def ijfact1_determinant ( n ):
#*****************************************************************************80
#
## IJFACT1_DETERMINANT computes the determinant of the IJFACT1 matrix.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 15 February 2015
#
# Author:
#
# John Burkardt
#
# Parameters:
#
# Input, integer N, the order of the matrix.
#
# Output, real VALUE, the determinant.
#
from r8_factorial import r8_factorial
value = 1.0
for i in range ( 1, n ):
value = value * r8_factorial ( i + 1 ) * r8_factorial ( n - i )
value = value * r8_factorial ( n + 1 )
return value
def ijfact1_determinant(n):
#*****************************************************************************80
#
## IJFACT1_DETERMINANT computes the determinant of the IJFACT1 matrix.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 15 February 2015
#
# Author:
#
# John Burkardt
#
# Parameters:
#
# Input, integer N, the order of the matrix.
#
# Output, real VALUE, the determinant.
#
from r8_factorial import r8_factorial
value = 1.0
for i in range(1, n):
value = value * r8_factorial(i + 1) * r8_factorial(n - i)
value = value * r8_factorial(n + 1)
return value
def laguerre_determinant ( n ):
#*****************************************************************************80
#
## LAGUERRE_DETERMINANT computes the determinant of the LAGUERRE matrix.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 17 February 2015
#
# Author:
#
# John Burkardt
#
# Parameters:
#
# Input, integer N, the order of the matrix.
#
# Output, real VALUE, the determinant.
#
from r8_factorial import r8_factorial
value = 1.0
p = + 1.0
for i in range ( 0, n ):
value = value * p / r8_factorial ( i )
p = - p
return value
def laguerre_determinant(n):
#*****************************************************************************80
#
## LAGUERRE_DETERMINANT computes the determinant of the LAGUERRE matrix.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 17 February 2015
#
# Author:
#
# John Burkardt
#
# Parameters:
#
# Input, integer N, the order of the matrix.
#
# Output, real VALUE, the determinant.
#
from r8_factorial import r8_factorial
value = 1.0
p = +1.0
for i in range(0, n):
value = value * p / r8_factorial(i)
p = -p
return value
def carry_inverse(n, alpha):
#*****************************************************************************80
#
## CARRY_INVERSE returns the inverse of the CARRY matrix.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 24 March 2015
#
# Author:
#
# John Burkardt
#
# Parameters:
#
# Input, integer ALPHA, the numeric base used in the addition.
#
# Input, integer ALPHA, the numeric base used in the addition.
#
# Output, real A(N,N), the matrix.
#
from diagonal import diagonal
from r8_factorial import r8_factorial
from r8mat_mm import r8mat_mm
v = carry_eigen_left(n, alpha)
d = carry_eigenvalues(n, alpha)
for i in range(0, n):
d[i] = 1.0 / d[i]
d_inv = diagonal(n, n, d)
u = carry_eigen_right(n, alpha)
dv = r8mat_mm(n, n, n, d_inv, v)
a = r8mat_mm(n, n, n, u, dv)
t = r8_factorial(n)
for j in range(0, n):
for i in range(0, n):
a[i, j] = a[i, j] / t
return a
def carry_inverse ( n, alpha ):
#*****************************************************************************80
#
## CARRY_INVERSE returns the inverse of the CARRY matrix.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 24 March 2015
#
# Author:
#
# John Burkardt
#
# Parameters:
#
# Input, integer ALPHA, the numeric base used in the addition.
#
# Input, integer ALPHA, the numeric base used in the addition.
#
# Output, real A(N,N), the matrix.
#
from diagonal import diagonal
from r8_factorial import r8_factorial
from r8mat_mm import r8mat_mm
v = carry_eigen_left ( n, alpha )
d = carry_eigenvalues ( n, alpha )
for i in range ( 0, n ):
d[i] = 1.0 / d[i]
d_inv = diagonal ( n, n, d )
u = carry_eigen_right ( n, alpha )
dv = r8mat_mm ( n, n, n, d_inv, v )
a = r8mat_mm ( n, n, n, u, dv )
t = r8_factorial ( n )
for j in range ( 0, n ):
for i in range ( 0, n ):
a[i,j] = a[i,j] / t
return a
def legendre_associated_normalized(n, m, x):
#*****************************************************************************80
#
## LEGENDRE_ASSOCIATED_NORMALIZED evaluates the associated Legendre functions.
#
# Discussion:
#
# The unnormalized associated Legendre functions P_N^M(X) have
# the property that
#
# Integral ( -1 <= X <= 1 ) ( P_N^M(X) )^2 dX
# = 2 * ( N + M )! / ( ( 2 * N + 1 ) * ( N - M )! )
#
# By dividing the function by the square root of this term,
# the normalized associated Legendre functions have norm 1.
#
# However, we plan to use these functions to build spherical
# harmonics, so we use a slightly different normalization factor of
#
# sqrt ( ( ( 2 * N + 1 ) * ( N - M )! ) / ( 4 * pi * ( N + M )! ) )
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 20 February 2015
#
# Author:
#
# John Burkardt
#
# Reference:
#
# Milton Abramowitz and Irene Stegun,
# Handbook of Mathematical Functions,
# US Department of Commerce, 1964.
#
# Parameters:
#
# Input, integer N, the maximum first index of the Legendre
# function, which must be at least 0.
#
# Input, integer M, the second index of the Legendre function,
# which must be at least 0, and no greater than N.
#
# Input, real X, the point at which the function is to be
# evaluated. X must satisfy -1 <= X <= 1.
#
# Output, real CX(1:N+1), the values of the first N+1 function.
#
import numpy as np
from r8_factorial import r8_factorial
if (m < 0):
print ''
print 'LEGENDRE_ASSOCIATED_NORMALIZED - Fatal error!'
print ' Input value of M is %d' % (m)
print ' but M must be nonnegative.'
if (n < m):
print ''
print 'LEGENDRE_ASSOCIATED_NORMALIZED - Fatal error!'
print ' Input value of M = %d' % (m)
print ' Input value of N = %d' % (n)
print ' but M must be less than or equal to N.'
if (x < -1.0):
print ''
print 'LEGENDRE_ASSOCIATED_NORMALIZED - Fatal error!'
print ' Input value of X = %f' % (x)
print ' but X must be no less than -1.'
if (1.0 < x):
print ''
print 'LEGENDRE_ASSOCIATED_NORMALIZED - Fatal error!'
print ' Input value of X = %f' % (x)
print ' but X must be no more than 1.'
cx = np.zeros(n + 1)
cx[m] = 1.0
somx2 = np.sqrt(1.0 - x * x)
fact = 1.0
for i in range(0, m):
cx[m] = -cx[m] * fact * somx2
fact = fact + 2.0
if (m != n):
cx[m + 1] = x * float(2 * m + 1) * cx[m]
for i in range(m + 2, n + 1):
cx[i] = ( float ( 2 * i - 1 ) * x * cx[i-1] \
+ float ( - i - m + 1 ) * cx[i-2] ) \
/ float ( i - m )
#
# Normalization.
#
for mm in range(m, n + 1):
factor = np.sqrt ( ( ( 2 * mm + 1 ) * r8_factorial ( mm - m ) ) \
/ ( 4.0 * np.pi * r8_factorial ( mm + m ) ) )
cx[mm] = cx[mm] * factor
return cx
def ijfact2 ( n ):
#*****************************************************************************80
#
## IJFACT2 returns the IJFACT2 matrix.
#
# Formula:
#
# A(I,J) = 1 / ( (I+J)! )
#
# Example:
#
# N = 4
#
# 1/2 1/6 1/24 1/120
# 1/6 1/24 1/120 1/720
# 1/24 1/120 1/720 1/5040
# 1/120 1/720 1/5040 1/40320
#
# Properties:
#
# A is symmetric: A' = A.
#
# Because A is symmetric, it is normal.
#
# Because A is normal, it is diagonalizable.
#
# A is a Hankel matrix: constant along anti-diagonals.
#
# A is integral: int ( A ) = A.
#
# The family of matrices is nested as a function of N.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 15 February 2015
#
# Author:
#
# John Burkardt
#
# Reference:
#
# MJC Gover,
# The explicit inverse of factorial Hankel matrices,
# Department of Mathematics, University of Bradford, 1993.
#
# Parameters:
#
# Input, integer N, the order of the matrix.
#
# Output, real A(N,N), the matrix.
#
import numpy as np
from r8_factorial import r8_factorial
a = np.zeros ( [ n, n ] )
for i in range ( 0, n ):
for j in range ( 0, n ):
a[i,j] = 1.0 / r8_factorial ( i + j + 2 )
return a
def simplex_coordinates2_test(m):
#*****************************************************************************80
#
## SIMPLEX_COORDINATES2_TEST tests SIMPLEX_COORDINATES2.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 28 June 2015
#
# Author:
#
# John Burkardt
#
# Parameters:
#
# Input, integer M, the spatial dimension.
#
import numpy as np
import platform
import r8_factorial
import r8mat_transpose_print
from simplex_volume import simplex_volume
print('')
print('SIMPLEX_COORDINATES2_TEST')
print(' Python version: %s' % (platform.python_version()))
print(' Test SIMPLEX_COORDINATES2')
x = simplex_coordinates2(m)
#r8mat_transpose_print.r8mat_transpose_print ( m, m + 1, x, ' Simplex vertex coordinates:' )
s = 0.0
for i in range(0, m):
s = s + (x[i, 0] - x[i, 1])**2
side = np.sqrt(s)
#volume = simplex_volume ( m, x )
volume2 = np.sqrt(m + 1) / r8_factorial.r8_factorial(m) / np.sqrt(
2.0**m) * side**m
print('')
print(' Side length = %g' % (side))
print(' Volume = %g' % (volume2))
print(' Expected volume = %g' % (volume2))
xtx = np.dot(np.transpose(x), x)
print(xtx)
#r8mat_transpose_print.r8mat_transpose_print ( m + 1, m + 1, xtx, ' Dot product matrix:' )
#
# Terminate.
#
print('')
print('SIMPLEX_COORDINATES2_TEST')
print(' Normal end of execution.')
return
def ijfact1(n):
#*****************************************************************************80
#
## IJFACT1 returns the IJFACT1 matrix.
#
# Formula:
#
# A(I,J) = (I+J)!
#
# Example:
#
# N = 4
#
# 2 6 24 120
# 6 24 120 720
# 24 120 720 5040
# 120 720 5040 40320
#
# Properties:
#
# A is symmetric: A' = A.
#
# Because A is symmetric, it is normal.
#
# Because A is normal, it is diagonalizable.
#
# A is a Hankel matrix: constant along anti-diagonals.
#
# A is integral: int ( A ) = A.
#
# The family of matrices is nested as a function of N.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 15 February 2015
#
# Author:
#
# John Burkardt
#
# Reference:
#
# MJC Gover,
# The explicit inverse of factorial Hankel matrices,
# Department of Mathematics, University of Bradford, 1993.
#
# Parameters:
#
# Input, integer N, the order of the matrix.
#
# Output, real A(N,N), the matrix.
#
import numpy as np
from r8_factorial import r8_factorial
a = np.zeros([n, n])
for i in range(0, n):
for j in range(0, n):
a[i, j] = r8_factorial(i + j + 2)
return a
def legendre_associated_normalized ( n, m, x ):
#*****************************************************************************80
#
## LEGENDRE_ASSOCIATED_NORMALIZED evaluates the associated Legendre functions.
#
# Discussion:
#
# The unnormalized associated Legendre functions P_N^M(X) have
# the property that
#
# Integral ( -1 <= X <= 1 ) ( P_N^M(X) )^2 dX
# = 2 * ( N + M )! / ( ( 2 * N + 1 ) * ( N - M )! )
#
# By dividing the function by the square root of this term,
# the normalized associated Legendre functions have norm 1.
#
# However, we plan to use these functions to build spherical
# harmonics, so we use a slightly different normalization factor of
#
# sqrt ( ( ( 2 * N + 1 ) * ( N - M )! ) / ( 4 * pi * ( N + M )! ) )
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 20 February 2015
#
# Author:
#
# John Burkardt
#
# Reference:
#
# Milton Abramowitz and Irene Stegun,
# Handbook of Mathematical Functions,
# US Department of Commerce, 1964.
#
# Parameters:
#
# Input, integer N, the maximum first index of the Legendre
# function, which must be at least 0.
#
# Input, integer M, the second index of the Legendre function,
# which must be at least 0, and no greater than N.
#
# Input, real X, the point at which the function is to be
# evaluated. X must satisfy -1 <= X <= 1.
#
# Output, real CX(1:N+1), the values of the first N+1 function.
#
import numpy as np
from r8_factorial import r8_factorial
if ( m < 0 ):
print ''
print 'LEGENDRE_ASSOCIATED_NORMALIZED - Fatal error!'
print ' Input value of M is %d' % ( m )
print ' but M must be nonnegative.'
if ( n < m ):
print ''
print 'LEGENDRE_ASSOCIATED_NORMALIZED - Fatal error!'
print ' Input value of M = %d' % ( m )
print ' Input value of N = %d' % ( n )
print ' but M must be less than or equal to N.'
if ( x < -1.0 ):
print ''
print 'LEGENDRE_ASSOCIATED_NORMALIZED - Fatal error!'
print ' Input value of X = %f' % ( x )
print ' but X must be no less than -1.'
if ( 1.0 < x ):
print ''
print 'LEGENDRE_ASSOCIATED_NORMALIZED - Fatal error!'
print ' Input value of X = %f' % ( x )
print ' but X must be no more than 1.'
cx = np.zeros ( n + 1 )
cx[m] = 1.0
somx2 = np.sqrt ( 1.0 - x * x )
fact = 1.0
for i in range ( 0, m ):
cx[m] = - cx[m] * fact * somx2
fact = fact + 2.0
if ( m != n ):
cx[m+1] = x * float ( 2 * m + 1 ) * cx[m]
for i in range ( m + 2, n + 1 ):
cx[i] = ( float ( 2 * i - 1 ) * x * cx[i-1] \
+ float ( - i - m + 1 ) * cx[i-2] ) \
/ float ( i - m )
#
# Normalization.
#
for mm in range ( m, n + 1 ):
factor = np.sqrt ( ( ( 2 * mm + 1 ) * r8_factorial ( mm - m ) ) \
/ ( 4.0 * np.pi * r8_factorial ( mm + m ) ) )
cx[mm] = cx[mm] * factor
return cx
def bernoulli_number3 ( n ):
#*****************************************************************************80
#
## BERNOULLI_NUMBER3 computes the value of the Bernoulli number B(N).
#
# Discussion:
#
# The Bernoulli numbers are rational.
#
# If we define the sum of the M-th powers of the first N integers as:
#
# SIGMA(M,N) = sum ( 0 <= I <= N ) I^M
#
# and let C(I,J) be the combinatorial coefficient:
#
# C(I,J) = I! / ( ( I - J )! * J! )
#
# then the Bernoulli numbers B(J) satisfy:
#
# SIGMA(M,N) = 1/(M+1) * sum ( 0 <= J <= M ) C(M+1,J) B(J) * (N+1)^(M+1-J)
#
# First values:
#
# B0 1 = 1.00000000000
# B1 -1/2 = -0.50000000000
# B2 1/6 = 1.66666666666
# B3 0 = 0
# B4 -1/30 = -0.03333333333
# B5 0 = 0
# B6 1/42 = 0.02380952380
# B7 0 = 0
# B8 -1/30 = -0.03333333333
# B9 0 = 0
# B10 5/66 = 0.07575757575
# B11 0 = 0
# B12 -691/2730 = -0.25311355311
# B13 0 = 0
# B14 7/6 = 1.16666666666
# B15 0 = 0
# B16 -3617/510 = -7.09215686274
# B17 0 = 0
# B18 43867/798 = 54.97117794486
# B19 0 = 0
# B20 -174611/330 = -529.12424242424
# B21 0 = 0
# B22 854513/138 = 6192.123
# B23 0 = 0
# B24 -236364091/2730 = -86580.257
# B25 0 = 0
# B26 8553103/6 = 1425517.16666
# B27 0 = 0
# B28 -23749461029/870 = -27298231.0678
# B29 0 = 0
# B30 8615841276005/14322 = 601580873.901
#
# Recursion:
#
# With C(N+1,K) denoting the standard binomial coefficient,
#
# B(0) = 1.0
# B(N) = - ( sum ( 0 <= K < N ) C(N+1,K) * B(K) ) / C(N+1,N)
#
# Special Values:
#
# Except for B(1), all Bernoulli numbers of odd index are 0.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 27 December 2014
#
# Author:
#
# John Burkardt
#
# Parameters:
#
# Input, integer N, the order of the Bernoulli number to compute.
#
# Output, real B, the desired Bernoulli number.
#
from r8_factorial import r8_factorial
itmax = 1000
r8_pi = 3.141592653589793
tol = 5.0E-07
if ( n < 0 ):
b = 0.0
elif ( n == 0 ):
b = 1.0
elif ( n == 1 ):
b = -0.5
elif ( n == 2 ):
b = 1.0 / 6.0
elif ( ( n % 2 ) == 1 ):
b = 0.0
else:
sum2 = 0.0
for i in range ( 1, itmax + 1 ):
term = 1.0 / float ( i ) ** n
sum2 = sum2 + term
if ( abs ( term ) < tol or abs ( term ) < tol * abs ( sum2 ) ):
break
b = 2.0 * sum2 * r8_factorial ( n ) / ( 2.0 * r8_pi ) ** n
if ( ( n % 4 ) == 0 ):
b = -b
return b