180 lines
3.8 KiB
Python
180 lines
3.8 KiB
Python
|
#!/usr/bin/python
|
||
|
|
# -*- coding: utf-8 -*-
|
||
|
|
|
||
|
|
from mpmath import mp
|
||
|
|
from mpmath import libmp
|
||
|
|
|
||
|
|
xrange = libmp.backend.xrange
|
||
|
|
|
||
|
|
def run_hessenberg(A, verbose = 0):
|
||
|
|
if verbose > 1:
|
||
|
|
print("original matrix (hessenberg):\n", A)
|
||
|
|
|
||
|
|
n = A.rows
|
||
|
|
|
||
|
|
Q, H = mp.hessenberg(A)
|
||
|
|
|
||
|
|
if verbose > 1:
|
||
|
|
print("Q:\n",Q)
|
||
|
|
print("H:\n",H)
|
||
|
|
|
||
|
|
B = Q * H * Q.transpose_conj()
|
||
|
|
|
||
|
|
eps = mp.exp(0.8 * mp.log(mp.eps))
|
||
|
|
|
||
|
|
err0 = 0
|
||
|
|
for x in xrange(n):
|
||
|
|
for y in xrange(n):
|
||
|
|
err0 += abs(A[y,x] - B[y,x])
|
||
|
|
err0 /= n * n
|
||
|
|
|
||
|
|
err1 = 0
|
||
|
|
for x in xrange(n):
|
||
|
|
for y in xrange(x + 2, n):
|
||
|
|
err1 += abs(H[y,x])
|
||
|
|
|
||
|
|
if verbose > 0:
|
||
|
|
print("difference (H):", err0, err1)
|
||
|
|
|
||
|
|
if verbose > 1:
|
||
|
|
print("B:\n", B)
|
||
|
|
|
||
|
|
assert err0 < eps
|
||
|
|
assert err1 == 0
|
||
|
|
|
||
|
|
|
||
|
|
def run_schur(A, verbose = 0):
|
||
|
|
if verbose > 1:
|
||
|
|
print("original matrix (schur):\n", A)
|
||
|
|
|
||
|
|
n = A.rows
|
||
|
|
|
||
|
|
Q, R = mp.schur(A)
|
||
|
|
|
||
|
|
if verbose > 1:
|
||
|
|
print("Q:\n", Q)
|
||
|
|
print("R:\n", R)
|
||
|
|
|
||
|
|
B = Q * R * Q.transpose_conj()
|
||
|
|
C = Q * Q.transpose_conj()
|
||
|
|
|
||
|
|
eps = mp.exp(0.8 * mp.log(mp.eps))
|
||
|
|
|
||
|
|
err0 = 0
|
||
|
|
for x in xrange(n):
|
||
|
|
for y in xrange(n):
|
||
|
|
err0 += abs(A[y,x] - B[y,x])
|
||
|
|
err0 /= n * n
|
||
|
|
|
||
|
|
err1 = 0
|
||
|
|
for x in xrange(n):
|
||
|
|
for y in xrange(n):
|
||
|
|
if x == y:
|
||
|
|
C[y,x] -= 1
|
||
|
|
err1 += abs(C[y,x])
|
||
|
|
err1 /= n * n
|
||
|
|
|
||
|
|
err2 = 0
|
||
|
|
for x in xrange(n):
|
||
|
|
for y in xrange(x + 1, n):
|
||
|
|
err2 += abs(R[y,x])
|
||
|
|
|
||
|
|
if verbose > 0:
|
||
|
|
print("difference (S):", err0, err1, err2)
|
||
|
|
|
||
|
|
if verbose > 1:
|
||
|
|
print("B:\n", B)
|
||
|
|
|
||
|
|
assert err0 < eps
|
||
|
|
assert err1 < eps
|
||
|
|
assert err2 == 0
|
||
|
|
|
||
|
|
def run_eig(A, verbose = 0):
|
||
|
|
if verbose > 1:
|
||
|
|
print("original matrix (eig):\n", A)
|
||
|
|
|
||
|
|
n = A.rows
|
||
|
|
|
||
|
|
E, EL, ER = mp.eig(A, left = True, right = True)
|
||
|
|
|
||
|
|
if verbose > 1:
|
||
|
|
print("E:\n", E)
|
||
|
|
print("EL:\n", EL)
|
||
|
|
print("ER:\n", ER)
|
||
|
|
|
||
|
|
eps = mp.exp(0.8 * mp.log(mp.eps))
|
||
|
|
|
||
|
|
err0 = 0
|
||
|
|
for i in xrange(n):
|
||
|
|
B = A * ER[:,i] - E[i] * ER[:,i]
|
||
|
|
err0 = max(err0, mp.mnorm(B))
|
||
|
|
|
||
|
|
B = EL[i,:] * A - EL[i,:] * E[i]
|
||
|
|
err0 = max(err0, mp.mnorm(B))
|
||
|
|
|
||
|
|
err0 /= n * n
|
||
|
|
|
||
|
|
if verbose > 0:
|
||
|
|
print("difference (E):", err0)
|
||
|
|
|
||
|
|
assert err0 < eps
|
||
|
|
|
||
|
|
#####################
|
||
|
|
|
||
|
|
def test_eig_dyn():
|
||
|
|
v = 0
|
||
|
|
for i in xrange(5):
|
||
|
|
n = 1 + int(mp.rand() * 5)
|
||
|
|
if mp.rand() > 0.5:
|
||
|
|
# real
|
||
|
|
A = 2 * mp.randmatrix(n, n) - 1
|
||
|
|
if mp.rand() > 0.5:
|
||
|
|
A *= 10
|
||
|
|
for x in xrange(n):
|
||
|
|
for y in xrange(n):
|
||
|
|
A[x,y] = int(A[x,y])
|
||
|
|
else:
|
||
|
|
A = (2 * mp.randmatrix(n, n) - 1) + 1j * (2 * mp.randmatrix(n, n) - 1)
|
||
|
|
if mp.rand() > 0.5:
|
||
|
|
A *= 10
|
||
|
|
for x in xrange(n):
|
||
|
|
for y in xrange(n):
|
||
|
|
A[x,y] = int(mp.re(A[x,y])) + 1j * int(mp.im(A[x,y]))
|
||
|
|
|
||
|
|
run_hessenberg(A, verbose = v)
|
||
|
|
run_schur(A, verbose = v)
|
||
|
|
run_eig(A, verbose = v)
|
||
|
|
|
||
|
|
def test_eig():
|
||
|
|
v = 0
|
||
|
|
AS = []
|
||
|
|
|
||
|
|
A = mp.matrix([[2, 1, 0], # jordan block of size 3
|
||
|
|
[0, 2, 1],
|
||
|
|
[0, 0, 2]])
|
||
|
|
AS.append(A)
|
||
|
|
AS.append(A.transpose())
|
||
|
|
|
||
|
|
A = mp.matrix([[2, 0, 0], # jordan block of size 2
|
||
|
|
[0, 2, 1],
|
||
|
|
[0, 0, 2]])
|
||
|
|
AS.append(A)
|
||
|
|
AS.append(A.transpose())
|
||
|
|
|
||
|
|
A = mp.matrix([[2, 0, 1], # jordan block of size 2
|
||
|
|
[0, 2, 0],
|
||
|
|
[0, 0, 2]])
|
||
|
|
AS.append(A)
|
||
|
|
AS.append(A.transpose())
|
||
|
|
|
||
|
|
A= mp.matrix([[0, 0, 1], # cyclic
|
||
|
|
[1, 0, 0],
|
||
|
|
[0, 1, 0]])
|
||
|
|
AS.append(A)
|
||
|
|
AS.append(A.transpose())
|
||
|
|
|
||
|
|
for A in AS:
|
||
|
|
run_hessenberg(A, verbose = v)
|
||
|
|
run_schur(A, verbose = v)
|
||
|
|
run_eig(A, verbose = v)
|