
     0.032982034883779341765683179672, \
     0.034765240645355877697180504643, \
     0.036463370085457289630452409788, \
     0.038072267584349556763638324928, \
     0.039587995891544093984807928149, \
     0.041006845759666398635110037009, \
     0.042325345020815822982505485403, \
     0.043540267083027590798964315704, \
     0.044648638825941395370332669517, \
     0.045647747876292608685885992609, \
     0.046535149245383696510395418747, \
     0.047308671312268919080604988339, \
     0.047966421137995131411052756195, \
     0.048506789097883847864090099146, \
     0.048928452820511989944709361549, \
     0.049230380423747560785043116988, \
     0.049411833039918178967039646117, \
     0.04947236662393102088866936042, \
     0.049411833039918178967039646117, \
     0.049230380423747560785043116988, \
     0.048928452820511989944709361549, \
     0.048506789097883847864090099146, \
     0.047966421137995131411052756195, \
     0.047308671312268919080604988339, \
     0.046535149245383696510395418747, \
     0.045647747876292608685885992609, \
     0.044648638825941395370332669517, \
     0.043540267083027590798964315704, \
     0.042325345020815822982505485403, \
     0.041006845759666398635110037009, \
     0.039587995891544093984807928149, \
     0.038072267584349556763638324928, \
     0.036463370085457289630452409788, \
     0.034765240645355877697180504643, \
     0.032982034883779341765683179672, \
     0.031118116622219817508215988557, \
     0.029178047208280526945551502154, \
     0.027166574359097933225189839439, \
     0.025088620553344986618630138068, \
     0.022949271004889933148942319562, \
     0.020753761258039090775341953421, \
     0.018507464460161270409260545805, \
     0.016215878410338338882283672975, \
     0.013884612616115610824866086368, \
     0.011519376076880041750750606149, \
     0.0091259686763266563540586454218, \
     0.0067102917659601362519069307298, \
     0.0042785083468637618660784110826, \
     0.0018398745955770841170924455540 \
    ] )

  elif ( n == 64 ):

    x = np.array ( [ \
     -0.99930504173577213945690562, \
     -0.99634011677195527934692450, \
     -0.99101337147674432073938238, \
     -0.98333625388462595693129930, \
     -0.97332682778991096374185351, \
     -0.96100879965205371891861412, \
     -0.94641137485840281606248149, \
     -0.92956917213193957582149015, \
     -0.91052213707850280575638067, \
     -0.88931544599511410585340404, \
     -0.86599939815409281976078339, \
     -0.8406292962525803627516915, \
     -0.8132653151227975597419233, \
     -0.7839723589433414076102205, \
     -0.7528199072605318966118638, \
     -0.7198818501716108268489402, \
     -0.6852363130542332425635584, \
     -0.6489654712546573398577612, \
     -0.6111553551723932502488530, \
     -0.5718956462026340342838781, \
     -0.5312794640198945456580139, \
     -0.4894031457070529574785263, \
     -0.4463660172534640879849477, \
     -0.4022701579639916036957668, \
     -0.3572201583376681159504426, \
     -0.3113228719902109561575127, \
     -0.2646871622087674163739642, \
     -0.2174236437400070841496487, \
     -0.1696444204239928180373136, \
     -0.1214628192961205544703765, \
     -0.0729931217877990394495429, \
     -0.0243502926634244325089558, \
     0.0243502926634244325089558, \
     0.0729931217877990394495429, \
     0.1214628192961205544703765, \
     0.1696444204239928180373136, \
     0.2174236437400070841496487, \
     0.2646871622087674163739642, \
     0.3113228719902109561575127, \
     0.3572201583376681159504426, \
     0.4022701579639916036957668, \
     0.4463660172534640879849477, \
     0.4894031457070529574785263, \
     0.5312794640198945456580139, \
     0.5718956462026340342838781, \
     0.6111553551723932502488530, \
     0.6489654712546573398577612, \
     0.6852363130542332425635584, \
     0.7198818501716108268489402, \
     0.7528199072605318966118638, \
     0.7839723589433414076102205, \
     0.8132653151227975597419233, \
     0.8406292962525803627516915, \
     0.8659993981540928197607834, \
     0.8893154459951141058534040, \
     0.9105221370785028057563807, \
     0.9295691721319395758214902, \
     0.9464113748584028160624815, \
     0.9610087996520537189186141, \
     0.9733268277899109637418535, \
     0.9833362538846259569312993, \
     0.9910133714767443207393824, \
     0.9963401167719552793469245, \
     0.9993050417357721394569056 \
    ] )

    w = np.array ( [ \
     0.0017832807216964329472960791450, \
     0.0041470332605624676352875357286, \
     0.006504457968978362856117360400, \
     0.008846759826363947723030914660, \
     0.011168139460131128818590493019, \
     0.013463047896718642598060766686, \
     0.015726030476024719321965995298, \
     0.017951715775697343085045302001, \
     0.020134823153530209372340316729, \
     0.022270173808383254159298330384, \
     0.024352702568710873338177550409, \
     0.026377469715054658671691792625, \
     0.028339672614259483227511305200, \
     0.030234657072402478867974059820, \
     0.032057928354851553585467504348, \
     0.033805161837141609391565482111, \
     0.035472213256882383810693146715, \
     0.037055128540240046040415101810, \
     0.038550153178615629128962496947, \
     0.039953741132720341386656926128, \
     0.041262563242623528610156297474, \
     0.042473515123653589007339767909, \
     0.043583724529323453376827860974, \
     0.044590558163756563060134710031, \
     0.045491627927418144479770996971, \
     0.046284796581314417295953249232, \
     0.046968182816210017325326285755, \
     0.047540165714830308662282206944, \
     0.04799938859645830772812617987, \
     0.04834476223480295716976952716, \
     0.04857546744150342693479906678, \
     0.04869095700913972038336539073, \
     0.04869095700913972038336539073, \
     0.04857546744150342693479906678, \
     0.04834476223480295716976952716, \
     0.04799938859645830772812617987, \
     0.047540165714830308662282206944, \
     0.046968182816210017325326285755, \
     0.046284796581314417295953249232, \
     0.045491627927418144479770996971, \
     0.044590558163756563060134710031, \
     0.043583724529323453376827860974, \
     0.042473515123653589007339767909, \
     0.041262563242623528610156297474, \
     0.039953741132720341386656926128, \
     0.038550153178615629128962496947, \
     0.037055128540240046040415101810, \
     0.035472213256882383810693146715, \
     0.033805161837141609391565482111, \
     0.032057928354851553585467504348, \
     0.030234657072402478867974059820, \
     0.028339672614259483227511305200, \
     0.026377469715054658671691792625, \
     0.024352702568710873338177550409, \
     0.022270173808383254159298330384, \
     0.020134823153530209372340316729, \
     0.017951715775697343085045302001, \
     0.015726030476024719321965995298, \
     0.013463047896718642598060766686, \
     0.011168139460131128818590493019, \
     0.008846759826363947723030914660, \
     0.006504457968978362856117360400, \
     0.0041470332605624676352875357286, \
     0.0017832807216964329472960791450 \
    ] )

  elif ( n == 65 ):

    x = np.array ( [ \
     -0.99932609707541287726569361, \
     -0.99645094806184916305579494, \
     -0.99128527617680166872182118, \
     -0.98383981218703494137763778, \
     -0.97413153983355116907496789, \
     -0.96218275471805523771198375, \
     -0.94802092816840750637376974, \
     -0.93167862822874933796567699, \
     -0.91319344054284626173654692, \
     -0.89260788050473893142328554, \
     -0.8699692949264070361941320, \
     -0.8453297528999302839424500, \
     -0.8187459259226514534339191, \
     -0.7902789574921218430473804, \
     -0.7599943224419997868739828, \
     -0.7279616763294246790119737, \
     -0.6942546952139916335526225, \
     -0.6589509061936251330409408, \
     -0.6221315090854002415825996, \
     -0.5838811896604873133271545, \
     -0.5442879248622271385455725, \
     -0.5034427804550068823410431, \
     -0.4614397015691450576978341, \
     -0.4183752966234090092641990, \
     -0.3743486151220660120087939, \
     -0.3294609198374864076452867, \
     -0.2838154539022487306176554, \
     -0.2375172033464168065707124, \
     -0.1906726556261427697749124, \
     -0.1433895546989751711312496, \
     -0.0957766532091975056522186, \
     -0.0479434623531718575225298, \
     0.0000000000000000000000000, \
     0.0479434623531718575225298, \
     0.0957766532091975056522186, \
     0.1433895546989751711312496, \
     0.1906726556261427697749124, \
     0.2375172033464168065707124, \
     0.2838154539022487306176554, \
     0.3294609198374864076452867, \
     0.3743486151220660120087939, \
     0.4183752966234090092641990, \
     0.4614397015691450576978341, \
     0.5034427804550068823410431, \
     0.5442879248622271385455725, \
     0.5838811896604873133271545, \
     0.6221315090854002415825996, \
     0.6589509061936251330409408, \
     0.6942546952139916335526225, \
     0.7279616763294246790119737, \
     0.7599943224419997868739828, \
     0.7902789574921218430473804, \
     0.8187459259226514534339191, \
     0.8453297528999302839424500, \
     0.8699692949264070361941320, \
     0.8926078805047389314232855, \
     0.9131934405428462617365469, \
     0.9316786282287493379656770, \
     0.9480209281684075063737697, \
     0.9621827547180552377119837, \
     0.9741315398335511690749679, \
     0.9838398121870349413776378, \
     0.9912852761768016687218212, \
     0.9964509480618491630557949, \
     0.9993260970754128772656936 \
    ] )

    w = np.array ( [ \
     0.0017292582513002508983395851463, \
     0.0040215241720037363470786599528, \
     0.0063079425789717545501888719039, \
     0.0085801482668814598936358121592, \
     0.0108326787895979686215140551272, \
     0.013060311639994846336168342922, \
     0.015257912146448310349265388145, \
     0.017420421997670248495365759969, \
     0.019542865836750062826837429313, \
     0.021620361284934062841654274667, \
     0.023648129691287236698780978994, \
     0.025621506938037758214084978694, \
     0.027535954088450343942499722327, \
     0.029387067789310668062644859210, \
     0.031170590380189142464431845777, \
     0.032882419676368574984049638008, \
     0.034518618398549058625221276859, \
     0.036075423225565273932166270524, \
     0.037549253448257709809772223198, \
     0.038936719204051197616673806364, \
     0.040234629273005533815446337743, \
     0.041439998417240293022686299233, \
     0.042550054246755802719217150803, \
     0.043562243595800486532284821661, \
     0.044474238395082974427323504000, \
     0.045283941026300230657128240574, \
     0.045989489146651696963893390818, \
     0.046589259972233498302255136790, \
     0.047081874010454522246006808290, \
     0.047466198232885503152644458740, \
     0.047741348681240621559038972227, \
     0.047906692500495862031347289176, \
     0.04796184939446661812070762137, \
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     0.046589259972233498302255136790, \
     0.045989489146651696963893390818, \
     0.045283941026300230657128240574, \
     0.044474238395082974427323504000, \
     0.043562243595800486532284821661, \
     0.042550054246755802719217150803, \
     0.041439998417240293022686299233, \
     0.040234629273005533815446337743, \
     0.038936719204051197616673806364, \
     0.037549253448257709809772223198, \
     0.036075423225565273932166270524, \
     0.034518618398549058625221276859, \
     0.032882419676368574984049638008, \
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     0.0063079425789717545501888719039, \
     0.0040215241720037363470786599528, \
     0.0017292582513002508983395851463 \
    ] )

  elif ( n == 127 ):

    x = np.array ( [ \
     -0.9998221304153061462673512, \
     -0.9990629343553118951383159, \
     -0.9976975661898046210744170, \
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     0.999062934355311895138316, \
     0.999822130415306146267351 \
    ] )

    w = np.array ( [ \
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     0.00045645726109586662791936519265 \
    ] )

  elif ( n == 128 ):

    x = np.array ( [ \
     -0.9998248879471319144736081, \
     -0.9990774599773758950119878, \
     -0.9977332486255140198821574, \
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     0.997733248625514019882157, \
     0.999077459977375895011988, \
     0.999824887947131914473608 \
    ] )

    w = np.array ( [ \
     0.00044938096029209037639429223999, \
     0.0010458126793403487793128516001, \
     0.0016425030186690295387908755948, \
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     0.0051901618326763302050707671348, \
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     0.006351663161707188787214327826, \
     0.006926892566898813563426670360, \
     0.007497981925634728687671962688, \
     0.008064589890486057972928598698, \
     0.008626377798616749704978843782, \
     0.009183009871660874334478743688, \
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     0.011877307372740279575891106926, \
     0.012396139543950922968821728197, \
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     0.013906964132951985244288007396, \
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     0.014873122602147314252385498520, \
     0.015343010768865144085990853741, \
     0.015803728659399346858965631687, \
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     0.016696557801589204589091507954, \
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     0.00044938096029209037639429223999 \
    ] )

  elif ( n == 129 ):

    x = np.array ( [ \
     -0.9998275818477487191077441, \
     -0.9990916504696409986514389, \
     -0.9977681080525852721429460, \
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     0.2636955832669005409666949, \
     0.2870162737574911929568755, \
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     0.3331373117387248575049982, \
     0.3559105174709357969672656, \
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     0.5294580601328034000099406, \
     0.5498801655714271702189358, \
     0.5699786721652138894754096, \
     0.5897417521489813916767844, \
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     0.6282153150457794374886895, \
     0.6469031566613704719753373, \
     0.6652103023962409818802202, \
     0.6831259786828258512462248, \
     0.7006396423293521790044710, \
     0.7177409867244055767721220, \
     0.7344199479022727047791516, \
     0.7506667104654910227632368, \
     0.7664717133611208816717785, \
     0.7818256555073413245387500, \
     0.7967195012670592680339606, \
     0.8111444857653120742087717, \
     0.8250921200473358809210133, \
     0.8385541960742664442975407, \
     0.8515227915535356930243826, \
     0.8639902746011257878940216, \
     0.875949308232943389203522, \
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     0.972149904842703429727416, \
     0.977547958299367247444781, \
     0.982370735251741311550742, \
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     0.997768108052585272142946, \
     0.999091650469640998651439, \
     0.999827581847748719107744 \
    ] )

    w = np.array ( [ \
     0.00044246794182939296923668005717, \
     0.00102972844619622394463273519315, \
     0.0016172530556785534682413679271, \
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     0.016891809595063204177526208819, \
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     0.00102972844619622394463273519315, \
     0.00044246794182939296923668005717 \
    ] )

  elif ( n == 255 ):

    x = np.array ( [ \
     -0.999955705317563751730191, \
     -0.999766621312000569367063, \
     -0.999426474680169959344386, \
     -0.998935241284654635142155, \
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     -0.970656338976880365477697, \
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     -0.924757572513824425220425, \
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    ] )

  elif ( n == 256 ):

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    ] )

    w = np.array ( [ \
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    ] )

  elif ( n == 257 ):

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    ] )

    w = np.array ( [ \
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     0.0080150571857480760504667455353, \
     0.0081266793714589108764118189068, \
     0.0082370919258701685661946145361, \
     0.0083462784144114279413811886655, \
     0.0084542225850084395379670551258, \
     0.0085609083705021941391459209280, \
     0.0086663198910404675908861979240, \
     0.0087704414564414858792445834744, \
     0.0088732575685293586050755892934, \
     0.0089747529234409331997949023068, \
     0.0090749124139037264846862498962, \
     0.0091737211314845944854270065178, \
     0.0092711643688088057725325917169, \
     0.0093672276217491880067391857021, \
     0.0094618965915850218253881576301, \
     0.0095551571871303607110514249099, \
     0.0096469955268314600363329731559, \
     0.0097373979408330030783691793250, \
     0.0098263509730128164423854701706, \
     0.0099138413829847720250916955489, \
     0.0099998561480695773850435626986, \
     0.0100843824652331611676814627839, \
     0.0101674077529923650568895461852, \
     0.0102489196532876585918958554047, \
     0.0103289060333225980974485876288, \
     0.0104073549873697559257355517893, \
     0.0104842548385428511997370260353, \
     0.0105595941405348182788823332058, \
     0.0106333616793215542382761147904, \
     0.0107055464748310917616231511294, \
     0.0107761377825779489945556541150, \
     0.0108451250952624130885928632830, \
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     0.0109782469015224934483083029166, \
     0.0110423615803254284301924654946, \
     0.0111048326374699756056269264803, \
     0.0111656507743308312328559850485, \
     0.0112248069383148083152535688671, \
     0.0112822923242082872447042603128, \
     0.0113380983754878447625379269120, \
     0.011392216785593866154247619654, \
     0.011444639499166951104119199270, \
     0.011495358713246929174010288914, \
     0.011544366878434306436012137033, \
     0.011591656700013970380783131035, \
     0.011637221139040985841125311445, \
     0.011681053413388320313049670635, \
     0.011723146998756342723302879656, \
     0.011763495629643945382264331878, \
     0.011802093300281144573421477037, \
     0.011838934265523020964443424791, \
     0.011874013041704866779344562066, \
     0.011907324407458412445505183140, \
     0.011938863404489011222535627643, \
     0.011968625338313666131272065445, \
     0.011996605778959789329711050159, \
     0.012022800561624589927558893338, \
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     0.012069817823327991167612855626, \
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     0.012142270770344990738801546574, \
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     0.012167663319667843366755737416, \
     0.012177643792842880196606249581, \
     0.012185811661083365425569178819, \
     0.012192165708627157605870499188, \
     0.012196704989693764053654538465, \
     0.012199428828625117371582840212, \
     0.01220033681998614507777289232, \
     0.012199428828625117371582840212, \
     0.012196704989693764053654538465, \
     0.012192165708627157605870499188, \
     0.012185811661083365425569178819, \
     0.012177643792842880196606249581, \
     0.012167663319667843366755737416, \
     0.012155871727121082685623083829, \
     0.012142270770344990738801546574, \
     0.012126862473800277391553601370, \
     0.012109649130964635027950450318, \
     0.012090633303991361438266420912, \
     0.012069817823327991167612855626, \
     0.012047205787294992091420946532, \
     0.012022800561624589927558893338, \
     0.011996605778959789329711050159, \
     0.011968625338313666131272065445, \
     0.011938863404489011222535627643, \
     0.011907324407458412445505183140, \
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     0.011838934265523020964443424791, \
     0.011802093300281144573421477037, \
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     0.011723146998756342723302879656, \
     0.011681053413388320313049670635, \
     0.011637221139040985841125311445, \
     0.011591656700013970380783131035, \
     0.011544366878434306436012137033, \
     0.011495358713246929174010288914, \
     0.011444639499166951104119199270, \
     0.011392216785593866154247619654, \
     0.0113380983754878447625379269120, \
     0.0112822923242082872447042603128, \
     0.0112248069383148083152535688671, \
     0.0111656507743308312328559850485, \
     0.0111048326374699756056269264803, \
     0.0110423615803254284301924654946, \
     0.0109782469015224934483083029166, \
     0.0109124981443345193856719616965, \
     0.0108451250952624130885928632830, \
     0.0107761377825779489945556541150, \
     0.0107055464748310917616231511294, \
     0.0106333616793215542382761147904, \
     0.0105595941405348182788823332058, \
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     0.0103289060333225980974485876288, \
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     0.0101674077529923650568895461852, \
     0.0100843824652331611676814627839, \
     0.0099998561480695773850435626986, \
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     0.0098263509730128164423854701706, \
     0.0097373979408330030783691793250, \
     0.0096469955268314600363329731559, \
     0.0095551571871303607110514249099, \
     0.0094618965915850218253881576301, \
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     0.0092711643688088057725325917169, \
     0.0091737211314845944854270065178, \
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     0.0089747529234409331997949023068, \
     0.0088732575685293586050755892934, \
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     0.0086663198910404675908861979240, \
     0.0085609083705021941391459209280, \
     0.0084542225850084395379670551258, \
     0.0083462784144114279413811886655, \
     0.0082370919258701685661946145361, \
     0.0081266793714589108764118189068, \
     0.0080150571857480760504667455353, \
     0.0079022419833580248574070864277, \
     0.0077882505564860261212726654404, \
     0.0076730998724067939537782250476, \
     0.0075568070709469658838993300454, \
     0.0074393894619338979090297315972, \
     0.0073208645226191563361371026044, \
     0.0072012498950770900730828552207, \
     0.0070805633835788707705149901066, \
     0.0069588229519423919043121805236, \
     0.0068360467208584215286561508406, \
     0.0067122529651934070221351960200, \
     0.0065874601112693336961737372300, \
     0.0064616867341210426397202932350, \
     0.0063349515547314166407936938524, \
     0.0062072734372448464599330978665, \
     0.0060786713861593931405204596709, \
     0.0059491645434980654366600347567, \
     0.0058187721859596348346566361185, \
     0.0056875137220494140577838938236, \
     0.0055554086891904284012033890901, \
     0.0054224767508154127788846727083, \
     0.0052887376934400710240953933529, \
     0.0051542114237180378340642003713, \
     0.0050189179654779878773297516544, \
     0.0048828774567433411142588306018, \
     0.0047461101467350184936945641585, \
     0.0046086363928577081326523656522, \
     0.0044704766576701092218388764046, \
     0.0043316515058396297504806208252, \
     0.0041921816010820254766367595011, \
     0.0040520877030864825223229951262, \
     0.0039113906644266662571543468015, \
     0.0037701114274582873548537007645, \
     0.0036282710212037760873102463983, \
     0.0034858905582247143702551557840, \
     0.0033429912314827618499065991316, \
     0.0031995943111899437356540290142, \
     0.0030557211416493711130936102459, \
     0.0029113931380877846359302447381, \
     0.0027666317834818283552560256501, \
     0.0026214586253808109266552781372, \
     0.0024758952727301488651840215879, \
     0.0023299633927021828561308282641, \
     0.0021836847075455253317921866057, \
     0.0020370809914723626741694800322, \
     0.0018901740676190104269878470891, \
     0.0017429858051468299509941139300, \
     0.0015955381166175133369701690235, \
     0.0014478529559255120065233994722, \
     0.0012999523174235227389668643832, \
     0.0011518582377826677880963146741, \
     0.00100359280467969441299468763292, \
     0.00085517818446696565626595950963, \
     0.00070663671051592291949335494247, \
     0.00055799120546880640169677292533, \
     0.00040926648283531339591138751432, \
     0.00026049995580176964436806680831, \
     0.00011191470145601756450862287886 \
    ] )

  else:

    print ( '' )
    print ( 'legendre_set - Fatal error!' )
    print ( '  Illegal value of N = %d' % ( n ) )
    print ( '  Legal values are 1 through 33, 63/64/65, 127/128/129 and 255/256/257.' )
    raise Exception ( 'legendre_set - Fatal error!' )

  return x, w

def legendre_set_test01 ( ):

#*****************************************************************************80
#
## legendre_set_test01() tests legendre_set().
#
#  Licensing:
#
#    This code is distributed under the MIT license.
#
#  Modified:
#
#    02 April 2015
#
#  Author:
#
#    John Burkardt
#
  import platform

  print ( '' )
  print ( 'legendre_set_test01():' )
  print ( '  Python version: %s' % ( platform.python_version ( ) ) )
  print ( '  legendre_set() returns points and weights of a Gauss-Legendre quadrature rule.' )

  for n in [ 3, 6, 11 ]:
# for n in [ 1, 2, 3 ]:

    x, w = legendre_set ( n )

    print ( '' )
    print ( '  Gauss-Legendre rule of order %d:' % ( n ) )
    print ( '' )
    print ( '        X            W' )
    print ( '' )
    for i in range ( 0, n ):
      print ( '  %24.16f  %24.16f' % ( x[i], w[i] ) )
#
#  Terminate.
#
  print ( '' )
  print ( 'legendre_set_test01():' )
  print ( '  Normal end of execution.' )
  return

def timestamp ( ):

#*****************************************************************************80
#
## timestamp() prints the date as a timestamp.
#
#  Licensing:
#
#    This code is distributed under the MIT license. 
#
#  Modified:
#
#    06 April 2013
#
#  Author:
#
#    John Burkardt
#
  import time

  t = time.time ( )
  print ( time.ctime ( t ) )

  return

if ( __name__ == '__main__' ):
  timestamp ( )
  cauchy_principal_value_test ( )
  timestamp ( )