358 lines
8.6 KiB
Python
358 lines
8.6 KiB
Python
#!/usr/bin/python
|
|
# -*- coding: utf-8 -*-
|
|
|
|
from mpmath import mp
|
|
from mpmath import libmp
|
|
|
|
xrange = libmp.backend.xrange
|
|
|
|
def run_eigsy(A, verbose = False):
|
|
if verbose:
|
|
print("original matrix:\n", str(A))
|
|
|
|
D, Q = mp.eigsy(A)
|
|
B = Q * mp.diag(D) * Q.transpose()
|
|
C = A - B
|
|
E = Q * Q.transpose() - mp.eye(A.rows)
|
|
|
|
if verbose:
|
|
print("eigenvalues:\n", D)
|
|
print("eigenvectors:\n", Q)
|
|
|
|
NC = mp.mnorm(C)
|
|
NE = mp.mnorm(E)
|
|
|
|
if verbose:
|
|
print("difference:", NC, "\n", C, "\n")
|
|
print("difference:", NE, "\n", E, "\n")
|
|
|
|
eps = mp.exp( 0.8 * mp.log(mp.eps))
|
|
|
|
assert NC < eps
|
|
assert NE < eps
|
|
|
|
return NC
|
|
|
|
def run_eighe(A, verbose = False):
|
|
if verbose:
|
|
print("original matrix:\n", str(A))
|
|
|
|
D, Q = mp.eighe(A)
|
|
B = Q * mp.diag(D) * Q.transpose_conj()
|
|
C = A - B
|
|
E = Q * Q.transpose_conj() - mp.eye(A.rows)
|
|
|
|
if verbose:
|
|
print("eigenvalues:\n", D)
|
|
print("eigenvectors:\n", Q)
|
|
|
|
NC = mp.mnorm(C)
|
|
NE = mp.mnorm(E)
|
|
|
|
if verbose:
|
|
print("difference:", NC, "\n", C, "\n")
|
|
print("difference:", NE, "\n", E, "\n")
|
|
|
|
eps = mp.exp( 0.8 * mp.log(mp.eps))
|
|
|
|
assert NC < eps
|
|
assert NE < eps
|
|
|
|
return NC
|
|
|
|
def run_svd_r(A, full_matrices = False, verbose = True):
|
|
|
|
m, n = A.rows, A.cols
|
|
|
|
eps = mp.exp(0.8 * mp.log(mp.eps))
|
|
|
|
if verbose:
|
|
print("original matrix:\n", str(A))
|
|
print("full", full_matrices)
|
|
|
|
U, S0, V = mp.svd_r(A, full_matrices = full_matrices)
|
|
|
|
S = mp.zeros(U.cols, V.rows)
|
|
for j in xrange(min(m, n)):
|
|
S[j,j] = S0[j]
|
|
|
|
if verbose:
|
|
print("U:\n", str(U))
|
|
print("S:\n", str(S0))
|
|
print("V:\n", str(V))
|
|
|
|
C = U * S * V - A
|
|
err = mp.mnorm(C)
|
|
if verbose:
|
|
print("C\n", str(C), "\n", err)
|
|
assert err < eps
|
|
|
|
D = V * V.transpose() - mp.eye(V.rows)
|
|
err = mp.mnorm(D)
|
|
if verbose:
|
|
print("D:\n", str(D), "\n", err)
|
|
assert err < eps
|
|
|
|
E = U.transpose() * U - mp.eye(U.cols)
|
|
err = mp.mnorm(E)
|
|
if verbose:
|
|
print("E:\n", str(E), "\n", err)
|
|
assert err < eps
|
|
|
|
def run_svd_c(A, full_matrices = False, verbose = True):
|
|
|
|
m, n = A.rows, A.cols
|
|
|
|
eps = mp.exp(0.8 * mp.log(mp.eps))
|
|
|
|
if verbose:
|
|
print("original matrix:\n", str(A))
|
|
print("full", full_matrices)
|
|
|
|
U, S0, V = mp.svd_c(A, full_matrices = full_matrices)
|
|
|
|
S = mp.zeros(U.cols, V.rows)
|
|
for j in xrange(min(m, n)):
|
|
S[j,j] = S0[j]
|
|
|
|
if verbose:
|
|
print("U:\n", str(U))
|
|
print("S:\n", str(S0))
|
|
print("V:\n", str(V))
|
|
|
|
C = U * S * V - A
|
|
err = mp.mnorm(C)
|
|
if verbose:
|
|
print("C\n", str(C), "\n", err)
|
|
assert err < eps
|
|
|
|
D = V * V.transpose_conj() - mp.eye(V.rows)
|
|
err = mp.mnorm(D)
|
|
if verbose:
|
|
print("D:\n", str(D), "\n", err)
|
|
assert err < eps
|
|
|
|
E = U.transpose_conj() * U - mp.eye(U.cols)
|
|
err = mp.mnorm(E)
|
|
if verbose:
|
|
print("E:\n", str(E), "\n", err)
|
|
assert err < eps
|
|
|
|
def run_gauss(qtype, a, b):
|
|
eps = 1e-5
|
|
|
|
d, e = mp.gauss_quadrature(len(a), qtype)
|
|
d -= mp.matrix(a)
|
|
e -= mp.matrix(b)
|
|
|
|
assert mp.mnorm(d) < eps
|
|
assert mp.mnorm(e) < eps
|
|
|
|
def irandmatrix(n, range = 10):
|
|
"""
|
|
random matrix with integer entries
|
|
"""
|
|
A = mp.matrix(n, n)
|
|
for i in xrange(n):
|
|
for j in xrange(n):
|
|
A[i,j]=int( (2 * mp.rand() - 1) * range)
|
|
return A
|
|
|
|
#######################
|
|
|
|
def test_eighe_fixed_matrix():
|
|
A = mp.matrix([[2, 3], [3, 5]])
|
|
run_eigsy(A)
|
|
run_eighe(A)
|
|
|
|
A = mp.matrix([[7, -11], [-11, 13]])
|
|
run_eigsy(A)
|
|
run_eighe(A)
|
|
|
|
A = mp.matrix([[2, 11, 7], [11, 3, 13], [7, 13, 5]])
|
|
run_eigsy(A)
|
|
run_eighe(A)
|
|
|
|
A = mp.matrix([[2, 0, 7], [0, 3, 1], [7, 1, 5]])
|
|
run_eigsy(A)
|
|
run_eighe(A)
|
|
|
|
#
|
|
|
|
A = mp.matrix([[2, 3+7j], [3-7j, 5]])
|
|
run_eighe(A)
|
|
|
|
A = mp.matrix([[2, -11j, 0], [+11j, 3, 29j], [0, -29j, 5]])
|
|
run_eighe(A)
|
|
|
|
A = mp.matrix([[2, 11 + 17j, 7 + 19j], [11 - 17j, 3, -13 + 23j], [7 - 19j, -13 - 23j, 5]])
|
|
run_eighe(A)
|
|
|
|
def test_eigsy_randmatrix():
|
|
N = 5
|
|
|
|
for a in xrange(10):
|
|
A = 2 * mp.randmatrix(N, N) - 1
|
|
|
|
for i in xrange(0, N):
|
|
for j in xrange(i + 1, N):
|
|
A[j,i] = A[i,j]
|
|
|
|
run_eigsy(A)
|
|
|
|
def test_eighe_randmatrix():
|
|
N = 5
|
|
|
|
for a in xrange(10):
|
|
A = (2 * mp.randmatrix(N, N) - 1) + 1j * (2 * mp.randmatrix(N, N) - 1)
|
|
|
|
for i in xrange(0, N):
|
|
A[i,i] = mp.re(A[i,i])
|
|
for j in xrange(i + 1, N):
|
|
A[j,i] = mp.conj(A[i,j])
|
|
|
|
run_eighe(A)
|
|
|
|
def test_eigsy_irandmatrix():
|
|
N = 4
|
|
R = 4
|
|
|
|
for a in xrange(10):
|
|
A=irandmatrix(N, R)
|
|
|
|
for i in xrange(0, N):
|
|
for j in xrange(i + 1, N):
|
|
A[j,i] = A[i,j]
|
|
|
|
run_eigsy(A)
|
|
|
|
def test_eighe_irandmatrix():
|
|
N = 4
|
|
R = 4
|
|
|
|
for a in xrange(10):
|
|
A=irandmatrix(N, R) + 1j * irandmatrix(N, R)
|
|
|
|
for i in xrange(0, N):
|
|
A[i,i] = mp.re(A[i,i])
|
|
for j in xrange(i + 1, N):
|
|
A[j,i] = mp.conj(A[i,j])
|
|
|
|
run_eighe(A)
|
|
|
|
def test_svd_r_rand():
|
|
for i in xrange(5):
|
|
full = mp.rand() > 0.5
|
|
m = 1 + int(mp.rand() * 10)
|
|
n = 1 + int(mp.rand() * 10)
|
|
A = 2 * mp.randmatrix(m, n) - 1
|
|
if mp.rand() > 0.5:
|
|
A *= 10
|
|
for x in xrange(m):
|
|
for y in xrange(n):
|
|
A[x,y]=int(A[x,y])
|
|
|
|
run_svd_r(A, full_matrices = full, verbose = False)
|
|
|
|
def test_svd_c_rand():
|
|
for i in xrange(5):
|
|
full = mp.rand() > 0.5
|
|
m = 1 + int(mp.rand() * 10)
|
|
n = 1 + int(mp.rand() * 10)
|
|
A = (2 * mp.randmatrix(m, n) - 1) + 1j * (2 * mp.randmatrix(m, n) - 1)
|
|
if mp.rand() > 0.5:
|
|
A *= 10
|
|
for x in xrange(m):
|
|
for y in xrange(n):
|
|
A[x,y]=int(mp.re(A[x,y])) + 1j * int(mp.im(A[x,y]))
|
|
|
|
run_svd_c(A, full_matrices=full, verbose=False)
|
|
|
|
def test_svd_test_case():
|
|
# a test case from Golub and Reinsch
|
|
# (see wilkinson/reinsch: handbook for auto. comp., vol ii-linear algebra, 134-151(1971).)
|
|
|
|
eps = mp.exp(0.8 * mp.log(mp.eps))
|
|
|
|
a = [[22, 10, 2, 3, 7],
|
|
[14, 7, 10, 0, 8],
|
|
[-1, 13, -1, -11, 3],
|
|
[-3, -2, 13, -2, 4],
|
|
[ 9, 8, 1, -2, 4],
|
|
[ 9, 1, -7, 5, -1],
|
|
[ 2, -6, 6, 5, 1],
|
|
[ 4, 5, 0, -2, 2]]
|
|
|
|
a = mp.matrix(a)
|
|
b = mp.matrix([mp.sqrt(1248), 20, mp.sqrt(384), 0, 0])
|
|
|
|
S = mp.svd_r(a, compute_uv = False)
|
|
S -= b
|
|
assert mp.mnorm(S) < eps
|
|
|
|
S = mp.svd_c(a, compute_uv = False)
|
|
S -= b
|
|
assert mp.mnorm(S) < eps
|
|
|
|
|
|
def test_gauss_quadrature_static():
|
|
a = [-0.57735027, 0.57735027]
|
|
b = [ 1, 1]
|
|
run_gauss("legendre", a , b)
|
|
|
|
a = [ -0.906179846, -0.538469310, 0, 0.538469310, 0.906179846]
|
|
b = [ 0.23692689, 0.47862867, 0.56888889, 0.47862867, 0.23692689]
|
|
run_gauss("legendre", a , b)
|
|
|
|
a = [ 0.06943184, 0.33000948, 0.66999052, 0.93056816]
|
|
b = [ 0.17392742, 0.32607258, 0.32607258, 0.17392742]
|
|
run_gauss("legendre01", a , b)
|
|
|
|
a = [-0.70710678, 0.70710678]
|
|
b = [ 0.88622693, 0.88622693]
|
|
run_gauss("hermite", a , b)
|
|
|
|
a = [ -2.02018287, -0.958572465, 0, 0.958572465, 2.02018287]
|
|
b = [ 0.01995324, 0.39361932, 0.94530872, 0.39361932, 0.01995324]
|
|
run_gauss("hermite", a , b)
|
|
|
|
a = [ 0.41577456, 2.29428036, 6.28994508]
|
|
b = [ 0.71109301, 0.27851773, 0.01038926]
|
|
run_gauss("laguerre", a , b)
|
|
|
|
def test_gauss_quadrature_dynamic(verbose = False):
|
|
n = 5
|
|
|
|
A = mp.randmatrix(2 * n, 1)
|
|
|
|
def F(x):
|
|
r = 0
|
|
for i in xrange(len(A) - 1, -1, -1):
|
|
r = r * x + A[i]
|
|
return r
|
|
|
|
def run(qtype, FW, R, alpha = 0, beta = 0):
|
|
X, W = mp.gauss_quadrature(n, qtype, alpha = alpha, beta = beta)
|
|
|
|
a = 0
|
|
for i in xrange(len(X)):
|
|
a += W[i] * F(X[i])
|
|
|
|
b = mp.quad(lambda x: FW(x) * F(x), R)
|
|
|
|
c = mp.fabs(a - b)
|
|
|
|
if verbose:
|
|
print(qtype, c, a, b)
|
|
|
|
assert c < 1e-5
|
|
|
|
run("legendre", lambda x: 1, [-1, 1])
|
|
run("legendre01", lambda x: 1, [0, 1])
|
|
run("hermite", lambda x: mp.exp(-x*x), [-mp.inf, mp.inf])
|
|
run("laguerre", lambda x: mp.exp(-x), [0, mp.inf])
|
|
run("glaguerre", lambda x: mp.sqrt(x)*mp.exp(-x), [0, mp.inf], alpha = 1 / mp.mpf(2))
|
|
run("chebyshev1", lambda x: 1/mp.sqrt(1-x*x), [-1, 1])
|
|
run("chebyshev2", lambda x: mp.sqrt(1-x*x), [-1, 1])
|
|
run("jacobi", lambda x: (1-x)**(1/mp.mpf(3)) * (1+x)**(1/mp.mpf(5)), [-1, 1], alpha = 1 / mp.mpf(3), beta = 1 / mp.mpf(5) )
|