def bernstein_matrix_inverse_test():
#*****************************************************************************80
#
## BERNSTEIN_MATRIX_INVERSE_TEST tests BERNSTEIN_MATRIX_INVERSE.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 03 December 2015
#
# Author:
#
# John Burkardt
#
import numpy as np
import platform
from bernstein_matrix import bernstein_matrix
from r8mat_is_identity import r8mat_is_identity
from r8mat_norm_fro import r8mat_norm_fro
print('')
print('BERNSTEIN_MATRIX_INVERSE_TEST')
print(' Python version: %s' % (platform.python_version()))
print(' BERNSTEIN_MATRIX returns a matrix A which transforms a')
print(' polynomial coefficient vector from the power basis to')
print(' the Bernstein basis.')
print(' BERNSTEIN_MATRIX_INVERSE computes the inverse B.')
print('')
print(' N ||A|| ||B|| ||I-A*B||')
print('')
for n in range(5, 16):
a = bernstein_matrix(n)
a_norm_frobenius = r8mat_norm_fro(n, n, a)
b = bernstein_matrix_inverse(n)
b_norm_frobenius = r8mat_norm_fro(n, n, b)
c = np.dot(a, b)
error_norm_frobenius = r8mat_is_identity(n, c)
print ( ' %4d %14g %14g %14g' % \
( n, a_norm_frobenius, b_norm_frobenius, error_norm_frobenius ) )
#
# Terminate.
#
print('')
print('BERNSTEIN_MATRIX_INVERSE TEST')
print(' Normal end of execution.')
return
def r8mat_is_inverse ( n, a, b ):
#*****************************************************************************80
#
## R8MAT_IS_INVERSE measures the error in a matrix inverse.
#
# Discussion:
#
# An R8MAT is a matrix of real values.
#
# This routine returns the sum of the Frobenius norms of
# A * B - I and B * A - I.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 23 March 2015
#
# Author:
#
# John Burkardt
#
# Parameters:
#
# Input, integer N, the order of the matrix.
#
# Input, real A(N,N), the matrix.
#
# Input, real B(M,N), the inverse matrix.
#
# Output, real VALUE, the Frobenius norm
# of the difference matrices A * B - I and B * A - I.
#
from identity import identity
from r8mat_mm import r8mat_mm
from r8mat_sub import r8mat_sub
from r8mat_norm_fro import r8mat_norm_fro
ab = r8mat_mm ( n, n, n, a, b )
ba = r8mat_mm ( n, n, n, b, a )
id = identity ( n, n )
d1 = r8mat_sub ( n, n, ab, id )
d2 = r8mat_sub ( n, n, ba, id )
value = r8mat_norm_fro ( n, n, d1 ) \
+ r8mat_norm_fro ( n, n, d2 )
return value
def r8mat_is_solution ( m, n, k, a, x, b ): #*****************************************************************************80 # ## R8MAT_IS_SOLUTION measures the error in a linear system solution. # # Discussion: # # An R8MAT is a matrix of real values. # # The system matrix A is an M x N matrix. # It is not required that A be invertible. # # The solution vector X is actually allowed to be an N x K matrix. # # The right hand side "vector" B is actually allowed to be an M x K matrix. # # This routine simply returns the Frobenius norm of the M x K matrix: # A * X - B. # # Licensing: # # This code is distributed under the GNU LGPL license. # # Modified: # # 02 March 2015 # # Author: # # John Burkardt # # Parameters: # # Input, integer M, N, K, the order of the matrices. # # Input, real A(M,N), X(N,K), B(M,K), the matrices. # # Output, real VALUE, the Frobenius norm # of the difference matrix A * X - B, which would be exactly zero # if X was the "solution" of the linear system. # from r8mat_norm_fro import r8mat_norm_fro from r8mat_mm import r8mat_mm from r8mat_sub import r8mat_sub # # AX = A*X # ax = r8mat_mm ( m, n, k, a, x ) # # AXMB = AX-B. # axmb = r8mat_sub ( m, k, ax, b ) # # Value = ||AX-B|\ # value = r8mat_norm_fro ( m, k, axmb ) return value
def r8mat_is_solution(m, n, k, a, x, b):
#*****************************************************************************80
#
## R8MAT_IS_SOLUTION measures the error in a linear system solution.
#
# Discussion:
#
# An R8MAT is a matrix of real values.
#
# The system matrix A is an M x N matrix.
# It is not required that A be invertible.
#
# The solution vector X is actually allowed to be an N x K matrix.
#
# The right hand side "vector" B is actually allowed to be an M x K matrix.
#
# This routine simply returns the Frobenius norm of the M x K matrix:
# A * X - B.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 02 March 2015
#
# Author:
#
# John Burkardt
#
# Parameters:
#
# Input, integer M, N, K, the order of the matrices.
#
# Input, real A(M,N), X(N,K), B(M,K), the matrices.
#
# Output, real VALUE, the Frobenius norm
# of the difference matrix A * X - B, which would be exactly zero
# if X was the "solution" of the linear system.
#
from r8mat_norm_fro import r8mat_norm_fro
from r8mat_mm import r8mat_mm
from r8mat_sub import r8mat_sub
#
# AX = A*X
#
ax = r8mat_mm(m, n, k, a, x)
#
# AXMB = AX-B.
#
axmb = r8mat_sub(m, k, ax, b)
#
# Value = ||AX-B|\
#
value = r8mat_norm_fro(m, k, axmb)
return value
def bernstein_matrix_determinant_test():
#*****************************************************************************80
#
## BERNSTEIN_MATRIX_DETERMINANT_TEST tests BERNSTEIN_MATRIX_DETERMINANT.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 03 December 2015
#
# Author:
#
# John Burkardt
#
import numpy as np
import platform
from bernstein_matrix import bernstein_matrix
from r8mat_is_identity import r8mat_is_identity
from r8mat_norm_fro import r8mat_norm_fro
print('')
print('BERNSTEIN_MATRIX_DETERMINANT_TEST')
print(' Python version: %s' % (platform.python_version()))
print(' BERNSTEIN_MATRIX_DETERMINANT computes the determinant of')
print(' the Bernstein matrix.')
print('')
print(' N ||A|| det(A) np.linalg.det(A)')
print('')
for n in range(5, 16):
a = bernstein_matrix(n)
a_norm_frobenius = r8mat_norm_fro(n, n, a)
d1 = bernstein_matrix_determinant(n)
d2 = np.linalg.det(a)
print ( ' %4d %14g %14g %14g' % \
( n, a_norm_frobenius, d1, d2 ) )
#
# Terminate.
#
print('')
print('BERNSTEIN_MATRIX_DETERMINANT TEST')
print(' Normal end of execution.')
return
def r8mat_is_plu ( m, n, a, p, l, u ): #*****************************************************************************80 # ## R8MAT_IS_PLU measures the error in a PLU factorization. # # Discussion: # # An R8MAT is a matrix of real values. # # This routine computes the Frobenius norm of # # A - P * L * U # # Licensing: # # This code is distributed under the GNU LGPL license. # # Modified: # # 23 March 2015 # # Author: # # John Burkardt # # Parameters: # # Input, integer M, N, the order of the matrix. # # Input, real A(M,N), the matrix. # # Input, real P(M,M), L(M,M), U(M,N), the PLU factors. # # Output, real VALUE, the Frobenius norm # of the difference matrix A - P * L * U. # from r8mat_mm import r8mat_mm from r8mat_sub import r8mat_sub from r8mat_norm_fro import r8mat_norm_fro lu = r8mat_mm ( m, m, n, l, u ) plu = r8mat_mm ( m, m, n, p, lu ) d = r8mat_sub ( m, n, a, plu ) value = r8mat_norm_fro ( m, n, d ) return value
def test_plu():
#*****************************************************************************80
#
## TEST_PLU tests the PLU factors.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 07 April 2014
#
# Author:
#
# John Burkardt
#
import numpy as np
from bodewig import bodewig
from bodewig import bodewig_plu
from borderband import borderband
from borderband import borderband_plu
from dif2 import dif2
from dif2 import dif2_plu
from gfpp import gfpp
from gfpp import gfpp_plu
from givens import givens
from givens import givens_plu
from i4_uniform_ab import i4_uniform_ab
from kms import kms
from kms import kms_plu
from lehmer import lehmer
from lehmer import lehmer_plu
from maxij import maxij
from maxij import maxij_plu
from minij import minij
from minij import minij_plu
from moler1 import moler1
from moler1 import moler1_plu
from moler3 import moler3
from moler3 import moler3_plu
from oto import oto
from oto import oto_plu
from pascal2 import pascal2
from pascal2 import pascal2_plu
from plu import plu
from plu import plu_plu
from r8_uniform_ab import r8_uniform_ab
from r8mat_is_plu import r8mat_is_plu
from r8mat_norm_fro import r8mat_norm_fro
from r8vec_uniform_ab import r8vec_uniform_ab
from vand2 import vand2
from vand2 import vand2_plu
from wilson import wilson
from wilson import wilson_plu
print ''
print 'TEST_PLU'
print ' A = a test matrix of order M by N'
print ' P, L, U are the PLU factors.'
print ''
print ' ||A|| = Frobenius norm of A.'
print ' ||A-PLU|| = Frobenius norm of A-P*L*U.'
print ''
print ' Title M N ',
print '||A|| ||A-PLU||'
print ''
#
# BODEWIG
#
title = 'BODEWIG'
m = 4
n = 4
a = bodewig()
p, l, u = bodewig_plu()
error_frobenius = r8mat_is_plu(m, n, a, p, l, u)
norm_a_frobenius = r8mat_norm_fro(m, n, a)
print ' %-20s %4d %4d %14g %14g' \
% ( title, m, n, norm_a_frobenius, error_frobenius )
#
# BORDERBAND
#
title = 'BORDERBAND'
m = 5
n = 5
a = borderband(n)
p, l, u = borderband_plu(n)
error_frobenius = r8mat_is_plu(m, n, a, p, l, u)
norm_a_frobenius = r8mat_norm_fro(m, n, a)
print ' %-20s %4d %4d %14g %14g' \
% ( title, m, n, norm_a_frobenius, error_frobenius )
#
# DIF2
#
title = 'DIF2'
m = 5
n = 5
a = dif2(m, n)
p, l, u = dif2_plu(n)
error_frobenius = r8mat_is_plu(m, n, a, p, l, u)
norm_a_frobenius = r8mat_norm_fro(m, n, a)
print ' %-20s %4d %4d %14g %14g' \
% ( title, m, n, norm_a_frobenius, error_frobenius )
#
# GFPP
#
title = 'GFPP'
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 = gfpp(n, alpha)
p, l, u = gfpp_plu(n, alpha)
error_frobenius = r8mat_is_plu(m, n, a, p, l, u)
norm_a_frobenius = r8mat_norm_fro(m, n, a)
print ' %-20s %4d %4d %14g %14g' \
% ( title, m, n, norm_a_frobenius, error_frobenius )
#
# GIVENS
#
title = 'GIVENS'
m = 5
n = 5
a = givens(m, n)
p, l, u = givens_plu(n)
error_frobenius = r8mat_is_plu(m, n, a, p, l, u)
norm_a_frobenius = r8mat_norm_fro(m, n, a)
print ' %-20s %4d %4d %14g %14g' \
% ( title, m, n, norm_a_frobenius, error_frobenius )
#
# KMS
#
title = 'KMS'
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 = kms(alpha, m, n)
p, l, u = kms_plu(alpha, n)
error_frobenius = r8mat_is_plu(m, n, a, p, l, u)
norm_a_frobenius = r8mat_norm_fro(m, n, a)
print ' %-20s %4d %4d %14g %14g' \
% ( title, m, n, norm_a_frobenius, error_frobenius )
#
# LEHMER
#
title = 'LEHMER'
m = 5
n = 5
a = lehmer(m, n)
p, l, u = lehmer_plu(n)
error_frobenius = r8mat_is_plu(m, n, a, p, l, u)
norm_a_frobenius = r8mat_norm_fro(m, n, a)
print ' %-20s %4d %4d %14g %14g' \
% ( title, m, n, norm_a_frobenius, error_frobenius )
#
# MAXIJ
#
title = 'MAXIJ'
m = 5
n = 5
a = maxij(m, n)
p, l, u = maxij_plu(n)
error_frobenius = r8mat_is_plu(m, n, a, p, l, u)
norm_a_frobenius = r8mat_norm_fro(m, n, a)
print ' %-20s %4d %4d %14g %14g' \
% ( title, m, n, norm_a_frobenius, error_frobenius )
#
# MINIJ
#
title = 'MINIJ'
m = 5
n = 5
a = minij(n, n)
p, l, u = minij_plu(n)
error_frobenius = r8mat_is_plu(m, n, a, p, l, u)
norm_a_frobenius = r8mat_norm_fro(m, n, a)
print ' %-20s %4d %4d %14g %14g' \
% ( title, m, n, norm_a_frobenius, error_frobenius )
#
# MOLER1
#
title = 'MOLER1'
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 = moler1(alpha, n, n)
p, l, u = moler1_plu(alpha, n)
error_frobenius = r8mat_is_plu(m, n, a, p, l, u)
norm_a_frobenius = r8mat_norm_fro(m, n, a)
print ' %-20s %4d %4d %14g %14g' \
% ( title, m, n, norm_a_frobenius, error_frobenius )
#
# MOLER3
#
title = 'MOLER3'
m = 5
n = 5
a = moler3(m, n)
p, l, u = moler3_plu(n)
error_frobenius = r8mat_is_plu(m, n, a, p, l, u)
norm_a_frobenius = r8mat_norm_fro(m, n, a)
print ' %-20s %4d %4d %14g %14g' \
% ( title, m, n, norm_a_frobenius, error_frobenius )
#
# OTO
#
title = 'OTO'
m = 5
n = 5
a = oto(m, n)
p, l, u = oto_plu(n)
error_frobenius = r8mat_is_plu(m, n, a, p, l, u)
norm_a_frobenius = r8mat_norm_fro(m, n, a)
print ' %-20s %4d %4d %14g %14g' \
% ( title, m, n, norm_a_frobenius, error_frobenius )
#
# PASCAL2
#
title = 'PASCAL2'
m = 5
n = 5
a = pascal2(n)
p, l, u = pascal2_plu(n)
error_frobenius = r8mat_is_plu(m, n, a, p, l, u)
norm_a_frobenius = r8mat_norm_fro(m, n, a)
print ' %-20s %4d %4d %14g %14g' \
% ( title, m, n, norm_a_frobenius, error_frobenius )
#
# PLU
#
title = 'PLU'
m = 5
n = 5
pivot = np.zeros(n)
seed = 123456789
for i in range(0, n):
i4_lo = i
i4_hi = n - 1
pivot[i], seed = i4_uniform_ab(i4_lo, i4_hi, seed)
a = plu(n, pivot)
p, l, u = plu_plu(n, pivot)
error_frobenius = r8mat_is_plu(m, n, a, p, l, u)
norm_a_frobenius = r8mat_norm_fro(m, n, a)
print ' %-20s %4d %4d %14g %14g' \
% ( title, m, n, norm_a_frobenius, error_frobenius )
#
# VAND2
#
title = 'VAND2'
m = 4
n = 4
r8_lo = -5.0
r8_hi = +5.0
seed = 123456789
x, seed = r8vec_uniform_ab(m, r8_lo, r8_hi, seed)
a = vand2(m, x)
p, l, u = vand2_plu(m, x)
error_frobenius = r8mat_is_plu(m, n, a, p, l, u)
norm_a_frobenius = r8mat_norm_fro(m, n, a)
print ' %-20s %4d %4d %14g %14g' \
% ( title, m, n, norm_a_frobenius, error_frobenius )
#
# WILSON
#
title = 'WILSON'
m = 4
n = 4
a = wilson()
p, l, u = wilson_plu()
error_frobenius = r8mat_is_plu(m, n, a, p, l, u)
norm_a_frobenius = r8mat_norm_fro(m, n, a)
print ' %-20s %4d %4d %14g %14g' \
% ( title, m, n, norm_a_frobenius, error_frobenius )
#
# Terminate.
#
print ''
print 'TEST_PLU:'
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 test_llt ( ):
#*****************************************************************************80
#
#% TEST_LLT tests LLT factors.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 09 April 2015
#
# Author:
#
# John Burkardt
#
from dif2 import dif2
from dif2 import dif2_llt
from givens import givens
from givens import givens_llt
from kershaw import kershaw
from kershaw import kershaw_llt
from lehmer import lehmer
from lehmer import lehmer_llt
from minij import minij
from minij import minij_llt
from moler1 import moler1
from moler1 import moler1_llt
from moler3 import moler3
from moler3 import moler3_llt
from oto import oto
from oto import oto_llt
from pascal2 import pascal2
from pascal2 import pascal2_llt
from wilson import wilson
from wilson import wilson_llt
from r8_uniform_ab import r8_uniform_ab
from r8mat_is_llt import r8mat_is_llt
from r8mat_norm_fro import r8mat_norm_fro
print ''
print 'TEST_LLT'
print ' A = a test matrix of order M by M'
print ' L is an M by N lower triangular Cholesky factor.'
print ''
print ' ||A|| = Frobenius norm of A.'
print ' ||A-LLT|| = Frobenius norm of A-L*L\'.'
print ''
print ' Title M N ',
print '||A|| ||A-LLT||'
print ''
#
# DIF2
#
title = 'DIF2'
m = 5
n = 5
a = dif2 ( m, n )
l = dif2_llt ( n )
error_frobenius = r8mat_is_llt ( m, n, a, l )
norm_a_frobenius = r8mat_norm_fro ( m, n, a )
print ' %-20s %4d %4d %14g %14g' \
% ( title, m, n, norm_a_frobenius, error_frobenius )
#
# GIVENS
#
title = 'GIVENS'
m = 5
n = 5
a = givens ( m, n )
l = givens_llt ( n )
error_frobenius = r8mat_is_llt ( m, n, a, l )
norm_a_frobenius = r8mat_norm_fro ( m, n, a )
print ' %-20s %4d %4d %14g %14g' \
% ( title, m, n, norm_a_frobenius, error_frobenius )
#
# KERSHAW
#
title = 'KERSHAW'
m = 4
n = 4
a = kershaw ( )
l = kershaw_llt ( )
error_frobenius = r8mat_is_llt ( m, n, a, l )
norm_a_frobenius = r8mat_norm_fro ( m, n, a )
print ' %-20s %4d %4d %14g %14g' \
% ( title, m, n, norm_a_frobenius, error_frobenius )
#
# LEHMER
#
title = 'LEHMER'
m = 5
n = 5
a = lehmer ( n, n )
l = lehmer_llt ( n )
error_frobenius = r8mat_is_llt ( m, n, a, l )
norm_a_frobenius = r8mat_norm_fro ( m, n, a )
print ' %-20s %4d %4d %14g %14g' \
% ( title, m, n, norm_a_frobenius, error_frobenius )
#
# MINIJ
#
title = 'MINIJ'
m = 5
n = 5
a = minij ( n, n )
l = minij_llt ( n )
error_frobenius = r8mat_is_llt ( m, n, a, l )
norm_a_frobenius = r8mat_norm_fro ( m, n, a )
print ' %-20s %4d %4d %14g %14g' \
% ( title, m, n, norm_a_frobenius, error_frobenius )
#
# MOLER1
#
title = 'MOLER1'
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 = moler1 ( alpha, m, n )
l = moler1_llt ( alpha, n )
error_frobenius = r8mat_is_llt ( m, n, a, l )
norm_a_frobenius = r8mat_norm_fro ( m, n, a )
print ' %-20s %4d %4d %14g %14g' \
% ( title, m, n, norm_a_frobenius, error_frobenius )
#
# MOLER3
#
title = 'MOLER3'
m = 5
n = 5
a = moler3 ( m, n )
l = moler3_llt ( n )
error_frobenius = r8mat_is_llt ( m, n, a, l )
norm_a_frobenius = r8mat_norm_fro ( m, n, a )
print ' %-20s %4d %4d %14g %14g' \
% ( title, m, n, norm_a_frobenius, error_frobenius )
#
# OTO
#
title = 'OTO'
m = 5
n = 5
a = oto ( m, n )
l = oto_llt ( n )
error_frobenius = r8mat_is_llt ( m, n, a, l )
norm_a_frobenius = r8mat_norm_fro ( m, n, a )
print ' %-20s %4d %4d %14g %14g' \
% ( title, m, n, norm_a_frobenius, error_frobenius )
#
# PASCAL2
#
title = 'PASCAL2'
m = 5
n = 5
a = pascal2 ( n )
l = pascal2_llt ( n )
error_frobenius = r8mat_is_llt ( m, n, a, l )
norm_a_frobenius = r8mat_norm_fro ( m, n, a )
print ' %-20s %4d %4d %14g %14g' \
% ( title, m, n, norm_a_frobenius, error_frobenius )
#
# WILSON
#
title = 'WILSON'
m = 4
n = 4
a = wilson ( )
l = wilson_llt ( )
error_frobenius = r8mat_is_llt ( m, n, a, l )
norm_a_frobenius = r8mat_norm_fro ( m, n, a )
print ' %-20s %4d %4d %14g %14g' \
% ( title, m, n, norm_a_frobenius, error_frobenius )
#
# Terminate.
#
print ''
print 'TEST_LLT:'
print ' Normal end of execution.'
return
def r8mat_is_eigen_left(n, k, a, x, lam):
#*****************************************************************************80
#
## R8MAT_IS_EIGEN_LEFT determines the error in a left eigensystem.
#
# Discussion:
#
# An R8MAT is a matrix of real values.
#
# This routine computes the Frobenius norm of
#
# X * A - LAMBDA * X
#
# where
#
# A is an N by N matrix,
# X is an K by N matrix (each of K columns is an eigenvector)
# LAMBDA is a K by K diagonal matrix of eigenvalues.
#
# This routine assumes that A, X and LAMBDA are all real.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 17 March 2015
#
# Author:
#
# John Burkardt
#
# Parameters:
#
# Input, integer N, the order of the matrix.
#
# Input, integer K, the number of eigenvectors.
# K is usually 1 or N.
#
# Input, real A(N,N), the matrix.
#
# Input, real X(K,N), the K eigenvectors.
#
# Input, real LAM(K), the K eigenvalues.
#
# Output, real VALUE, the Frobenius norm of X * A - LAM * X.
#
from r8mat_mm import r8mat_mm
from r8mat_norm_fro import r8mat_norm_fro
c = r8mat_mm(k, n, n, x, a)
for i in range(0, k):
for j in range(0, n):
c[i, j] = c[i, j] - lam[i] * x[i, j]
value = r8mat_norm_fro(k, n, c)
return value
def test_plu ( ):
#*****************************************************************************80
#
## TEST_PLU tests the PLU factors.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 07 April 2014
#
# Author:
#
# John Burkardt
#
import numpy as np
from bodewig import bodewig
from bodewig import bodewig_plu
from borderband import borderband
from borderband import borderband_plu
from dif2 import dif2
from dif2 import dif2_plu
from gfpp import gfpp
from gfpp import gfpp_plu
from givens import givens
from givens import givens_plu
from i4_uniform_ab import i4_uniform_ab
from kms import kms
from kms import kms_plu
from lehmer import lehmer
from lehmer import lehmer_plu
from maxij import maxij
from maxij import maxij_plu
from minij import minij
from minij import minij_plu
from moler1 import moler1
from moler1 import moler1_plu
from moler3 import moler3
from moler3 import moler3_plu
from oto import oto
from oto import oto_plu
from pascal2 import pascal2
from pascal2 import pascal2_plu
from plu import plu
from plu import plu_plu
from r8_uniform_ab import r8_uniform_ab
from r8mat_is_plu import r8mat_is_plu
from r8mat_norm_fro import r8mat_norm_fro
from r8vec_uniform_ab import r8vec_uniform_ab
from vand2 import vand2
from vand2 import vand2_plu
from wilson import wilson
from wilson import wilson_plu
print ''
print 'TEST_PLU'
print ' A = a test matrix of order M by N'
print ' P, L, U are the PLU factors.'
print ''
print ' ||A|| = Frobenius norm of A.'
print ' ||A-PLU|| = Frobenius norm of A-P*L*U.'
print ''
print ' Title M N ',
print '||A|| ||A-PLU||'
print ''
#
# BODEWIG
#
title = 'BODEWIG'
m = 4
n = 4
a = bodewig ( )
p, l, u = bodewig_plu ( )
error_frobenius = r8mat_is_plu ( m, n, a, p, l, u )
norm_a_frobenius = r8mat_norm_fro ( m, n, a )
print ' %-20s %4d %4d %14g %14g' \
% ( title, m, n, norm_a_frobenius, error_frobenius )
#
# BORDERBAND
#
title = 'BORDERBAND'
m = 5
n = 5
a = borderband ( n )
p, l, u = borderband_plu ( n )
error_frobenius = r8mat_is_plu ( m, n, a, p, l, u )
norm_a_frobenius = r8mat_norm_fro ( m, n, a )
print ' %-20s %4d %4d %14g %14g' \
% ( title, m, n, norm_a_frobenius, error_frobenius )
#
# DIF2
#
title = 'DIF2'
m = 5
n = 5
a = dif2 ( m, n )
p, l, u = dif2_plu ( n )
error_frobenius = r8mat_is_plu ( m, n, a, p, l, u )
norm_a_frobenius = r8mat_norm_fro ( m, n, a )
print ' %-20s %4d %4d %14g %14g' \
% ( title, m, n, norm_a_frobenius, error_frobenius )
#
# GFPP
#
title = 'GFPP'
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 = gfpp ( n, alpha )
p, l, u = gfpp_plu ( n, alpha )
error_frobenius = r8mat_is_plu ( m, n, a, p, l, u )
norm_a_frobenius = r8mat_norm_fro ( m, n, a )
print ' %-20s %4d %4d %14g %14g' \
% ( title, m, n, norm_a_frobenius, error_frobenius )
#
# GIVENS
#
title = 'GIVENS'
m = 5
n = 5
a = givens ( m, n )
p, l, u = givens_plu ( n )
error_frobenius = r8mat_is_plu ( m, n, a, p, l, u )
norm_a_frobenius = r8mat_norm_fro ( m, n, a )
print ' %-20s %4d %4d %14g %14g' \
% ( title, m, n, norm_a_frobenius, error_frobenius )
#
# KMS
#
title = 'KMS'
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 = kms ( alpha, m, n )
p, l, u = kms_plu ( alpha, n )
error_frobenius = r8mat_is_plu ( m, n, a, p, l, u )
norm_a_frobenius = r8mat_norm_fro ( m, n, a )
print ' %-20s %4d %4d %14g %14g' \
% ( title, m, n, norm_a_frobenius, error_frobenius )
#
# LEHMER
#
title = 'LEHMER'
m = 5
n = 5
a = lehmer ( m, n )
p, l, u = lehmer_plu ( n )
error_frobenius = r8mat_is_plu ( m, n, a, p, l, u )
norm_a_frobenius = r8mat_norm_fro ( m, n, a )
print ' %-20s %4d %4d %14g %14g' \
% ( title, m, n, norm_a_frobenius, error_frobenius )
#
# MAXIJ
#
title = 'MAXIJ'
m = 5
n = 5
a = maxij ( m, n )
p, l, u = maxij_plu ( n )
error_frobenius = r8mat_is_plu ( m, n, a, p, l, u )
norm_a_frobenius = r8mat_norm_fro ( m, n, a )
print ' %-20s %4d %4d %14g %14g' \
% ( title, m, n, norm_a_frobenius, error_frobenius )
#
# MINIJ
#
title = 'MINIJ'
m = 5
n = 5
a = minij ( n, n )
p, l, u = minij_plu ( n )
error_frobenius = r8mat_is_plu ( m, n, a, p, l, u )
norm_a_frobenius = r8mat_norm_fro ( m, n, a )
print ' %-20s %4d %4d %14g %14g' \
% ( title, m, n, norm_a_frobenius, error_frobenius )
#
# MOLER1
#
title = 'MOLER1'
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 = moler1 ( alpha, n, n )
p, l, u = moler1_plu ( alpha, n )
error_frobenius = r8mat_is_plu ( m, n, a, p, l, u )
norm_a_frobenius = r8mat_norm_fro ( m, n, a )
print ' %-20s %4d %4d %14g %14g' \
% ( title, m, n, norm_a_frobenius, error_frobenius )
#
# MOLER3
#
title = 'MOLER3'
m = 5
n = 5
a = moler3 ( m, n )
p, l, u = moler3_plu ( n )
error_frobenius = r8mat_is_plu ( m, n, a, p, l, u )
norm_a_frobenius = r8mat_norm_fro ( m, n, a )
print ' %-20s %4d %4d %14g %14g' \
% ( title, m, n, norm_a_frobenius, error_frobenius )
#
# OTO
#
title = 'OTO'
m = 5
n = 5
a = oto ( m, n )
p, l, u = oto_plu ( n )
error_frobenius = r8mat_is_plu ( m, n, a, p, l, u )
norm_a_frobenius = r8mat_norm_fro ( m, n, a )
print ' %-20s %4d %4d %14g %14g' \
% ( title, m, n, norm_a_frobenius, error_frobenius )
#
# PASCAL2
#
title = 'PASCAL2'
m = 5
n = 5
a = pascal2 ( n )
p, l, u = pascal2_plu ( n )
error_frobenius = r8mat_is_plu ( m, n, a, p, l, u )
norm_a_frobenius = r8mat_norm_fro ( m, n, a )
print ' %-20s %4d %4d %14g %14g' \
% ( title, m, n, norm_a_frobenius, error_frobenius )
#
# PLU
#
title = 'PLU'
m = 5
n = 5
pivot = np.zeros ( n )
seed = 123456789
for i in range ( 0, n ):
i4_lo = i
i4_hi = n - 1
pivot[i], seed = i4_uniform_ab ( i4_lo, i4_hi, seed )
a = plu ( n, pivot )
p, l, u = plu_plu ( n, pivot )
error_frobenius = r8mat_is_plu ( m, n, a, p, l, u )
norm_a_frobenius = r8mat_norm_fro ( m, n, a )
print ' %-20s %4d %4d %14g %14g' \
% ( title, m, n, norm_a_frobenius, error_frobenius )
#
# VAND2
#
title = 'VAND2'
m = 4
n = 4
r8_lo = -5.0
r8_hi = +5.0
seed = 123456789
x, seed = r8vec_uniform_ab ( m, r8_lo, r8_hi, seed )
a = vand2 ( m, x )
p, l, u = vand2_plu ( m, x )
error_frobenius = r8mat_is_plu ( m, n, a, p, l, u )
norm_a_frobenius = r8mat_norm_fro ( m, n, a )
print ' %-20s %4d %4d %14g %14g' \
% ( title, m, n, norm_a_frobenius, error_frobenius )
#
# WILSON
#
title = 'WILSON'
m = 4
n = 4
a = wilson ( )
p, l, u = wilson_plu ( )
error_frobenius = r8mat_is_plu ( m, n, a, p, l, u )
norm_a_frobenius = r8mat_norm_fro ( m, n, a )
print ' %-20s %4d %4d %14g %14g' \
% ( title, m, n, norm_a_frobenius, error_frobenius )
#
# Terminate.
#
print ''
print 'TEST_PLU:'
print ' Normal end of execution.'
return
def test_solution ( ):
#*****************************************************************************80
#
## TEST_SOLUTION tests the linear solution computations.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 10 March 2015
#
# Author:
#
# John Burkardt
#
import numpy as np
from a123 import a123
from a123 import a123_rhs
from a123 import a123_solution
from bodewig import bodewig
from bodewig import bodewig_rhs
from bodewig import bodewig_solution
from dif2 import dif2
from dif2 import dif2_rhs
from dif2 import dif2_solution
from frank import frank
from frank import frank_rhs
from frank import frank_solution
from poisson import poisson
from poisson import poisson_rhs
from poisson import poisson_solution
from r8mat_is_solution import r8mat_is_solution
from r8mat_norm_fro import r8mat_norm_fro
from wilk03 import wilk03
from wilk03 import wilk03_rhs
from wilk03 import wilk03_solution
from wilk04 import wilk04
from wilk04 import wilk04_rhs
from wilk04 import wilk04_solution
from wilson import wilson
from wilson import wilson_rhs
from wilson import wilson_solution
print ''
print 'TEST_SOLUTION'
print ' Compute the Frobenius norm of the solution error:'
print ' A * X - B'
print ' given MxN matrix A, NxK solution X, MxK right hand side B.'
print ''
print ' Title M N K ||A||',
print ' ||A*X-B||'
print ''
#
# A123 matrix.
#
title = 'A123'
m = 3
n = 3
k = 1
a = a123 ( )
b = a123_rhs ( )
x = a123_solution ( )
error_frobenius = r8mat_is_solution ( m, n, k, a, x, b )
norm_frobenius = r8mat_norm_fro ( n, n, a )
print ' %-20s %4d %4d %4d %14g %14g' \
% ( title, m, n, k, norm_frobenius, error_frobenius )
#
# BODEWIG matrix.
#
title = 'BODEWIG'
m = 4
n = 4
k = 1
a = bodewig ( )
b = bodewig_rhs ( )
x = bodewig_solution ( )
error_frobenius = r8mat_is_solution ( m, n, k, a, x, b )
norm_frobenius = r8mat_norm_fro ( n, n, a )
print ' %-20s %4d %4d %4d %14g %14g' \
% ( title, m, n, k, norm_frobenius, error_frobenius )
#
# DIF2 matrix.
#
title = 'DIF2'
m = 10
n = 10
k = 2
a = dif2 ( m, n )
b = dif2_rhs ( m, k )
x = dif2_solution ( n, k )
error_frobenius = r8mat_is_solution ( m, n, k, a, x, b )
norm_frobenius = r8mat_norm_fro ( n, n, a )
print ' %-20s %4d %4d %4d %14g %14g' \
% ( title, m, n, k, norm_frobenius, error_frobenius )
#
# FRANK matrix.
#
title = 'FRANK'
m = 10
n = 10
k = 2
a = frank ( n )
b = frank_rhs ( m, k )
x = frank_solution ( n, k )
error_frobenius = r8mat_is_solution ( m, n, k, a, x, b )
norm_frobenius = r8mat_norm_fro ( n, n, a )
print ' %-20s %4d %4d %4d %14g %14g' \
% ( title, m, n, k, norm_frobenius, error_frobenius )
#
# POISSON matrix.
#
title = 'POISSON'
nrow = 4
ncol = 5
m = nrow * ncol
n = nrow * ncol
k = 1
a = poisson ( nrow, ncol )
b = poisson_rhs ( nrow, ncol, k )
x = poisson_solution ( nrow, ncol, k )
error_frobenius = r8mat_is_solution ( m, n, k, a, x, b )
norm_frobenius = r8mat_norm_fro ( n, n, a )
print ' %-20s %4d %4d %4d %14g %14g' \
% ( title, m, n, k, norm_frobenius, error_frobenius )
#
# WILK03 matrix.
#
title = 'WILK03'
m = 3
n = 3
k = 1
a = wilk03 ( )
b = wilk03_rhs ( )
x = wilk03_solution ( )
error_frobenius = r8mat_is_solution ( m, n, k, a, x, b )
norm_frobenius = r8mat_norm_fro ( n, n, a )
print ' %-20s %4d %4d %4d %14g %14g' \
% ( title, m, n, k, norm_frobenius, error_frobenius )
#
# WILK04 matrix.
#
title = 'WILK04'
m = 4
n = 4
k = 1
a = wilk04 ( )
b = wilk04_rhs ( )
x = wilk04_solution ( )
error_frobenius = r8mat_is_solution ( m, n, k, a, x, b )
norm_frobenius = r8mat_norm_fro ( n, n, a )
print ' %-20s %4d %4d %4d %14g %14g' \
% ( title, m, n, k, norm_frobenius, error_frobenius )
#
# WILSON matrix.
#
title = 'WILSON'
m = 4
n = 4
k = 1
a = wilson ( )
b = wilson_rhs ( )
x = wilson_solution ( )
error_frobenius = r8mat_is_solution ( m, n, k, a, x, b )
norm_frobenius = r8mat_norm_fro ( n, n, a )
print ' %-20s %4d %4d %4d %14g %14g' \
% ( title, m, n, k, norm_frobenius, error_frobenius )
#
# Terminate.
#
print ''
print 'TEST_SOLUTION'
print ' Normal end of execution.'
return
def r8mat_is_eigen_left ( n, k, a, x, lam ):
#*****************************************************************************80
#
## R8MAT_IS_EIGEN_LEFT determines the error in a left eigensystem.
#
# Discussion:
#
# An R8MAT is a matrix of real values.
#
# This routine computes the Frobenius norm of
#
# X * A - LAMBDA * X
#
# where
#
# A is an N by N matrix,
# X is an K by N matrix (each of K columns is an eigenvector)
# LAMBDA is a K by K diagonal matrix of eigenvalues.
#
# This routine assumes that A, X and LAMBDA are all real.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 17 March 2015
#
# Author:
#
# John Burkardt
#
# Parameters:
#
# Input, integer N, the order of the matrix.
#
# Input, integer K, the number of eigenvectors.
# K is usually 1 or N.
#
# Input, real A(N,N), the matrix.
#
# Input, real X(K,N), the K eigenvectors.
#
# Input, real LAM(K), the K eigenvalues.
#
# Output, real VALUE, the Frobenius norm of X * A - LAM * X.
#
from r8mat_mm import r8mat_mm
from r8mat_norm_fro import r8mat_norm_fro
c = r8mat_mm ( k, n, n, x, a )
for i in range ( 0, k ):
for j in range ( 0, n ):
c[i,j] = c[i,j] - lam[i] * x[i,j]
value = r8mat_norm_fro ( k, n, c )
return value
def r8mat_is_eigen_right ( n, k, a, x, lam ):
#*****************************************************************************80
#
## R8MAT_IS_EIGEN_RIGHT determines the error in a right eigensystem.
#
# Discussion:
#
# An R8MAT is a matrix of real values.
#
# This routine computes the Frobenius norm of
#
# A * X - X * LAMBDA
#
# where
#
# A is an N by N matrix,
# X is an N by K matrix (each of K columns is an eigenvector)
# LAMBDA is a K by K diagonal matrix of eigenvalues.
#
# This routine assumes that A, X and LAMBDA are all real.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 11 March 2015
#
# Author:
#
# John Burkardt
#
# Parameters:
#
# Input, integer N, the order of the matrix.
#
# Input, integer K, the number of eigenvectors.
# K is usually 1 or N.
#
# Input, real A(N,N), the matrix.
#
# Input, real X(N,K), the K eigenvectors.
#
# Input, real LAM(K), the K eigenvalues.
#
# Output, real VALUE, the Frobenius norm
# of the difference matrix A * X - X * LAM, which would be exactly zero
# if X and LAM were exact eigenvectors and eigenvalues of A.
#
from r8mat_mm import r8mat_mm
from r8mat_norm_fro import r8mat_norm_fro
c = r8mat_mm ( n, n, k, a, x )
for j in range ( 0, k ):
for i in range ( 0, n ):
c[i,j] = c[i,j] - lam[j] * x[i,j]
value = r8mat_norm_fro ( n, k, c )
return value
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
def r8mat_is_eigen_right(n, k, a, x, lam):
#*****************************************************************************80
#
## R8MAT_IS_EIGEN_RIGHT determines the error in a right eigensystem.
#
# Discussion:
#
# An R8MAT is a matrix of real values.
#
# This routine computes the Frobenius norm of
#
# A * X - X * LAMBDA
#
# where
#
# A is an N by N matrix,
# X is an N by K matrix (each of K columns is an eigenvector)
# LAMBDA is a K by K diagonal matrix of eigenvalues.
#
# This routine assumes that A, X and LAMBDA are all real.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 11 March 2015
#
# Author:
#
# John Burkardt
#
# Parameters:
#
# Input, integer N, the order of the matrix.
#
# Input, integer K, the number of eigenvectors.
# K is usually 1 or N.
#
# Input, real A(N,N), the matrix.
#
# Input, real X(N,K), the K eigenvectors.
#
# Input, real LAM(K), the K eigenvalues.
#
# Output, real VALUE, the Frobenius norm
# of the difference matrix A * X - X * LAM, which would be exactly zero
# if X and LAM were exact eigenvectors and eigenvalues of A.
#
from r8mat_mm import r8mat_mm
from r8mat_norm_fro import r8mat_norm_fro
c = r8mat_mm(n, n, k, a, x)
for j in range(0, k):
for i in range(0, n):
c[i, j] = c[i, j] - lam[j] * x[i, j]
value = r8mat_norm_fro(n, k, c)
return value
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
