def r8_normal_01_test():
#*****************************************************************************80
#
## R8_NORMAL_01_TEST tests R8_NORMAL_01.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 26 July 2014
#
# Author:
#
# John Burkardt
#
from r8_normal_01 import r8_normal_01
seed = 123456789
test_num = 20
print ''
print 'R8_NORMAL_01_TEST'
print ' R8_NORMAL_01 generates normally distributed'
print ' random values.'
print ' Using initial random number seed = %d' % (seed)
print ''
for test in range(0, test_num):
x, seed = r8_normal_01(seed)
print ' %f' % (x)
#
# Terminate.
#
print ''
print 'R8_NORMAL_01_TEST'
print ' Normal end of execution.'
return
def r8vec_normal_ab ( n, mu, sigma, seed ):
#*****************************************************************************80
#
## R8VEC_NORMAL_AB returns a scaled pseudonormal R8VEC.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 04 March 2015
#
# Author:
#
# John Burkardt
#
# Parameters:
#
# Input, integer N, the number of entries in the vector.
#
# Input, real MU, the average value of the PDF.
#
# Input, real SIGMA, the standard deviation of the PDF.
#
# Input, integer SEED, a seed for the random number generator.
#
# Output, real X(N), the vector of pseudorandom values.
#
# Output, integer SEED, an updated seed for the random number generator.
#
import numpy as np
from r8_normal_01 import r8_normal_01
x = np.zeros ( n )
for i in range ( 0, n ):
xi, seed = r8_normal_01 ( seed )
x[i] = mu + sigma * xi
return x, seed
def r8_normal_01_test ( ):
#*****************************************************************************80
#
## R8_NORMAL_01_TEST tests R8_NORMAL_01.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 26 July 2014
#
# Author:
#
# John Burkardt
#
from r8_normal_01 import r8_normal_01
seed = 123456789
test_num = 20
print ''
print 'R8_NORMAL_01_TEST'
print ' R8_NORMAL_01 generates normally distributed'
print ' random values.'
print ' Using initial random number seed = %d' % ( seed )
print ''
for test in range ( 0, test_num ):
x, seed = r8_normal_01 ( seed )
print ' %f' % ( x )
#
# Terminate.
#
print ''
print 'R8_NORMAL_01_TEST'
print ' Normal end of execution.'
return
def r8mat_normal_ab ( m, n, mu, sigma, seed ):
#*****************************************************************************80
#
## R8MAT_NORMAL_AB returns a scaled pseudonormal R8MAT.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 04 March 2015
#
# Author:
#
# John Burkardt
#
# Parameters:
#
# Input, integer M, N, the number of rows and columns.
#
# Input, real MU, SIGMA, the mean and standard deviation.
#
# Input, integer SEED, a seed for the random number generator.
#
# Output, real X(M,N), the pseudorandom values.
#
# Output, integer SEED, an updated seed for the random number generator.
#
import numpy as np
from r8_normal_01 import r8_normal_01
x = np.zeros ( ( m, n ) )
for i in range ( 0, m ):
for j in range ( 0, n ):
xi, seed = r8_normal_01 ( seed )
x[i,j] = mu + sigma * xi
return x, seed
def r8mat_normal_01 ( m, n, seed ):
#*****************************************************************************80
#
## R8MAT_NORMAL_01 returns a unit pseudonormal R8MAT.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 04 March 2015
#
# Author:
#
# John Burkardt
#
# Parameters:
#
# Input, integer M, N, the number of rows and columns.
#
# Input, integer SEED, a seed for the random number generator.
#
# Output, real X(M,N), the pseudorandom values.
#
# Output, integer SEED, an updated seed for the random number generator.
#
import numpy as np
from r8_normal_01 import r8_normal_01
x = np.zeros ( ( m, n ) )
for i in range ( 0, m ):
for j in range ( 0, n ):
x[i,j], seed = r8_normal_01 ( seed )
return x, seed
def r8vec_normal_01 ( n, seed ):
#*****************************************************************************80
#
## R8VEC_NORMAL_01 returns a unit pseudonormal R8VEC.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 04 March 2015
#
# Author:
#
# John Burkardt
#
# Parameters:
#
# Input, integer N, the number of entries in the vector.
#
# Input, integer SEED, a seed for the random number generator.
#
# Output, real X(N), the vector of pseudorandom values.
#
# Output, integer SEED, an updated seed for the random number generator.
#
import numpy as np
from r8_normal_01 import r8_normal_01
x = np.zeros ( n )
for i in range ( 0, n ):
x[i], seed = r8_normal_01 ( seed )
return x, seed
def r8_normal_ab ( a, b, seed ): #*****************************************************************************80 # ## R8_NORMAL_AB returns a scaled pseudonormal R8. # # Licensing: # # This code is distributed under the GNU LGPL license. # # Modified: # # 25 July 2014 # # Author: # # John Burkardt # # Parameters: # # Input, real A, the mean of the normal PDF. # # Input, real B, the standard deviation of the normal PDF. # # Input, integer SEED, a seed for the random number generator. # # Output, real X, a sample of the standard normal PDF. # # Output, integer SEED, an updated seed for the random number generator. # from r8_normal_01 import r8_normal_01 x, seed = r8_normal_01 ( seed ) x = a + b * x return x, seed
def orth_random ( n, key ):
#*****************************************************************************80
#
## ORTH_RANDOM returns an ORTH_RANDOM matrix.
#
# Properties:
#
# The inverse of A is equal to A'.
#
# A is orthogonal: A * A' = A' * A = I.
#
# Because A is orthogonal, it is normal: A' * A = A * A'.
#
# Columns and rows of A have unit Euclidean norm.
#
# Distinct pairs of columns of A are orthogonal.
#
# Distinct pairs of rows of A are orthogonal.
#
# The L2 vector norm of A*x = the L2 vector norm of x for any vector x.
#
# The L2 matrix norm of A*B = the L2 matrix norm of B for any matrix B.
#
# det ( A ) = +1 or -1.
#
# A is unimodular.
#
# All the eigenvalues of A have modulus 1.
#
# All singular values of A are 1.
#
# All entries of A are between -1 and 1.
#
# Discussion:
#
# Thanks to Eugene Petrov, B I Stepanov Institute of Physics,
# National Academy of Sciences of Belarus, for convincingly
# pointing out the severe deficiencies of an earlier version of
# this routine.
#
# Essentially, the computation involves saving the Q factor of the
# QR factorization of a matrix whose entries are normally distributed.
# However, it is only necessary to generate this matrix a column at
# a time, since it can be shown that when it comes time to annihilate
# the subdiagonal elements of column K, these (transformed) elements of
# column K are still normally distributed random values. Hence, there
# is no need to generate them at the beginning of the process and
# transform them K-1 times.
#
# For computational efficiency, the individual Householder transformations
# could be saved, as recommended in the reference, instead of being
# accumulated into an explicit matrix format.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 17 March 2015
#
# Author:
#
# John Burkardt
#
# Reference:
#
# Pete Stewart,
# Efficient Generation of Random Orthogonal Matrices With an Application
# to Condition Estimators,
# SIAM Journal on Numerical Analysis,
# Volume 17, Number 3, June 1980, pages 403-409.
#
# Parameters:
#
# Input, integer N, the order of A.
#
# Input, integer KEY, a positive value that selects the data.
#
# Output, real A(N,N), the matrix.
#
import numpy as np
from r8_normal_01 import r8_normal_01
from r8mat_house_axh import r8mat_house_axh
from r8vec_house_column import r8vec_house_column
#
# Start with A = the identity matrix.
#
a = np.zeros ( ( n, n ) )
for i in range ( 0, n ):
a[i,i] = 1.0
#
# Now behave as though we were computing the QR factorization of
# some other random matrix. Generate the N elements of the first column,
# compute the Householder matrix H1 that annihilates the subdiagonal elements,
# and set A := A * H1' = A * H.
#
# On the second step, generate the lower N-1 elements of the second column,
# compute the Householder matrix H2 that annihilates them,
# and set A := A * H2' = A * H2 = H1 * H2.
#
# On the N-1 step, generate the lower 2 elements of column N-1,
# compute the Householder matrix HN-1 that annihilates them, and
# and set A := A * H(N-1)' = A * H(N-1) = H1 * H2 * ... * H(N-1).
# This is our random orthogonal matrix.
#
x_col = np.zeros ( n )
seed = key
for j in range ( 0, n - 1 ):
for i in range ( 0, j ):
x_col[i] = 0.0
for i in range ( j, n ):
x_col[i], seed = r8_normal_01 ( seed )
#
# Compute the vector V that defines a Householder transformation matrix
# H(V) that annihilates the subdiagonal elements of X.
#
v = r8vec_house_column ( n, x_col, j )
#
# Postmultiply the matrix A by H'(V) = H(V).
#
a = r8mat_house_axh ( n, a, v )
return a
def orth_random(n, key):
#*****************************************************************************80
#
## ORTH_RANDOM returns an ORTH_RANDOM matrix.
#
# Properties:
#
# The inverse of A is equal to A'.
#
# A is orthogonal: A * A' = A' * A = I.
#
# Because A is orthogonal, it is normal: A' * A = A * A'.
#
# Columns and rows of A have unit Euclidean norm.
#
# Distinct pairs of columns of A are orthogonal.
#
# Distinct pairs of rows of A are orthogonal.
#
# The L2 vector norm of A*x = the L2 vector norm of x for any vector x.
#
# The L2 matrix norm of A*B = the L2 matrix norm of B for any matrix B.
#
# det ( A ) = +1 or -1.
#
# A is unimodular.
#
# All the eigenvalues of A have modulus 1.
#
# All singular values of A are 1.
#
# All entries of A are between -1 and 1.
#
# Discussion:
#
# Thanks to Eugene Petrov, B I Stepanov Institute of Physics,
# National Academy of Sciences of Belarus, for convincingly
# pointing out the severe deficiencies of an earlier version of
# this routine.
#
# Essentially, the computation involves saving the Q factor of the
# QR factorization of a matrix whose entries are normally distributed.
# However, it is only necessary to generate this matrix a column at
# a time, since it can be shown that when it comes time to annihilate
# the subdiagonal elements of column K, these (transformed) elements of
# column K are still normally distributed random values. Hence, there
# is no need to generate them at the beginning of the process and
# transform them K-1 times.
#
# For computational efficiency, the individual Householder transformations
# could be saved, as recommended in the reference, instead of being
# accumulated into an explicit matrix format.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 17 March 2015
#
# Author:
#
# John Burkardt
#
# Reference:
#
# Pete Stewart,
# Efficient Generation of Random Orthogonal Matrices With an Application
# to Condition Estimators,
# SIAM Journal on Numerical Analysis,
# Volume 17, Number 3, June 1980, pages 403-409.
#
# Parameters:
#
# Input, integer N, the order of A.
#
# Input, integer KEY, a positive value that selects the data.
#
# Output, real A(N,N), the matrix.
#
import numpy as np
from r8_normal_01 import r8_normal_01
from r8mat_house_axh import r8mat_house_axh
from r8vec_house_column import r8vec_house_column
#
# Start with A = the identity matrix.
#
a = np.zeros((n, n))
for i in range(0, n):
a[i, i] = 1.0
#
# Now behave as though we were computing the QR factorization of
# some other random matrix. Generate the N elements of the first column,
# compute the Householder matrix H1 that annihilates the subdiagonal elements,
# and set A := A * H1' = A * H.
#
# On the second step, generate the lower N-1 elements of the second column,
# compute the Householder matrix H2 that annihilates them,
# and set A := A * H2' = A * H2 = H1 * H2.
#
# On the N-1 step, generate the lower 2 elements of column N-1,
# compute the Householder matrix HN-1 that annihilates them, and
# and set A := A * H(N-1)' = A * H(N-1) = H1 * H2 * ... * H(N-1).
# This is our random orthogonal matrix.
#
x_col = np.zeros(n)
seed = key
for j in range(0, n - 1):
for i in range(0, j):
x_col[i] = 0.0
for i in range(j, n):
x_col[i], seed = r8_normal_01(seed)
#
# Compute the vector V that defines a Householder transformation matrix
# H(V) that annihilates the subdiagonal elements of X.
#
v = r8vec_house_column(n, x_col, j)
#
# Postmultiply the matrix A by H'(V) = H(V).
#
a = r8mat_house_axh(n, a, v)
return a
