def r8mat_is_null_left(m, n, a, x):
#*****************************************************************************80
#
## R8MAT_IS_NULL_LEFT determines if x is a left null vector of matrix A.
#
# Discussion:
#
# The nonzero M vector x is a left null vector of the MxN matrix A if
#
# x'*A = A'*x = 0
#
# If A is a square matrix, then this implies that A is singular.
#
# If A is a square matrix, this implies that 0 is an eigenvalue of A,
# and that x is an associated eigenvector.
#
# This routine returns 0 if x is exactly a left null vector of A.
#
# It returns a "huge" value if x is the zero vector.
#
# Otherwise, it returns the L2 norm of A' * x divided by the L2 norm of x:
#
# ERROR_L2 = NORM_L2 ( A' * x ) / NORM_L2 ( x )
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 07 March 2015
#
# Author:
#
# John Burkardt
#
# Parameters:
#
# Input, integer M, N, the row and column dimensions of
# the matrix. M and N must be positive.
#
# Input, real A(M,N), the matrix.
#
# Input, real X(M), the vector.
#
# Output, real VALUE, the result.
# 0.0 indicates that X is exactly a left null vector.
# A "huge" value indicates that ||x|| = 0;
# Otherwise, the value returned is a relative error ||A'*x||/||x||.
#
from r8mat_mtv import r8mat_mtv
from r8vec_norm_l2 import r8vec_norm_l2
#
# X_NORM
#
x_norm = r8vec_norm_l2(m, x)
#
# ATX = A'*X
#
atx = r8mat_mtv(m, n, a, x)
#
# ATX_NORM
#
atx_norm = r8vec_norm_l2(n, atx)
#
# Value
#
value = atx_norm / x_norm
return value
def parameter_spiral_test():
#*****************************************************************************80
#
## PARAMETER_SPIRAL_TEST monitors solution norms over time for various values of NU, RHO.
#
# Location:
#
# http://people.sc.fsu.edu/~jburkardt/py_src/navier_stokes_2d_exact/parameter_spiral_test.py
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 31 January 2015
#
# Author:
#
# John Burkardt
#
import numpy as np
from r8vec_norm_l2 import r8vec_norm_l2
from r8vec_uniform_ab import r8vec_uniform_ab
from uvp_spiral import uvp_spiral
print ''
print 'PARAMETER_SPIRAL_TEST'
print ' Spiral Flow'
print ' Monitor solution norms over time for various'
print ' values of NU, RHO.'
n = 1000
xy_lo = 0.0
xy_hi = 1.0
seed = 123456789
x, seed = r8vec_uniform_ab(n, xy_lo, xy_hi, seed)
y, seed = r8vec_uniform_ab(n, xy_lo, xy_hi, seed)
#
# Vary RHO.
#
print ''
print ' RHO affects the pressure scaling.'
print ''
print ' RHO NU T ||U|| ||V|| ||P||'
print ''
nu = 1.0
rho = 1.0
for j in range(0, 3):
for k in range(0, 6):
t = k / 5.0
u, v, p = uvp_spiral(nu, rho, n, x, y, t)
u_norm = r8vec_norm_l2(n, u) / n
v_norm = r8vec_norm_l2(n, v) / n
p_norm = r8vec_norm_l2(n, p) / n
print ' %10.4g %10.4g %8.4g %10.4g %10.4g %10.4g' \
% ( rho, nu, t, u_norm, v_norm, p_norm )
print ''
rho = rho / 100.0
#
# Vary NU.
#
print ''
print ' NU affects the time scaling.'
print ''
print ' RHO NU T ||U|| ||V|| ||P||'
print ''
nu = 1.0
rho = 1.0
for i in range(0, 4):
for k in range(0, 6):
t = k / 5.0
u, v, p = uvp_spiral(nu, rho, n, x, y, t)
u_norm = r8vec_norm_l2(n, u) / n
v_norm = r8vec_norm_l2(n, v) / n
p_norm = r8vec_norm_l2(n, p) / n
print ' %10.4g %10.4g %8.4g %10.4g %10.4g %10.4g' \
% ( rho, nu, t, u_norm, v_norm, p_norm )
print ''
nu = nu / 10.0
print ''
print 'PARAMETER_TAYLOR_TEST:'
print ' Normal end of execution.'
return
def test_null_right():
#*****************************************************************************80
#
## TEST_NULL_RIGHT tests right null vectors.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 12 March 2015
#
# Author:
#
# John Burkardt
#
from a123 import a123
from a123 import a123_null_right
from archimedes import archimedes
from archimedes import archimedes_null_right
from cheby_diff1 import cheby_diff1
from cheby_diff1 import cheby_diff1_null_right
from creation import creation
from creation import creation_null_right
from dif1 import dif1
from dif1 import dif1_null_right
from dif1cyclic import dif1cyclic
from dif1cyclic import dif1cyclic_null_right
from dif2cyclic import dif2cyclic
from dif2cyclic import dif2cyclic_null_right
from fibonacci1 import fibonacci1
from fibonacci1 import fibonacci1_null_right
from hamming import hamming
from hamming import hamming_null_right
from line_adj import line_adj
from line_adj import line_adj_null_right
from moler2 import moler2
from moler2 import moler2_null_right
from neumann import neumann
from neumann import neumann_null_right
from one import one
from one import one_null_right
from r8_uniform_ab import r8_uniform_ab
from r8mat_is_null_right import r8mat_is_null_right
from r8mat_norm_fro import r8mat_norm_fro
from r8vec_norm_l2 import r8vec_norm_l2
from ring_adj import ring_adj
from ring_adj import ring_adj_null_right
from rosser1 import rosser1
from rosser1 import rosser1_null_right
from zero import zero
from zero import zero_null_right
print ''
print 'TEST_NULL_RIGHT'
print ' A = a test matrix of order M by N'
print ' x = an N vector, candidate for a right null vector.'
print ''
print ' ||A|| = Frobenius norm of A.'
print ' ||x|| = L2 norm of x.'
print ' ||A*x||/||x|| = L2 norm of A*x over L2 norm of x.'
print ''
print ' Title M N ',
print '||A|| ||x|| ||A*x||/||x||'
print ''
#
# A123
#
title = 'A123'
m = 3
n = 3
a = a123()
x = a123_null_right()
error_l2 = r8mat_is_null_right(m, n, a, x)
norm_a_frobenius = r8mat_norm_fro(m, n, a)
norm_x_l2 = r8vec_norm_l2(n, x)
print ' %-20s %4d %4d %14g %14g %10.2g' \
% ( title, m, n, norm_a_frobenius, norm_x_l2, error_l2 )
#
# ARCHIMEDES
#
title = 'ARCHIMEDES'
m = 7
n = 8
a = archimedes()
x = archimedes_null_right()
error_l2 = r8mat_is_null_right(m, n, a, x)
norm_a_frobenius = r8mat_norm_fro(m, n, a)
norm_x_l2 = r8vec_norm_l2(n, x)
print ' %-20s %4d %4d %14g %14g %10.2g' \
% ( title, m, n, norm_a_frobenius, norm_x_l2, error_l2 )
#
# CHEBY_DIFF1
#
title = 'CHEBY_DIFF1'
m = 5
n = m
a = cheby_diff1(n)
x = cheby_diff1_null_right(m, n)
error_l2 = r8mat_is_null_right(m, n, a, x)
norm_a_frobenius = r8mat_norm_fro(m, n, a)
norm_x_l2 = r8vec_norm_l2(n, x)
print ' %-20s %4d %4d %14g %14g %10.2g' \
% ( title, m, n, norm_a_frobenius, norm_x_l2, error_l2 )
#
# CREATION
#
title = 'CREATION'
m = 5
n = m
a = creation(m, n)
x = creation_null_right(m, n)
error_l2 = r8mat_is_null_right(m, n, a, x)
norm_a_frobenius = r8mat_norm_fro(m, n, a)
norm_x_l2 = r8vec_norm_l2(n, x)
print ' %-20s %4d %4d %14g %14g %10.2g' \
% ( title, m, n, norm_a_frobenius, norm_x_l2, error_l2 )
#
# DIF1
# Only has null vectors for N odd.
#
title = 'DIF1'
m = 5
n = m
a = dif1(m, n)
x = dif1_null_right(m, n)
error_l2 = r8mat_is_null_right(m, n, a, x)
norm_a_frobenius = r8mat_norm_fro(m, n, a)
norm_x_l2 = r8vec_norm_l2(n, x)
print ' %-20s %4d %4d %14g %14g %10.2g' \
% ( title, m, n, norm_a_frobenius, norm_x_l2, error_l2 )
#
# DIF1CYCLIC
#
title = 'DIF1CYCLIC'
m = 5
n = m
a = dif1cyclic(n)
x = dif1cyclic_null_right(m, n)
error_l2 = r8mat_is_null_right(m, n, a, x)
norm_a_frobenius = r8mat_norm_fro(m, n, a)
norm_x_l2 = r8vec_norm_l2(n, x)
print ' %-20s %4d %4d %14g %14g %10.2g' \
% ( title, m, n, norm_a_frobenius, norm_x_l2, error_l2 )
#
# DIF2CYCLIC
#
title = 'DIF2CYCLIC'
m = 5
n = m
a = dif2cyclic(n)
x = dif2cyclic_null_right(m, n)
error_l2 = r8mat_is_null_right(m, n, a, x)
norm_a_frobenius = r8mat_norm_fro(m, n, a)
norm_x_l2 = r8vec_norm_l2(n, x)
print ' %-20s %4d %4d %14g %14g %10.2g' \
% ( title, m, n, norm_a_frobenius, norm_x_l2, error_l2 )
#
# FIBONACCI1
#
title = 'FIBONACCI1'
m = 5
n = m
r8_lo = -5.0
r8_hi = +5.0
seed = 123456789
f1, seed = r8_uniform_ab(r8_lo, r8_hi, seed)
f2, seed = r8_uniform_ab(r8_lo, r8_hi, seed)
a = fibonacci1(n, f1, f2)
x = fibonacci1_null_right(m, n, f1, f2)
error_l2 = r8mat_is_null_right(m, n, a, x)
norm_a_frobenius = r8mat_norm_fro(m, n, a)
norm_x_l2 = r8vec_norm_l2(n, x)
print ' %-20s %4d %4d %14g %14g %10.2g' \
% ( title, m, n, norm_a_frobenius, norm_x_l2, error_l2 )
#
# HAMMING
#
title = 'HAMMING'
m = 5
n = (2**m) - 1
a = hamming(m, n)
x = hamming_null_right(m, n)
error_l2 = r8mat_is_null_right(m, n, a, x)
norm_a_frobenius = r8mat_norm_fro(m, n, a)
norm_x_l2 = r8vec_norm_l2(n, x)
print ' %-20s %4d %4d %14g %14g %10.2g' \
% ( title, m, n, norm_a_frobenius, norm_x_l2, error_l2 )
#
# LINE_ADJ
#
title = 'LINE_ADJ'
m = 7
n = m
a = line_adj(n)
x = line_adj_null_right(m, n)
error_l2 = r8mat_is_null_right(m, n, a, x)
norm_a_frobenius = r8mat_norm_fro(m, n, a)
norm_x_l2 = r8vec_norm_l2(n, x)
print ' %-20s %4d %4d %14g %14g %10.2g' \
% ( title, m, n, norm_a_frobenius, norm_x_l2, error_l2 )
#
# MOLER2
#
title = 'MOLER2'
m = 5
n = 5
a = moler2()
x = moler2_null_right()
error_l2 = r8mat_is_null_right(m, n, a, x)
norm_a_frobenius = r8mat_norm_fro(m, n, a)
norm_x_l2 = r8vec_norm_l2(n, x)
print ' %-20s %4d %4d %14g %14g %10.2g' \
% ( title, m, n, norm_a_frobenius, norm_x_l2, error_l2 )
#
# NEUMANN
#
title = 'NEUMANN'
row_num = 5
col_num = 5
m = row_num * col_num
n = row_num * col_num
a = neumann(row_num, col_num)
x = neumann_null_right(row_num, col_num)
error_l2 = r8mat_is_null_right(m, n, a, x)
norm_a_frobenius = r8mat_norm_fro(m, n, a)
norm_x_l2 = r8vec_norm_l2(n, x)
print ' %-20s %4d %4d %14g %14g %10.2g' \
% ( title, m, n, norm_a_frobenius, norm_x_l2, error_l2 )
#
# ONE
#
title = 'ONE'
m = 5
n = 5
a = one(m, n)
x = one_null_right(m, n)
error_l2 = r8mat_is_null_right(m, n, a, x)
norm_a_frobenius = r8mat_norm_fro(m, n, a)
norm_x_l2 = r8vec_norm_l2(n, x)
print ' %-20s %4d %4d %14g %14g %10.2g' \
% ( title, m, n, norm_a_frobenius, norm_x_l2, error_l2 )
#
# RING_ADJ
# N must be a multiple of 4 for there to be a null vector.
#
title = 'RING_ADJ'
m = 12
n = 12
a = ring_adj(m, n)
x = ring_adj_null_right(m, n)
error_l2 = r8mat_is_null_right(m, n, a, x)
norm_a_frobenius = r8mat_norm_fro(m, n, a)
norm_x_l2 = r8vec_norm_l2(n, x)
print ' %-20s %4d %4d %14g %14g %10.2g' \
% ( title, m, n, norm_a_frobenius, norm_x_l2, error_l2 )
#
# ROSSER1
#
title = 'ROSSER1'
m = 8
n = 8
a = rosser1()
x = rosser1_null_right()
error_l2 = r8mat_is_null_right(m, n, a, x)
norm_a_frobenius = r8mat_norm_fro(m, n, a)
norm_x_l2 = r8vec_norm_l2(n, x)
print ' %-20s %4d %4d %14g %14g %10.2g' \
% ( title, m, n, norm_a_frobenius, norm_x_l2, error_l2 )
#
# ZERO
#
title = 'ZERO'
m = 5
n = 5
a = zero(m, n)
x = zero_null_right(m, n)
error_l2 = r8mat_is_null_right(m, n, a, x)
norm_a_frobenius = r8mat_norm_fro(m, n, a)
norm_x_l2 = r8vec_norm_l2(n, x)
print ' %-20s %4d %4d %14g %14g %10.2g' \
% ( title, m, n, norm_a_frobenius, norm_x_l2, error_l2 )
#
# Terminate.
#
print ''
print 'TEST_NULL_RIGHT:'
print ' Normal end of execution.'
return
def parameter_spiral_test ( ):
#*****************************************************************************80
#
## PARAMETER_SPIRAL_TEST monitors solution norms over time for various values of NU, RHO.
#
# Location:
#
# http://people.sc.fsu.edu/~jburkardt/py_src/navier_stokes_2d_exact/parameter_spiral_test.py
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 31 January 2015
#
# Author:
#
# John Burkardt
#
import numpy as np
from r8vec_norm_l2 import r8vec_norm_l2
from r8vec_uniform_ab import r8vec_uniform_ab
from uvp_spiral import uvp_spiral
print ''
print 'PARAMETER_SPIRAL_TEST'
print ' Spiral Flow'
print ' Monitor solution norms over time for various'
print ' values of NU, RHO.'
n = 1000
xy_lo = 0.0
xy_hi = 1.0
seed = 123456789
x, seed = r8vec_uniform_ab ( n, xy_lo, xy_hi, seed )
y, seed = r8vec_uniform_ab ( n, xy_lo, xy_hi, seed )
#
# Vary RHO.
#
print ''
print ' RHO affects the pressure scaling.'
print ''
print ' RHO NU T ||U|| ||V|| ||P||'
print ''
nu = 1.0
rho = 1.0
for j in range ( 0, 3 ):
for k in range ( 0, 6 ):
t = k / 5.0
u, v, p = uvp_spiral ( nu, rho, n, x, y, t )
u_norm = r8vec_norm_l2 ( n, u ) / n
v_norm = r8vec_norm_l2 ( n, v ) / n
p_norm = r8vec_norm_l2 ( n, p ) / n
print ' %10.4g %10.4g %8.4g %10.4g %10.4g %10.4g' \
% ( rho, nu, t, u_norm, v_norm, p_norm )
print ''
rho = rho / 100.0
#
# Vary NU.
#
print ''
print ' NU affects the time scaling.'
print ''
print ' RHO NU T ||U|| ||V|| ||P||'
print ''
nu = 1.0;
rho = 1.0;
for i in range ( 0, 4 ):
for k in range ( 0, 6 ):
t = k / 5.0
u, v, p = uvp_spiral ( nu, rho, n, x, y, t )
u_norm = r8vec_norm_l2 ( n, u ) / n
v_norm = r8vec_norm_l2 ( n, v ) / n
p_norm = r8vec_norm_l2 ( n, p ) / n
print ' %10.4g %10.4g %8.4g %10.4g %10.4g %10.4g' \
% ( rho, nu, t, u_norm, v_norm, p_norm )
print ''
nu = nu / 10.0
print ''
print 'PARAMETER_TAYLOR_TEST:'
print ' Normal end of execution.'
return
def r8mat_is_null_left ( m, n, a, x ): #*****************************************************************************80 # ## R8MAT_IS_NULL_LEFT determines if x is a left null vector of matrix A. # # Discussion: # # The nonzero M vector x is a left null vector of the MxN matrix A if # # x'*A = A'*x = 0 # # If A is a square matrix, then this implies that A is singular. # # If A is a square matrix, this implies that 0 is an eigenvalue of A, # and that x is an associated eigenvector. # # This routine returns 0 if x is exactly a left null vector of A. # # It returns a "huge" value if x is the zero vector. # # Otherwise, it returns the L2 norm of A' * x divided by the L2 norm of x: # # ERROR_L2 = NORM_L2 ( A' * x ) / NORM_L2 ( x ) # # Licensing: # # This code is distributed under the GNU LGPL license. # # Modified: # # 07 March 2015 # # Author: # # John Burkardt # # Parameters: # # Input, integer M, N, the row and column dimensions of # the matrix. M and N must be positive. # # Input, real A(M,N), the matrix. # # Input, real X(M), the vector. # # Output, real VALUE, the result. # 0.0 indicates that X is exactly a left null vector. # A "huge" value indicates that ||x|| = 0; # Otherwise, the value returned is a relative error ||A'*x||/||x||. # from r8mat_mtv import r8mat_mtv from r8vec_norm_l2 import r8vec_norm_l2 # # X_NORM # x_norm = r8vec_norm_l2 ( m, x ) # # ATX = A'*X # atx = r8mat_mtv ( m, n, a, x ) # # ATX_NORM # atx_norm = r8vec_norm_l2 ( n, atx ) # # Value # value = atx_norm / x_norm return value
def helmert2 ( n, x ):
#*****************************************************************************80
#
## HELMERT2 returns the HELMERT2 matrix.
#
# Formula:
#
# Row 1 = the vector, divided by its L2 norm.
#
# Row 2 is computed by the requirements that it be orthogonal to row 1,
# be nonzero only from columns 1 to 2, and have a negative diagonal.
#
# Row 3 is computed by the requirements that it be orthogonal to
# rows 1 and 2, be nonzero only from columns 1 to 3, and have a
# negative diagonal, and so on.
#
# Properties:
#
# The first row of A should be the vector X, divided by its L2 norm.
#
# A is orthogonal: A' * A = A * A' = I.
#
# A is not symmetric: A' ~= A.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 26 March 2015
#
# Author:
#
# John Burkardt
#
# Reference:
#
# HO Lancaster,
# The Helmert Matrices,
# American Mathematical Monthly,
# Volume 72, 1965, pages 4-12.
#
# Parameters:
#
# Input, integer N, the order of A.
#
# Input, real X(N), the vector that defines the first row.
# X must not have 0 L2 norm, and its first entry must not be 0.
#
# Output, real A(N,N), the matrix.
#
import numpy as np
from r8vec_norm_l2 import r8vec_norm_l2
from sys import exit
a = np.zeros ( ( n, n ) )
x_norm_l2 = r8vec_norm_l2 ( n, x )
if ( x_norm_l2 == 0.0 ):
print ''
print 'HELMERT2 - Fatal error!'
print ' Input vector has zero L2 norm.'
exit ( 'HELMERT2 - Fatal error!' );
if ( x[0] == 0.0 ):
print ''
print 'HELMERT2 - Fatal error!'
print ' Input vector has X[0] = 0.'
exit ( 'HELMERT2 - Fatal error!' );
w = np.zeros ( n )
for i in range ( 0, n ):
w[i] = ( x[i] / x_norm_l2 ) ** 2
s = np.zeros ( n )
s[0] = w[0]
for i in range ( 1, n ):
s[i] = s[i-1] + w[i]
for j in range ( 0, n ):
a[0,j] = np.sqrt ( w[j] )
for i in range ( 1, n ):
for j in range ( 0, i ):
a[i,j] = np.sqrt ( w[i] * w[j] / ( s[i] * s[i-1] ) )
a[i,i] = - np.sqrt ( s[i-1] / s[i] )
return a
def test_null_right ( ):
#*****************************************************************************80
#
## TEST_NULL_RIGHT tests right null vectors.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 12 March 2015
#
# Author:
#
# John Burkardt
#
from a123 import a123
from a123 import a123_null_right
from archimedes import archimedes
from archimedes import archimedes_null_right
from cheby_diff1 import cheby_diff1
from cheby_diff1 import cheby_diff1_null_right
from creation import creation
from creation import creation_null_right
from dif1 import dif1
from dif1 import dif1_null_right
from dif1cyclic import dif1cyclic
from dif1cyclic import dif1cyclic_null_right
from dif2cyclic import dif2cyclic
from dif2cyclic import dif2cyclic_null_right
from fibonacci1 import fibonacci1
from fibonacci1 import fibonacci1_null_right
from hamming import hamming
from hamming import hamming_null_right
from line_adj import line_adj
from line_adj import line_adj_null_right
from moler2 import moler2
from moler2 import moler2_null_right
from neumann import neumann
from neumann import neumann_null_right
from one import one
from one import one_null_right
from r8_uniform_ab import r8_uniform_ab
from r8mat_is_null_right import r8mat_is_null_right
from r8mat_norm_fro import r8mat_norm_fro
from r8vec_norm_l2 import r8vec_norm_l2
from ring_adj import ring_adj
from ring_adj import ring_adj_null_right
from rosser1 import rosser1
from rosser1 import rosser1_null_right
from zero import zero
from zero import zero_null_right
print ''
print 'TEST_NULL_RIGHT'
print ' A = a test matrix of order M by N'
print ' x = an N vector, candidate for a right null vector.'
print ''
print ' ||A|| = Frobenius norm of A.'
print ' ||x|| = L2 norm of x.'
print ' ||A*x||/||x|| = L2 norm of A*x over L2 norm of x.'
print ''
print ' Title M N ',
print '||A|| ||x|| ||A*x||/||x||'
print ''
#
# A123
#
title = 'A123'
m = 3
n = 3
a = a123 ( )
x = a123_null_right ( )
error_l2 = r8mat_is_null_right ( m, n, a, x )
norm_a_frobenius = r8mat_norm_fro ( m, n, a )
norm_x_l2 = r8vec_norm_l2 ( n, x )
print ' %-20s %4d %4d %14g %14g %10.2g' \
% ( title, m, n, norm_a_frobenius, norm_x_l2, error_l2 )
#
# ARCHIMEDES
#
title = 'ARCHIMEDES'
m = 7
n = 8
a = archimedes ( )
x = archimedes_null_right ( )
error_l2 = r8mat_is_null_right ( m, n, a, x )
norm_a_frobenius = r8mat_norm_fro ( m, n, a )
norm_x_l2 = r8vec_norm_l2 ( n, x )
print ' %-20s %4d %4d %14g %14g %10.2g' \
% ( title, m, n, norm_a_frobenius, norm_x_l2, error_l2 )
#
# CHEBY_DIFF1
#
title = 'CHEBY_DIFF1'
m = 5
n = m
a = cheby_diff1 ( n )
x = cheby_diff1_null_right ( m, n )
error_l2 = r8mat_is_null_right ( m, n, a, x )
norm_a_frobenius = r8mat_norm_fro ( m, n, a )
norm_x_l2 = r8vec_norm_l2 ( n, x )
print ' %-20s %4d %4d %14g %14g %10.2g' \
% ( title, m, n, norm_a_frobenius, norm_x_l2, error_l2 )
#
# CREATION
#
title = 'CREATION'
m = 5
n = m
a = creation ( m, n )
x = creation_null_right ( m, n )
error_l2 = r8mat_is_null_right ( m, n, a, x )
norm_a_frobenius = r8mat_norm_fro ( m, n, a )
norm_x_l2 = r8vec_norm_l2 ( n, x )
print ' %-20s %4d %4d %14g %14g %10.2g' \
% ( title, m, n, norm_a_frobenius, norm_x_l2, error_l2 )
#
# DIF1
# Only has null vectors for N odd.
#
title = 'DIF1'
m = 5
n = m
a = dif1 ( m, n )
x = dif1_null_right ( m, n )
error_l2 = r8mat_is_null_right ( m, n, a, x )
norm_a_frobenius = r8mat_norm_fro ( m, n, a )
norm_x_l2 = r8vec_norm_l2 ( n, x )
print ' %-20s %4d %4d %14g %14g %10.2g' \
% ( title, m, n, norm_a_frobenius, norm_x_l2, error_l2 )
#
# DIF1CYCLIC
#
title = 'DIF1CYCLIC'
m = 5
n = m
a = dif1cyclic ( n )
x = dif1cyclic_null_right ( m, n )
error_l2 = r8mat_is_null_right ( m, n, a, x )
norm_a_frobenius = r8mat_norm_fro ( m, n, a )
norm_x_l2 = r8vec_norm_l2 ( n, x )
print ' %-20s %4d %4d %14g %14g %10.2g' \
% ( title, m, n, norm_a_frobenius, norm_x_l2, error_l2 )
#
# DIF2CYCLIC
#
title = 'DIF2CYCLIC'
m = 5
n = m
a = dif2cyclic ( n )
x = dif2cyclic_null_right ( m, n )
error_l2 = r8mat_is_null_right ( m, n, a, x )
norm_a_frobenius = r8mat_norm_fro ( m, n, a )
norm_x_l2 = r8vec_norm_l2 ( n, x )
print ' %-20s %4d %4d %14g %14g %10.2g' \
% ( title, m, n, norm_a_frobenius, norm_x_l2, error_l2 )
#
# FIBONACCI1
#
title = 'FIBONACCI1'
m = 5
n = m
r8_lo = -5.0
r8_hi = +5.0
seed = 123456789
f1, seed = r8_uniform_ab ( r8_lo, r8_hi, seed )
f2, seed = r8_uniform_ab ( r8_lo, r8_hi, seed )
a = fibonacci1 ( n, f1, f2 )
x = fibonacci1_null_right ( m, n, f1, f2 )
error_l2 = r8mat_is_null_right ( m, n, a, x )
norm_a_frobenius = r8mat_norm_fro ( m, n, a )
norm_x_l2 = r8vec_norm_l2 ( n, x )
print ' %-20s %4d %4d %14g %14g %10.2g' \
% ( title, m, n, norm_a_frobenius, norm_x_l2, error_l2 )
#
# HAMMING
#
title = 'HAMMING'
m = 5
n = ( 2 ** m ) - 1
a = hamming ( m, n )
x = hamming_null_right ( m, n )
error_l2 = r8mat_is_null_right ( m, n, a, x )
norm_a_frobenius = r8mat_norm_fro ( m, n, a )
norm_x_l2 = r8vec_norm_l2 ( n, x )
print ' %-20s %4d %4d %14g %14g %10.2g' \
% ( title, m, n, norm_a_frobenius, norm_x_l2, error_l2 )
#
# LINE_ADJ
#
title = 'LINE_ADJ'
m = 7
n = m
a = line_adj ( n )
x = line_adj_null_right ( m, n )
error_l2 = r8mat_is_null_right ( m, n, a, x )
norm_a_frobenius = r8mat_norm_fro ( m, n, a )
norm_x_l2 = r8vec_norm_l2 ( n, x )
print ' %-20s %4d %4d %14g %14g %10.2g' \
% ( title, m, n, norm_a_frobenius, norm_x_l2, error_l2 )
#
# MOLER2
#
title = 'MOLER2'
m = 5
n = 5
a = moler2 ( )
x = moler2_null_right ( )
error_l2 = r8mat_is_null_right ( m, n, a, x )
norm_a_frobenius = r8mat_norm_fro ( m, n, a )
norm_x_l2 = r8vec_norm_l2 ( n, x )
print ' %-20s %4d %4d %14g %14g %10.2g' \
% ( title, m, n, norm_a_frobenius, norm_x_l2, error_l2 )
#
# NEUMANN
#
title = 'NEUMANN'
row_num = 5
col_num = 5
m = row_num * col_num
n = row_num * col_num
a = neumann ( row_num, col_num )
x = neumann_null_right ( row_num, col_num )
error_l2 = r8mat_is_null_right ( m, n, a, x )
norm_a_frobenius = r8mat_norm_fro ( m, n, a )
norm_x_l2 = r8vec_norm_l2 ( n, x )
print ' %-20s %4d %4d %14g %14g %10.2g' \
% ( title, m, n, norm_a_frobenius, norm_x_l2, error_l2 )
#
# ONE
#
title = 'ONE'
m = 5
n = 5
a = one ( m, n )
x = one_null_right ( m, n )
error_l2 = r8mat_is_null_right ( m, n, a, x )
norm_a_frobenius = r8mat_norm_fro ( m, n, a )
norm_x_l2 = r8vec_norm_l2 ( n, x )
print ' %-20s %4d %4d %14g %14g %10.2g' \
% ( title, m, n, norm_a_frobenius, norm_x_l2, error_l2 )
#
# RING_ADJ
# N must be a multiple of 4 for there to be a null vector.
#
title = 'RING_ADJ'
m = 12
n = 12
a = ring_adj ( m, n )
x = ring_adj_null_right ( m, n )
error_l2 = r8mat_is_null_right ( m, n, a, x )
norm_a_frobenius = r8mat_norm_fro ( m, n, a )
norm_x_l2 = r8vec_norm_l2 ( n, x )
print ' %-20s %4d %4d %14g %14g %10.2g' \
% ( title, m, n, norm_a_frobenius, norm_x_l2, error_l2 )
#
# ROSSER1
#
title = 'ROSSER1'
m = 8
n = 8
a = rosser1 ( )
x = rosser1_null_right ( )
error_l2 = r8mat_is_null_right ( m, n, a, x )
norm_a_frobenius = r8mat_norm_fro ( m, n, a )
norm_x_l2 = r8vec_norm_l2 ( n, x )
print ' %-20s %4d %4d %14g %14g %10.2g' \
% ( title, m, n, norm_a_frobenius, norm_x_l2, error_l2 )
#
# ZERO
#
title = 'ZERO'
m = 5
n = 5
a = zero ( m, n )
x = zero_null_right ( m, n )
error_l2 = r8mat_is_null_right ( m, n, a, x )
norm_a_frobenius = r8mat_norm_fro ( m, n, a )
norm_x_l2 = r8vec_norm_l2 ( n, x )
print ' %-20s %4d %4d %14g %14g %10.2g' \
% ( title, m, n, norm_a_frobenius, norm_x_l2, error_l2 )
#
# Terminate.
#
print ''
print 'TEST_NULL_RIGHT:'
print ' Normal end of execution.'
return
def helmert2(n, x):
#*****************************************************************************80
#
## HELMERT2 returns the HELMERT2 matrix.
#
# Formula:
#
# Row 1 = the vector, divided by its L2 norm.
#
# Row 2 is computed by the requirements that it be orthogonal to row 1,
# be nonzero only from columns 1 to 2, and have a negative diagonal.
#
# Row 3 is computed by the requirements that it be orthogonal to
# rows 1 and 2, be nonzero only from columns 1 to 3, and have a
# negative diagonal, and so on.
#
# Properties:
#
# The first row of A should be the vector X, divided by its L2 norm.
#
# A is orthogonal: A' * A = A * A' = I.
#
# A is not symmetric: A' ~= A.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 26 March 2015
#
# Author:
#
# John Burkardt
#
# Reference:
#
# HO Lancaster,
# The Helmert Matrices,
# American Mathematical Monthly,
# Volume 72, 1965, pages 4-12.
#
# Parameters:
#
# Input, integer N, the order of A.
#
# Input, real X(N), the vector that defines the first row.
# X must not have 0 L2 norm, and its first entry must not be 0.
#
# Output, real A(N,N), the matrix.
#
import numpy as np
from r8vec_norm_l2 import r8vec_norm_l2
from sys import exit
a = np.zeros((n, n))
x_norm_l2 = r8vec_norm_l2(n, x)
if (x_norm_l2 == 0.0):
print ''
print 'HELMERT2 - Fatal error!'
print ' Input vector has zero L2 norm.'
exit('HELMERT2 - Fatal error!')
if (x[0] == 0.0):
print ''
print 'HELMERT2 - Fatal error!'
print ' Input vector has X[0] = 0.'
exit('HELMERT2 - Fatal error!')
w = np.zeros(n)
for i in range(0, n):
w[i] = (x[i] / x_norm_l2)**2
s = np.zeros(n)
s[0] = w[0]
for i in range(1, n):
s[i] = s[i - 1] + w[i]
for j in range(0, n):
a[0, j] = np.sqrt(w[j])
for i in range(1, n):
for j in range(0, i):
a[i, j] = np.sqrt(w[i] * w[j] / (s[i] * s[i - 1]))
a[i, i] = -np.sqrt(s[i - 1] / s[i])
return a
def test_null_left ( ):
#*****************************************************************************80
#
## TEST_NULL_LEFT tests left null vectors.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 17 March 2015
#
# Author:
#
# John Burkardt
#
from a123 import a123
from a123 import a123_null_left
from cheby_diff1 import cheby_diff1
from cheby_diff1 import cheby_diff1_null_left
from creation import creation
from creation import creation_null_left
from dif1 import dif1
from dif1 import dif1_null_left
from dif1cyclic import dif1cyclic
from dif1cyclic import dif1cyclic_null_left
from dif2cyclic import dif2cyclic
from dif2cyclic import dif2cyclic_null_left
from eberlein import eberlein
from eberlein import eberlein_null_left
from fibonacci1 import fibonacci1
from fibonacci1 import fibonacci1_null_left
from lauchli import lauchli
from lauchli import lauchli_null_left
from line_adj import line_adj
from line_adj import line_adj_null_left
from moler2 import moler2
from moler2 import moler2_null_left
from one import one
from one import one_null_left
from r8_uniform_ab import r8_uniform_ab
from r8mat_is_null_left import r8mat_is_null_left
from r8mat_norm_fro import r8mat_norm_fro
from r8vec_norm_l2 import r8vec_norm_l2
from ring_adj import ring_adj
from ring_adj import ring_adj_null_left
from rosser1 import rosser1
from rosser1 import rosser1_null_left
from zero import zero
from zero import zero_null_left
print ''
print 'TEST_NULL_LEFT'
print ' A = a test matrix of order M by N'
print ' x = an M vector, candidate for a left null vector.'
print ''
print ' ||A|| = Frobenius norm of A.'
print ' ||x|| = L2 norm of x.'
print ' ||A''*x||/||x|| = L2 norm of A\'*x over L2 norm of x.'
print ''
print ' Title M N ',
print '||A|| ||x|| ||A\'*x||/||x||'
print ''
#
# A123
#
title = 'A123'
m = 3
n = 3
a = a123 ( )
x = a123_null_left ( )
error_l2 = r8mat_is_null_left ( m, n, a, x )
norm_a_frobenius = r8mat_norm_fro ( m, n, a )
norm_x_l2 = r8vec_norm_l2 ( m, x )
print ' %-20s %4d %4d %14g %14g %10.2g' \
% ( title, m, n, norm_a_frobenius, norm_x_l2, error_l2 )
#
# CHEBY_DIFF1
#
title = 'CHEBY_DIFF1'
m = 5
n = m
a = cheby_diff1 ( n )
x = cheby_diff1_null_left ( m, n )
error_l2 = r8mat_is_null_left ( m, n, a, x )
norm_a_frobenius = r8mat_norm_fro ( m, n, a )
norm_x_l2 = r8vec_norm_l2 ( m, x )
print ' %-20s %4d %4d %14g %14g %10.2g' \
% ( title, m, n, norm_a_frobenius, norm_x_l2, error_l2 )
#
# CREATION
#
title = 'CREATION'
m = 5
n = m
a = creation ( m, n )
x = creation_null_left ( m, n )
error_l2 = r8mat_is_null_left ( m, n, a, x )
norm_a_frobenius = r8mat_norm_fro ( m, n, a )
norm_x_l2 = r8vec_norm_l2 ( m, x )
print ' %-20s %4d %4d %14g %14g %10.2g' \
% ( title, m, n, norm_a_frobenius, norm_x_l2, error_l2 )
#
# DIF1
#
title = 'DIF1'
m = 5
n = 5
a = dif1 ( m, n )
x = dif1_null_left ( m, n )
error_l2 = r8mat_is_null_left ( m, n, a, x )
norm_a_frobenius = r8mat_norm_fro ( m, n, a )
norm_x_l2 = r8vec_norm_l2 ( m, x )
print ' %-20s %4d %4d %14g %14g %10.2g' \
% ( title, m, n, norm_a_frobenius, norm_x_l2, error_l2 )
#
# DIF1CYCLIC
#
title = 'DIF1CYCLIC'
m = 5
n = m
a = dif1cyclic ( n )
x = dif1cyclic_null_left ( m, n )
error_l2 = r8mat_is_null_left ( m, n, a, x )
norm_a_frobenius = r8mat_norm_fro ( m, n, a )
norm_x_l2 = r8vec_norm_l2 ( m, x )
print ' %-20s %4d %4d %14g %14g %10.2g' \
% ( title, m, n, norm_a_frobenius, norm_x_l2, error_l2 )
#
# DIF2CYCLIC
#
title = 'DIF2CYCLIC'
m = 5
n = 5
a = dif2cyclic ( n )
x = dif2cyclic_null_left ( m, n )
error_l2 = r8mat_is_null_left ( m, n, a, x )
norm_a_frobenius = r8mat_norm_fro ( m, n, a )
norm_x_l2 = r8vec_norm_l2 ( m, x )
print ' %-20s %4d %4d %14g %14g %10.2g' \
% ( title, m, n, norm_a_frobenius, norm_x_l2, error_l2 )
#
# EBERLEIN
#
title = 'EBERLEIN'
m = 5
n = 5
r8_lo = -5.0
r8_hi = +5.0
seed = 123456789
alpha, seed = r8_uniform_ab ( r8_lo, r8_hi, seed )
a = eberlein ( alpha, n )
x = eberlein_null_left ( m, n )
error_l2 = r8mat_is_null_left ( m, n, a, x )
norm_a_frobenius = r8mat_norm_fro ( m, n, a )
norm_x_l2 = r8vec_norm_l2 ( m, x )
print ' %-20s %4d %4d %14g %14g %10.2g' \
% ( title, m, n, norm_a_frobenius, norm_x_l2, error_l2 )
#
# FIBONACCI1
#
title = 'FIBONACCI1'
m = 5
n = m
r8_lo = -5.0
r8_hi = +5.0
seed = 123456789
f1, seed = r8_uniform_ab ( r8_lo, r8_hi, seed )
f2, seed = r8_uniform_ab ( r8_lo, r8_hi, seed )
a = fibonacci1 ( n, f1, f2 )
x = fibonacci1_null_left ( m, n, f1, f2 )
error_l2 = r8mat_is_null_left ( m, n, a, x )
norm_a_frobenius = r8mat_norm_fro ( m, n, a )
norm_x_l2 = r8vec_norm_l2 ( m, x )
print ' %-20s %4d %4d %14g %14g %10.2g' \
% ( title, m, n, norm_a_frobenius, norm_x_l2, error_l2 )
#
# LAUCHLI
#
title = 'LAUCHLI'
m = 6
n = m - 1
r8_lo = -5.0
r8_hi = +5.0
seed = 123456789
alpha, seed = r8_uniform_ab ( r8_lo, r8_hi, seed )
a = lauchli ( alpha, m, n )
x = lauchli_null_left ( alpha, m, n )
error_l2 = r8mat_is_null_left ( m, n, a, x )
norm_a_frobenius = r8mat_norm_fro ( m, n, a )
norm_x_l2 = r8vec_norm_l2 ( m, x )
print ' %-20s %4d %4d %14g %14g %10.2g' \
% ( title, m, n, norm_a_frobenius, norm_x_l2, error_l2 )
#
# LINE_ADJ
#
title = 'LINE_ADJ'
m = 7
n = m
a = line_adj ( n )
x = line_adj_null_left ( m, n )
error_l2 = r8mat_is_null_left ( m, n, a, x )
norm_a_frobenius = r8mat_norm_fro ( m, n, a )
norm_x_l2 = r8vec_norm_l2 ( m, x )
print ' %-20s %4d %4d %14g %14g %10.2g' \
% ( title, m, n, norm_a_frobenius, norm_x_l2, error_l2 )
#
# MOLER2
#
title = 'MOLER2'
m = 5
n = m
a = moler2 ( )
x = moler2_null_left ( )
error_l2 = r8mat_is_null_left ( m, n, a, x )
norm_a_frobenius = r8mat_norm_fro ( m, n, a )
norm_x_l2 = r8vec_norm_l2 ( m, x )
print ' %-20s %4d %4d %14g %14g %10.2g' \
% ( title, m, n, norm_a_frobenius, norm_x_l2, error_l2 )
#
# ONE
#
title = 'ONE'
m = 5
n = 5
a = one ( m, n )
x = one_null_left ( m, n )
error_l2 = r8mat_is_null_left ( m, n, a, x )
norm_a_frobenius = r8mat_norm_fro ( m, n, a )
norm_x_l2 = r8vec_norm_l2 ( m, x )
print ' %-20s %4d %4d %14g %14g %10.2g' \
% ( title, m, n, norm_a_frobenius, norm_x_l2, error_l2 )
#
# RING_ADJ
# N must be a multiple of 4 for there to be a null vector.
#
title = 'RING_ADJ'
m = 12
n = 12
a = ring_adj ( m, n )
x = ring_adj_null_left ( m, n )
error_l2 = r8mat_is_null_left ( m, n, a, x )
norm_a_frobenius = r8mat_norm_fro ( m, n, a )
norm_x_l2 = r8vec_norm_l2 ( m, x )
print ' %-20s %4d %4d %14g %14g %10.2g' \
% ( title, m, n, norm_a_frobenius, norm_x_l2, error_l2 )
#
# ROSSER1
#
title = 'ROSSER1'
m = 8
n = 8
a = rosser1 ( )
x = rosser1_null_left ( )
error_l2 = r8mat_is_null_left ( m, n, a, x )
norm_a_frobenius = r8mat_norm_fro ( m, n, a )
norm_x_l2 = r8vec_norm_l2 ( m, x )
print ' %-20s %4d %4d %14g %14g %10.2g' \
% ( title, m, n, norm_a_frobenius, norm_x_l2, error_l2 )
#
# ZERO
#
title = 'ZERO'
m = 5
n = 5
a = zero ( m, n )
x = zero_null_left ( m, n )
error_l2 = r8mat_is_null_left ( m, n, a, x )
norm_a_frobenius = r8mat_norm_fro ( m, n, a )
norm_x_l2 = r8vec_norm_l2 ( m, x )
print ' %-20s %4d %4d %14g %14g %10.2g' \
% ( title, m, n, norm_a_frobenius, norm_x_l2, error_l2 )
#
# Terminate.
#
print ''
print 'TEST_NULL_LEFT:'
print ' Normal end of execution.'
return
