#! /usr/bin/env python
#
def i4vec_print ( n, a, title ):

#*****************************************************************************80
#
## I4VEC_PRINT prints an I4VEC.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    31 August 2014
#
#  Author:
#
#    John Burkardt
#
#  Parameters:
#
#    Input, integer N, the dimension of the vector.
#
#    Input, integer A(N), the vector to be printed.
#
#    Input, string TITLE, a title.
#
  print ( '' )
  print ( title )
  print ( '' )
  for i in range ( 0, n ):
    print ( '%6d  %6d' % ( i, a[i] ) )

  return

def i4vec_print_test ( ):

#*****************************************************************************80
#
## I4VEC_PRINT_TEST tests I4VEC_PRINT.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    25 September 2016
#
#  Author:
#
#    John Burkardt
#
  import numpy as np
  import platform

  print ( '' )
  print ( 'I4VEC_PRINT_TEST' )
  print ( '  Python version: %s' % ( platform.python_version ( ) ) )
  print ( '  I4VEC_PRINT prints an I4VEC.' )

  n = 4
  v = np.array ( [ 91, 92, 93, 94 ], dtype = np.int32 )
  i4vec_print ( n, v, '  Here is an I4VEC:' )
#
#  Terminate.
#
  print ( '' )
  print ( 'I4VEC_PRINT_TEST:' )
  print ( '  Normal end of execution.' )
  return

def i4vec_transpose_print ( n, a, title ):

#*****************************************************************************80
#
## I4VEC_TRANSPOSE_PRINT prints an I4VEC "transposed".
#
#  Example:
#
#    A = (/ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 /)
#    TITLE = 'My vector:  '
#
#    My vector:
#
#       1    2    3    4    5
#       6    7    8    9   10
#      11
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    02 June 2015
#
#  Author:
#
#    John Burkardt
#
#  Parameters:
#
#    Input, integer N, the number of components of the vector.
#
#    Input, integer A(N), the vector to be printed.
#
#    Input, string TITLE, a title.
#
  if ( 0 < len ( title ) ):
    print ( '' )
    print ( title )

  if ( 0 < n ):
    for i in range ( 0, n ):
      print ( '%8d' % ( a[i] ) ),
      if ( ( i + 1 ) % 10 == 0 or i == n - 1 ):
        print ( '' )
  else:
    print ( '  (empty vector)' )

  return

def i4vec_transpose_print_test ( ):

#*****************************************************************************80
#
## I4VEC_TRANSPOSE_PRINT_TEST tests I4VEC_TRANSPOSE_PRINT.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    07 April 2015
#
#  Author:
#
#    John Burkardt
#
  import numpy as np
  import platform

  print ( '' )
  print ( 'I4VEC_TRANSPOSE_PRINT_TEST' )
  print ( '  Python version: %s' % ( platform.python_version ( ) ) )
  print ( '  I4VEC_TRANSPOSE_PRINT prints an I4VEC' )
  print ( '  with 5 entries to a row, and an optional title.' )

  n = 12
  a = np.zeros ( n, dtype = np.int32 )
  
  for i in range ( 0, n ):
    a[i] = i + 1

  i4vec_transpose_print ( n, a, '  My array:  ' )
#
#  Terminate.
#
  print ( '' )
  print ( 'I4VEC_TRANSPOSE_PRINT_TEST:' )
  print ( '  Normal end of execution.' )
  return

def i4vec_uniform_ab ( n, a, b, seed ):

#*****************************************************************************80
#
## I4VEC_UNIFORM_AB returns a scaled pseudorandom I4VEC.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license. 
#
#  Modified:
#
#    05 April 2013
#
#  Author:
#
#    John Burkardt
#
#  Reference:
#
#    Paul Bratley, Bennett Fox, Linus Schrage,
#    A Guide to Simulation,
#    Second Edition,
#    Springer, 1987,
#    ISBN: 0387964673,
#    LC: QA76.9.C65.B73.
#
#    Bennett Fox,
#    Algorithm 647:
#    Implementation and Relative Efficiency of Quasirandom
#    Sequence Generators,
#    ACM Transactions on Mathematical Software,
#    Volume 12, Number 4, December 1986, pages 362-376.
#
#    Pierre L'Ecuyer,
#    Random Number Generation,
#    in Handbook of Simulation,
#    edited by Jerry Banks,
#    Wiley, 1998,
#    ISBN: 0471134031,
#    LC: T57.62.H37.
#
#    Peter Lewis, Allen Goodman, James Miller,
#    A Pseudo-Random Number Generator for the System/360,
#    IBM Systems Journal,
#    Volume 8, Number 2, 1969, pages 136-143.
#
#  Parameters:
#
#    Input, integer N, the number of entries in the vector.
#
#    Input, integer A, B, the minimum and maximum acceptable values.
#
#    Input, integer SEED, a seed for the random number generator.
#
#    Output, integer C(N), the randomly chosen integer vector.
#
#    Output, integer SEED, the updated seed.
#
  import numpy as np
  from sys import exit

  i4_huge = 2147483647

  seed = int ( seed  )

  if ( seed < 0 ):
    seed = seed + i4_huge

  if ( seed == 0 ):
    print ( '' )
    print ( 'I4VEC_UNIFORM_AB - Fatal error!' )
    print ( '  Input SEED = 0!' )
    exit ( 'I4VEC_UNIFORM_AB - Fatal error!' )

  a = round ( a )
  b = round ( b )

  c = np.zeros ( n, dtype = np.int32 )

  for i in range ( 0, n ):

    k = ( seed // 127773 )

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

    seed = ( seed % i4_huge )

    if ( seed < 0 ):
      seed = seed + i4_huge

    r = seed * 4.656612875E-10
#
#  Scale R to lie between A-0.5 and B+0.5.
#
    r = ( 1.0 - r ) * ( min ( a, b ) - 0.5 ) \
      +         r   * ( max ( a, b ) + 0.5 )
#
#  Use rounding to convert R to an integer between A and B.
#
    value = round ( r )

    value = max ( value, min ( a, b ) )
    value = min ( value, max ( a, b ) )

    c[i] = value

  return c, seed

def i4vec_uniform_ab_test ( ):

#*****************************************************************************80
#
## I4VEC_UNIFORM_AB_TEST tests I4VEC_UNIFORM_AB.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license. 
#
#  Modified:
#
#    27 October 2014
#
#  Author:
#
#    John Burkardt
#
  import platform

  n = 20
  a = -100
  b = 200
  seed = 123456789

  print ( '' )
  print ( 'I4VEC_UNIFORM_AB_TEST' )
  print ( '  Python version: %s' % ( platform.python_version ( ) ) )
  print ( '  I4VEC_UNIFORM_AB computes pseudorandom values' )
  print ( '  in an interval [A,B].' )
  print ( '' )
  print ( '  The lower endpoint A = %d' % ( a ) )
  print ( '  The upper endpoint B = %d' % ( b ) )
  print ( '  The initial seed is %d' % ( seed ) )
  print ( '' )

  v, seed = i4vec_uniform_ab ( n, a, b, seed )

  i4vec_print ( n, v, '  The random vector:' )
#
#  Terminate.
#
  print ( '' )
  print ( 'I4VEC_UNIFORM_AB_TEST:' )
  print ( '  Normal end of execution.' )
  return

def monomial_value ( m, n, e, x ):

#*****************************************************************************80
#
## MONOMIAL_VALUE evaluates a monomial.
#
#  Discussion:
#
#    This routine evaluates a monomial of the form
#
#      product ( 1 <= i <= m ) x(i)^e(i)
#
#    The combination 0.0^0, if encountered, is treated as 1.0.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license. 
#
#  Modified:
#
#    07 April 2015
#
#  Author:
#
#    John Burkardt
#
#  Parameters:
#
#    Input, integer M, the spatial dimension.
#
#    Input, integer N, the number of evaluation points.
#
#    Input, integer E(M), the exponents.
#
#    Input, real X(M,N), the point coordinates.
#
#    Output, real V(N), the monomial values.
#
  import numpy as np

  v = np.ones ( n )

  for i in range ( 0, m ):
    if ( 0 != e[i] ):
      for j in range ( 0, n ):
        v[j] = v[j] * x[i,j] ** e[i]

  return v

def monomial_value_test ( ):

#*****************************************************************************80
#
## MONOMIAL_VALUE_TEST tests MONOMIAL_VALUE on sets of data in various dimensions.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    07 April 2015
#
#  Author:
#
#    John Burkardt
#
  import platform

  print ( '' )
  print ( 'MONOMIAL_VALUE_TEST' )
  print ( '  Python version: %s' % ( platform.python_version ( ) ) )
  print ( '  Use monomial_value() to evaluate some monomials' )
  print ( '  in dimensions 1 through 3.' )

  e_min = -3
  e_max = 6
  n = 5
  seed = 123456789
  x_min = -2.0
  x_max = +10.0

  for m in range ( 1, 4 ):

    print ( '' )
    print ( '  Spatial dimension M =  %d' % ( m ) )

    e, seed = i4vec_uniform_ab ( m, e_min, e_max, seed )
    i4vec_transpose_print ( m, e, '  Exponents:' )
    x, seed = r8mat_uniform_ab ( m, n, x_min, x_max, seed )
#
#  To make checking easier, make the X values integers.
#
    for i in range ( 0, m ):
      for j in range ( 0, n ):
        x[i,j] = round ( x[i,j] )

    v = monomial_value ( m, n, e, x )

    print ( '' )
    print ( '   V(X)         ' ),
    for i in range ( 0, m ):
      print ( '      X(%d)' % ( i ) ),
    print ( '' )
    print ( '' )
    for j in range ( 0, n ):
      print ( '%14.6g  ' % ( v[j] ) ),
      for i in range ( 0, m ):
        print ( '%10.4f' % ( x[i,j] ) ),
      print ( '' )
#
#  Terminate.
#
  print ( '' )
  print ( 'MONOMIAL_VALUE_TEST' )
  print ( '  Normal end of execution.' )
  return

def r8mat_print ( m, n, a, title ):

#*****************************************************************************80
#
## R8MAT_PRINT prints an R8MAT.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    31 August 2014
#
#  Author:
#
#    John Burkardt
#
#  Parameters:
#
#    Input, integer M, the number of rows in A.
#
#    Input, integer N, the number of columns in A.
#
#    Input, real A(M,N), the matrix.
#
#    Input, string TITLE, a title.
#
  r8mat_print_some ( m, n, a, 0, 0, m - 1, n - 1, title )

  return

def r8mat_print_test ( ):

#*****************************************************************************80
#
## R8MAT_PRINT_TEST tests R8MAT_PRINT.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    10 February 2015
#
#  Author:
#
#    John Burkardt
#
  import numpy as np
  import platform

  print ( '' )
  print ( 'R8MAT_PRINT_TEST' )
  print ( '  Python version: %s' % ( platform.python_version ( ) ) )
  print ( '  R8MAT_PRINT prints an R8MAT.' )

  m = 4
  n = 6
  v = np.array ( [ \
    [ 11.0, 12.0, 13.0, 14.0, 15.0, 16.0 ], 
    [ 21.0, 22.0, 23.0, 24.0, 25.0, 26.0 ], 
    [ 31.0, 32.0, 33.0, 34.0, 35.0, 36.0 ], 
    [ 41.0, 42.0, 43.0, 44.0, 45.0, 46.0 ] ], dtype = np.float64 )
  r8mat_print ( m, n, v, '  Here is an R8MAT:' )
#
#  Terminate.
#
  print ( '' )
  print ( 'R8MAT_PRINT_TEST:' )
  print ( '  Normal end of execution.' )
  return

def r8mat_print_some ( m, n, a, ilo, jlo, ihi, jhi, title ):

#*****************************************************************************80
#
## R8MAT_PRINT_SOME prints out a portion of an R8MAT.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    10 February 2015
#
#  Author:
#
#    John Burkardt
#
#  Parameters:
#
#    Input, integer M, N, the number of rows and columns of the matrix.
#
#    Input, real A(M,N), an M by N matrix to be printed.
#
#    Input, integer ILO, JLO, the first row and column to print.
#
#    Input, integer IHI, JHI, the last row and column to print.
#
#    Input, string TITLE, a title.
#
  incx = 5

  print ( '' )
  print ( title )

  if ( m <= 0 or n <= 0 ):
    print ( '' )
    print ( '  (None)' )
    return

  for j2lo in range ( max ( jlo, 0 ), min ( jhi + 1, n ), incx ):

    j2hi = j2lo + incx - 1
    j2hi = min ( j2hi, n )
    j2hi = min ( j2hi, jhi )
    
    print ( '' )
    print ( '  Col: ' ),

    for j in range ( j2lo, j2hi + 1 ):
      print ( '%7d       ' % ( j ) ),

    print ( '' )
    print ( '  Row' )

    i2lo = max ( ilo, 0 )
    i2hi = min ( ihi, m )

    for i in range ( i2lo, i2hi + 1 ):

      print ( '%7d :' % ( i ) ),
      
      for j in range ( j2lo, j2hi + 1 ):
        print ( '%12g  ' % ( a[i,j] ) ),

      print ( '' )

  return

def r8mat_print_some_test ( ):

#*****************************************************************************80
#
## R8MAT_PRINT_SOME_TEST tests R8MAT_PRINT_SOME.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    31 October 2014
#
#  Author:
#
#    John Burkardt
#
  import numpy as np
  import platform

  print ( '' )
  print ( 'R8MAT_PRINT_SOME_TEST' )
  print ( '  Python version: %s' % ( platform.python_version ( ) ) )
  print ( '  R8MAT_PRINT_SOME prints some of an R8MAT.' )

  m = 4
  n = 6
  v = np.array ( [ \
    [ 11.0, 12.0, 13.0, 14.0, 15.0, 16.0 ], 
    [ 21.0, 22.0, 23.0, 24.0, 25.0, 26.0 ], 
    [ 31.0, 32.0, 33.0, 34.0, 35.0, 36.0 ], 
    [ 41.0, 42.0, 43.0, 44.0, 45.0, 46.0 ] ], dtype = np.float64 )
  r8mat_print_some ( m, n, v, 0, 3, 2, 5, '  Here is an R8MAT:' )
#
#  Terminate.
#
  print ( '' )
  print ( 'R8MAT_PRINT_SOME_TEST:' )
  print ( '  Normal end of execution.' )
  return

def r8mat_transpose_print ( m, n, a, title ):

#*****************************************************************************80
#
## R8MAT_TRANSPOSE_PRINT prints an R8MAT, transposed.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    31 August 2014
#
#  Author:
#
#    John Burkardt
#
#  Parameters:
#
#    Input, integer M, the number of rows in A.
#
#    Input, integer N, the number of columns in A.
#
#    Input, real A(M,N), the matrix.
#
#    Input, string TITLE, a title.
#
  r8mat_transpose_print_some ( m, n, a, 0, 0, m - 1, n - 1, title )

  return

def r8mat_transpose_print_test ( ):

#*****************************************************************************80
#
## R8MAT_TRANSPOSE_PRINT_TEST tests R8MAT_TRANSPOSE_PRINT.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    31 October 2014
#
#  Author:
#
#    John Burkardt
#
  import numpy as np
  import platform

  print ( '' )
  print ( 'R8MAT_TRANSPOSE_PRINT_TEST' )
  print ( '  Python version: %s' % ( platform.python_version ( ) ) )
  print ( '  R8MAT_TRANSPOSE_PRINT prints an R8MAT.' )

  m = 4
  n = 3
  v = np.array ( [ \
    [ 11.0, 12.0, 13.0 ], 
    [ 21.0, 22.0, 23.0 ], 
    [ 31.0, 32.0, 33.0 ], 
    [ 41.0, 42.0, 43.0 ] ], dtype = np.float64 )
  r8mat_transpose_print ( m, n, v, '  Here is an R8MAT, transposed:' )
#
#  Terminate.
#
  print ( '' )
  print ( 'R8MAT_TRANSPOSE_PRINT_TEST:' )
  print ( '  Normal end of execution.' )
  return

def r8mat_transpose_print_some ( m, n, a, ilo, jlo, ihi, jhi, title ):

#*****************************************************************************80
#
## R8MAT_TRANSPOSE_PRINT_SOME prints a portion of an R8MAT, transposed.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    13 November 2014
#
#  Author:
#
#    John Burkardt
#
#  Parameters:
#
#    Input, integer M, N, the number of rows and columns of the matrix.
#
#    Input, real A(M,N), an M by N matrix to be printed.
#
#    Input, integer ILO, JLO, the first row and column to print.
#
#    Input, integer IHI, JHI, the last row and column to print.
#
#    Input, string TITLE, a title.
#
  incx = 5

  print ( '' )
  print ( title )

  if ( m <= 0 or n <= 0 ):
    print ( '' )
    print ( '  (None)' )
    return

  for i2lo in range ( max ( ilo, 0 ), min ( ihi, m - 1 ), incx ):

    i2hi = i2lo + incx - 1
    i2hi = min ( i2hi, m - 1 )
    i2hi = min ( i2hi, ihi )
    
    print ( '' )
    print ( '  Row: ' ),

    for i in range ( i2lo, i2hi + 1 ):
      print ( '%7d       ' % ( i ) ),

    print ( '' )
    print ( '  Col' )

    j2lo = max ( jlo, 0 )
    j2hi = min ( jhi, n - 1 )

    for j in range ( j2lo, j2hi + 1 ):

      print ( '%7d :' % ( j ) ),
      
      for i in range ( i2lo, i2hi + 1 ):
        print ( '%12g  ' % ( a[i,j] ) ),

      print ( '' )

  return

def r8mat_transpose_print_some_test ( ):

#*****************************************************************************80
#
## R8MAT_TRANSPOSE_PRINT_SOME_TEST tests R8MAT_TRANSPOSE_PRINT_SOME.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    31 October 2014
#
#  Author:
#
#    John Burkardt
#
  import numpy as np
  import platform

  print ( '' )
  print ( 'R8MAT_TRANSPOSE_PRINT_SOME_TEST' )
  print ( '  Python version: %s' % ( platform.python_version ( ) ) )
  print ( '  R8MAT_TRANSPOSE_PRINT_SOME prints some of an R8MAT, transposed.' )

  m = 4
  n = 6
  v = np.array ( [ \
    [ 11.0, 12.0, 13.0, 14.0, 15.0, 16.0 ], 
    [ 21.0, 22.0, 23.0, 24.0, 25.0, 26.0 ], 
    [ 31.0, 32.0, 33.0, 34.0, 35.0, 36.0 ], 
    [ 41.0, 42.0, 43.0, 44.0, 45.0, 46.0 ] ], dtype = np.float64 )
  r8mat_transpose_print_some ( m, n, v, 0, 3, 2, 5, '  R8MAT, rows 0:2, cols 3:5:' )
#
#  Terminate.
#
  print ( '' )
  print ( 'R8MAT_TRANSPOSE_PRINT_SOME_TEST:' )
  print ( '  Normal end of execution.' )
  return

def r8mat_uniform_01 ( m, n, seed ):

#*****************************************************************************80
#
## R8MAT_UNIFORM_01 returns a unit pseudorandom R8MAT.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license. 
#
#  Modified:
#
#    08 April 2013
#
#  Author:
#
#    John Burkardt
#
#  Reference:
#
#    Paul Bratley, Bennett Fox, Linus Schrage,
#    A Guide to Simulation,
#    Second Edition,
#    Springer, 1987,
#    ISBN: 0387964673,
#    LC: QA76.9.C65.B73.
#
#    Bennett Fox,
#    Algorithm 647:
#    Implementation and Relative Efficiency of Quasirandom
#    Sequence Generators,
#    ACM Transactions on Mathematical Software,
#    Volume 12, Number 4, December 1986, pages 362-376.
#
#    Pierre L'Ecuyer,
#    Random Number Generation,
#    in Handbook of Simulation,
#    edited by Jerry Banks,
#    Wiley, 1998,
#    ISBN: 0471134031,
#    LC: T57.62.H37.
#
#    Peter Lewis, Allen Goodman, James Miller,
#    A Pseudo-Random Number Generator for the System/360,
#    IBM Systems Journal,
#    Volume 8, Number 2, 1969, pages 136-143.
#
#  Parameters:
#
#    Input, integer M, N, the number of rows and columns in the array.
#
#    Input, integer SEED, the integer "seed" used to generate
#    the output random number.
#
#    Output, real R(M,N), an array of random values between 0 and 1.
#
#    Output, integer SEED, the updated seed.  This would
#    normally be used as the input seed on the next call.
#
  import numpy as np
  from sys import exit

  i4_huge = 2147483647

  seed = int ( seed )

  if ( seed < 0 ):
    seed = seed + i4_huge

  if ( seed == 0 ):
    print ( '' )
    print ( 'R8MAT_UNIFORM_01 - Fatal error!' )
    print ( '  Input SEED = 0!' )
    exit ( 'R8MAT_UNIFORM_01 - Fatal error!' )

  r = np.zeros ( [ m, n ] )

  for j in range ( 0, n ):
    for i in range ( 0, m ):

      k = ( seed // 127773 )

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

      seed = ( seed % i4_huge )

      if ( seed < 0 ):
        seed = seed + i4_huge

      r[i,j] = seed * 4.656612875E-10

  return r, seed

def r8mat_uniform_01_test ( ):

#*****************************************************************************80
#
## R8MAT_UNIFORM_01_TEST tests R8MAT_UNIFORM_01.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license. 
#
#  Modified:
#
#    31 October 2014
#
#  Author:
#
#    John Burkardt
#
  import numpy as np
  import platform

  m = 5
  n = 4
  seed = 123456789

  print ( '' )
  print ( 'R8MAT_UNIFORM_01_TEST' )
  print ( '  Python version: %s' % ( platform.python_version ( ) ) )
  print ( '  R8MAT_UNIFORM_01 computes a random R8MAT.' )
  print ( '' )
  print ( '  0 <= X <= 1' )
  print ( '  Initial seed is %d' % ( seed ) )

  v, seed = r8mat_uniform_01 ( m, n, seed )

  r8mat_print ( m, n, v, '  Random R8MAT:' )
#
#  Terminate.
#
  print ( '' )
  print ( 'R8MAT_UNIFORM_01_TEST:' )
  print ( '  Normal end of execution.' )
  return

def r8mat_uniform_ab ( m, n, a, b, seed ):

#*****************************************************************************80
#
## R8MAT_UNIFORM_AB returns a scaled pseudorandom R8MAT.
#
#  Discussion:
#
#    An R8MAT is an array of R8's.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license. 
#
#  Modified:
#
#    08 April 2013
#
#  Author:
#
#    John Burkardt
#
#  Reference:
#
#    Paul Bratley, Bennett Fox, Linus Schrage,
#    A Guide to Simulation,
#    Second Edition,
#    Springer, 1987,
#    ISBN: 0387964673,
#    LC: QA76.9.C65.B73.
#
#    Bennett Fox,
#    Algorithm 647:
#    Implementation and Relative Efficiency of Quasirandom
#    Sequence Generators,
#    ACM Transactions on Mathematical Software,
#    Volume 12, Number 4, December 1986, pages 362-376.
#
#    Pierre L'Ecuyer,
#    Random Number Generation,
#    in Handbook of Simulation,
#    edited by Jerry Banks,
#    Wiley, 1998,
#    ISBN: 0471134031,
#    LC: T57.62.H37.
#
#    Peter Lewis, Allen Goodman, James Miller,
#    A Pseudo-Random Number Generator for the System/360,
#    IBM Systems Journal,
#    Volume 8, Number 2, 1969, pages 136-143.
#
#  Parameters:
#
#    Input, integer M, N, the number of rows and columns in the array.
#
#    Input, real A, B, the range of the pseudorandom values.
#
#    Input, integer SEED, the integer "seed" used to generate
#    the output random number.
#
#    Output, real R(M,N), an array of random values between 0 and 1.
#
#    Output, integer SEED, the updated seed.  This would
#    normally be used as the input seed on the next call.
#
  import numpy
  from sys import exit

  i4_huge = 2147483647

  seed = int ( seed )

  if ( seed < 0 ):
    seed = seed + i4_huge

  if ( seed == 0 ):
    print ( '' )
    print ( 'R8MAT_UNIFORM_AB - Fatal error!' )
    print ( '  Input SEED = 0!' )
    exit ( 'R8MAT_UNIFORM_AB - Fatal error!' )

  r = numpy.zeros ( [ m, n ] )

  for j in range ( 0, n ):
    for i in range ( 0, m ):

      k = ( seed // 127773 )

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

      seed = ( seed % i4_huge )

      if ( seed < 0 ):
        seed = seed + i4_huge

      r[i,j] = a + ( b - a ) * seed * 4.656612875E-10

  return r, seed

def r8mat_uniform_ab_test ( ):

#*****************************************************************************80
#
## R8MAT_UNIFORM_AB_TEST tests R8MAT_UNIFORM_AB.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license. 
#
#  Modified:
#
#    31 October 2014
#
#  Author:
#
#    John Burkardt
#
  import numpy as np
  import platform

  m = 5
  n = 4
  a = -1.0
  b = +5.0
  seed = 123456789

  print ( '' )
  print ( 'R8MAT_UNIFORM_AB_TEST' )
  print ( '  Python version: %s' % ( platform.python_version ( ) ) )
  print ( '  R8MAT_UNIFORM_AB computes a random R8MAT.' )
  print ( '' )
  print ( '  %g <= X <= %g' % ( a, b ) )
  print ( '  Initial seed is %d' % ( seed ) )

  v, seed = r8mat_uniform_ab ( m, n, a, b, seed )

  r8mat_print ( m, n, v, '  Random R8MAT:' )
#
#  Terminate.
#
  print ( '' )
  print ( 'R8MAT_UNIFORM_AB_TEST:' )
  print ( '  Normal end of execution.' )
  return

def square01_area ( ):

#*****************************************************************************80
#
## SQUARE01_AREA returns the area of the unit square in 2D.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license. 
#
#  Modified:
#
#    23 June 2015
#
#  Author:
#
#    John Burkardt
#
#  Parameters:
#
#    Output, real VALUE, the area.
#
  value = 1.0;

  return value

def square01_area_test ( ) :

#*****************************************************************************80
#
## SQUARE01_AREA tests SQUARE01_AREA.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    23 June 2015
#
#  Author:
#
#    John Burkardt
#
  import platform

  print ( '' )
  print ( 'SQUARE01_AREA_TEST' )
  print ( '  Python version: %s' % ( platform.python_version ( ) ) )
  print ( '  SQUARE01_AREA returns the area of the unit square.' )

  value = square01_area ( )

  print ( '' )
  print ( '  SQUARE01_AREA() = %g' % ( value ) )
#
#  Terminate.
#
  print ( '' )
  print ( 'SQUARE01_AREA_TEST' )
  print ( '  Normal end of execution.' )
  return

def square01_monomial_integral ( e ):

#*****************************************************************************80
#
## SQUARE01_MONOMIAL_INTEGRAL: integrals over the unit square in 2D.
#
#  Discussion:
#
#    The integration region is 
#
#      0 <= X <= 1,
#      0 <= Y <= 1.
#
#    The monomial is F(X,Y) = X^E(1) * Y^E(2).
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license. 
#
#  Modified:
#
#    23 June 2015
#
#  Author:
#
#    John Burkardt
#
#  Reference:
#
#    Philip Davis, Philip Rabinowitz,
#    Methods of Numerical Integration,
#    Second Edition,
#    Academic Press, 1984, page 263.
#
#  Parameters:
#
#    Input, integer E(2), the exponents.  Each exponent must be nonnegative.
#
#    Output, real INTEGRAL, the integral.
#
  from sys import exit

  m = 2

  if ( e[0] < 0 or e[1] < 0 ):
    print ( '' )
    print ( 'SQUARE01_MONOMIAL_INTEGRAL - Fatal error!' )
    print ( '  All exponents must be nonnegative.' )
    exit ( 'SQUARE01_MONOMIAL_INTEGRAL - Fatal error!' )

  integral = 1.0
  for i in range ( 0, m ):
    integral = integral / float ( e[i] + 1 )

  return integral

def square01_monomial_integral_test ( ):

#*****************************************************************************80
#
## SQUARE01_MONOMIAL_INTEGRAL_TEST tests SQUARE01_MONOMIAL_INTEGRAL.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    23 June 2015
#
#  Author:
#
#    John Burkardt
#
  import numpy as np
  import platform

  m = 2
  n = 4192
  test_num = 20

  print ( '' )
  print ( 'SQUARE01_MONOMIAL_INTEGRAL_TEST' )
  print ( '  Python version: %s' % ( platform.python_version ( ) ) )
  print ( '  SQUARE01_MONOMIAL_INTEGRAL returns the integral of a monomial' )
  print ( '  over the interior of the unit square in 2D.' )
  print ( '  Compare with a Monte Carlo estimate.' )
#
#  Get sample points.
#
  seed = 123456789
  x, seed = square01_sample ( n, seed )

  print ( '' )
  print ( '  Number of sample points used is %d' % ( n ) )
#
#  Randomly choose exponents.
#
  print ( '' )
  print ( '  Ex  Ey     MC-Estimate           Exact      Error' )
  print ( '' )

  for test in range ( 0, test_num ):

    e, seed = i4vec_uniform_ab ( m, 0, 7, seed )

    value = monomial_value ( m, n, e, x )

    result = square01_area ( ) * np.sum ( value ) / float ( n )
    exact = square01_monomial_integral ( e )
    error = abs ( result - exact )

    print ( '  %2d  %2d  %14.6g  %14.6g  %10.2g' \
      % ( e[0], e[1], result, exact, error ) )
#
#  Terminate.
#
  print ( '' )
  print ( 'SQUARE01_MONOMIAL_INTEGRAL_TEST:' )
  print ( '  Normal end of execution.' )
  return

def square01_monte_carlo_test ( ):

#*****************************************************************************80
#
## SQUARE01_MONTE_CARLO_TEST estimates integrals over the unit square in 2D.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    08 November 2016
#
#  Author:
#
#    John Burkardt
#
  import numpy as np
  import platform

  e_test = np.array ( [ \
    [ 0, 0 ], \
    [ 2, 0 ], \
    [ 0, 2 ], \
    [ 4, 0 ], \
    [ 2, 2 ], \
    [ 0, 4 ], \
    [ 6, 0 ] ] )

  print ( '' )
  print ( 'SQUARE01_MONTE_CARLO_TEST' )
  print ( '  Python version: %s' % ( platform.python_version ( ) ) )
  print ( '  Use SQUARE01_SAMPLE to estimate integrals ' )
  print ( '  along the interior of the unit square in 2D.' )

  seed = 123456789

  print ( '' ),
  print ( '         N' ),
  print ( '           1' ),
  print ( '              X^2' ),
  print ( '             Y^2' ),
  print ( '             X^4' ),
  print ( '           X^2Y^2' ),
  print ( '             Y^4' ),
  print ( '           X^6\n' ),
  print ( '' )

  n = 1
  e = np.zeros ( 2 )

  while ( n <= 65536 ):

    x, seed = square01_sample ( n, seed )
    print ( '  %8d' % ( n ) ),

    for j in range ( 0, 7 ):

      e = e_test[j,0:2]

      value = monomial_value ( 2, n, e, x )

      result = square01_area ( ) * np.sum ( value[0:n] ) / float ( n )

      print ( '  %14.6g' % ( result ) ),

    print ( '' )

    n = 2 * n

  print ( '' )
  print ( '     Exact' )

  for j in range ( 0, 7 ):

    e = e_test[j,0:2]

    result = square01_monomial_integral ( e )
    print ( '  %14.6g', result ),

  print ( '' )

#
#  Terminate.
#
  print ( '' )
  print ( 'SQUARE01_MONTE_CARLO_TEST:' )
  print ( '  Normal end of execution.' )
  return

def square01_sample ( n, seed ):

#*****************************************************************************80
#
## SQUARE01_SAMPLE samples points in the unit square in 2D.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license. 
#
#  Modified:
#
#    23 June 2015
#
#  Author:
#
#    John Burkardt
#
#  Parameters:
#
#    Input, integer N, the number of points.
#
#    Input/output, integer SEED, a seed for the random 
#    number generator.
#
#    Output, real X(2,N), the points.
#
  m = 2

  x, seed = r8mat_uniform_01 ( m, n, seed )

  return x, seed

def square01_sample_test ( ):

#*****************************************************************************80
#
## SQUARE01_SAMPLE_TEST tests SQUARE01_SAMPLE.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    23 June 2015
#
#  Author:
#
#    John Burkardt
#
  import platform

  print ( '' )
  print ( 'SQUARE01_SAMPLE_TEST' )
  print ( '  Python version: %s' % ( platform.python_version ( ) ) )
  print ( '  SQUARE01_SAMPLE samples the unit square.' )

  m = 2
  n = 10
  seed = 123456789

  x, seed = square01_sample ( n, seed )

  r8mat_transpose_print ( m, n, x, '  Sample points in the unit square.' )
#
#  Terminate.
#
  print ( '' )
  print ( 'SQUARE01_SAMPLE_TEST' )
  print ( '  Normal end of execution.' )
  return

def timestamp ( ):

#*****************************************************************************80
#
## TIMESTAMP prints the date as a timestamp.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license. 
#
#  Modified:
#
#    06 April 2013
#
#  Author:
#
#    John Burkardt
#
#  Parameters:
#
#    None
#
  import time

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

  return None

def timestamp_test ( ):

#*****************************************************************************80
#
## TIMESTAMP_TEST tests TIMESTAMP.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license. 
#
#  Modified:
#
#    03 December 2014
#
#  Author:
#
#    John Burkardt
#
#  Parameters:
#
#    None
#
  import platform

  print ( '' )
  print ( 'TIMESTAMP_TEST:' )
  print ( '  Python version: %s' % ( platform.python_version ( ) ) )
  print ( '  TIMESTAMP prints a timestamp of the current date and time.' )
  print ( '' )

  timestamp ( )
#
#  Terminate.
#
  print ( '' )
  print ( 'TIMESTAMP_TEST:' )
  print ( '  Normal end of execution.' )
  return

def square_monte_carlo_test ( ):

#*****************************************************************************80
#
## SQUARE_MONTE_CARLO_TEST tests the SQUARE_MONTE_CARLO library.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license. 
#
#  Modified:
#
#    08 November 2016
#
#  Author:
#
#    John Burkardt
#
  import platform

  print ( '' )
  print ( 'SQUARE_MONTE_CARLO_TEST' )
  print ( '  Python version: %s' % ( platform.python_version ( ) ) )
  print ( '  Test the SQUARE_MONTE_CARLO library.' )
#
#  Utility functions.
#
  i4vec_print_test ( )
  i4vec_transpose_print_test ( )
  i4vec_uniform_ab_test ( )
  r8mat_print_test ( )
  r8mat_print_some_test ( )
  r8mat_transpose_print_test ( )
  r8mat_transpose_print_some_test ( )
  r8mat_uniform_01_test ( )
  r8mat_uniform_ab_test ( )
#
#  Library functions.
#
  monomial_value_test ( )
  square01_area_test ( )
  square01_monomial_integral_test ( )
  square01_monte_carlo_test ( )
  square01_sample_test ( )
#
#  Terminate.
#
  print ( '' )
  print ( 'SQUARE_MONTE_CARLO_TEST:' )
  print ( '  Normal end of execution.' )
  return

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