def lu(A):
"""
A -> P, L, U
LU factorisation of a square matrix A. L is the lower, U the upper part.
P is the permutation matrix indicating the row swaps.
P*A = L*U
If you need efficiency, use the low-level method LU_decomp instead, it's
much more memory efficient.
"""
# get factorization
A, p = LU_decomp(A.copy())
n = A.rows
L = matrix(n)
U = matrix(n)
for i in xrange(n):
for j in xrange(n):
if i > j:
L[i, j] = A[i, j]
elif i == j:
L[i, j] = 1
U[i, j] = A[i, j]
else:
U[i, j] = A[i, j]
# calculate permutation matrix
P = eye(n)
for k in xrange(len(p)):
swap_row(P, k, p[k])
return P, L, U
def lu(A):
"""
A -> P, L, U
LU factorisation of a square matrix A. L is the lower, U the upper part.
P is the permutation matrix indicating the row swaps.
P*A = L*U
If you need efficiency, use the low-level method LU_decomp instead, it's
much more memory efficient.
"""
# get factorization
A, p = LU_decomp(A)
n = A.rows
L = matrix(n)
U = matrix(n)
for i in xrange(n):
for j in xrange(n):
if i > j:
L[i,j] = A[i,j]
elif i == j:
L[i,j] = 1
U[i,j] = A[i,j]
else:
U[i,j] = A[i,j]
# calculate permutation matrix
P = eye(n)
for k in xrange(len(p)):
swap_row(P, k, p[k])
return P, L, U
def exp_pade(a):
"""Exponential of a matrix using Pade approximants.
See G. H. Golub, C. F. van Loan 'Matrix Computations',
third Ed., page 572
TODO:
- find a good estimate for q
- reduce the number of matrix multiplications to improve
performance
"""
def eps_pade(p):
return mpf(2)**(3-2*p) * factorial(p)**2/(factorial(2*p)**2 * (2*p + 1))
q = 4
extraq = 8
while 1:
if eps_pade(q) < eps:
break
q += 1
q += extraq
j = max(1, int(log(mnorm(a,'inf'),2)))
extra = q
mp.dps += extra
try:
a = a/2**j
na = a.rows
den = eye(na)
num = eye(na)
x = eye(na)
c = mpf(1)
for k in range(1, q+1):
c *= mpf(q - k + 1)/((2*q - k + 1) * k)
x = a*x
cx = c*x
num += cx
den += (-1)**k * cx
f = lu_solve_mat(den, num)
for k in range(j):
f = f*f
finally:
mp.dps -= extra
return f
def exp_pade(a):
"""Exponential of a matrix using Pade approximants.
See G. H. Golub, C. F. van Loan 'Matrix Computations',
third Ed., page 572
TODO:
- find a good estimate for q
- reduce the number of matrix multiplications to improve
performance
"""
def eps_pade(p):
return mpf(2)**(3-2*p) * factorial(p)**2/(factorial(2*p)**2 * (2*p + 1))
q = 4
extraq = 8
while 1:
if eps_pade(q) < eps:
break
q += 1
q += extraq
j = max(1, int(log(mnorm(a,'inf'),2)))
extra = q
mp.dps += extra
try:
a = a/2**j
na = a.rows
den = eye(na)
num = eye(na)
x = eye(na)
c = mpf(1)
for k in range(1, q+1):
c *= mpf(q - k + 1)/((2*q - k + 1) * k)
x = a*x
cx = c*x
num += cx
den += (-1)**k * cx
f = lu_solve_mat(den, num)
for k in range(j):
f = f*f
finally:
mp.dps -= extra
return f