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 L_solve(L, b, p=None):
"""
Solve the lower part of a LU factorized matrix for y.
"""
L.rows == L.cols, 'need n*n matrix'
n = L.rows
assert len(b) == n
b = copy(b)
if p: # swap b according to p
for k in xrange(0, len(p)):
swap_row(b, k, p[k])
# solve
for i in xrange(1, n):
for j in xrange(i):
b[i] -= L[i, j] * b[j]
return b
def L_solve(L, b, p=None):
"""
Solve the lower part of a LU factorized matrix for y.
"""
assert L.rows == L.cols, 'need n*n matrix'
n = L.rows
assert len(b) == n
b = copy(b)
if p: # swap b according to p
for k in xrange(0, len(p)):
swap_row(b, k, p[k])
# solve
for i in xrange(1, n):
for j in xrange(i):
b[i] -= L[i,j] * b[j]
return b
def LU_decomp(A, overwrite=False, use_cache=True):
"""
LU-factorization of a n*n matrix using the Gauss algorithm.
Returns L and U in one matrix and the pivot indices.
Use overwrite to specify whether A will be overwritten with L and U.
"""
if not A.rows == A.cols:
raise ValueError('need n*n matrix')
# get from cache if possible
if use_cache and isinstance(A, matrix) and A._LU:
return A._LU
if not overwrite:
orig = A
A = A.copy()
tol = absmin(mnorm(A,1) * eps) # each pivot element has to be bigger
n = A.rows
p = [None]*(n - 1)
for j in xrange(n - 1):
# pivoting, choose max(abs(reciprocal row sum)*abs(pivot element))
biggest = 0
for k in xrange(j, n):
s = fsum([absmin(A[k,l]) for l in xrange(j, n)])
if absmin(s) <= tol:
raise ZeroDivisionError('matrix is numerically singular')
current = 1/s * absmin(A[k,j])
if current > biggest: # TODO: what if equal?
biggest = current
p[j] = k
# swap rows according to p
swap_row(A, j, p[j])
if absmin(A[j,j]) <= tol:
raise ZeroDivisionError('matrix is numerically singular')
# calculate elimination factors and add rows
for i in xrange(j + 1, n):
A[i,j] /= A[j,j]
for k in xrange(j + 1, n):
A[i,k] -= A[i,j]*A[j,k]
if absmin(A[n - 1,n - 1]) <= tol:
raise ZeroDivisionError('matrix is numerically singular')
# cache decomposition
if not overwrite and isinstance(orig, matrix):
orig._LU = (A, p)
return A, p
def LU_decomp(A, overwrite=False, use_cache=True):
"""
LU-factorization of a n*n matrix using the Gauss algorithm.
Returns L and U in one matrix and the pivot indices.
Use overwrite to specify whether A will be overwritten with L and U.
"""
if not A.rows == A.cols:
raise ValueError('need n*n matrix')
# get from cache if possible
if use_cache and isinstance(A, matrix) and A._LU:
return A._LU
if not overwrite:
orig = A
A = A.copy()
tol = absmin(mnorm(A,1) * eps) # each pivot element has to be bigger
n = A.rows
p = [None]*(n - 1)
for j in xrange(n - 1):
# pivoting, choose max(abs(reciprocal row sum)*abs(pivot element))
biggest = 0
for k in xrange(j, n):
s = fsum([absmin(A[k,l]) for l in xrange(j, n)])
if absmin(s) <= tol:
raise ZeroDivisionError('matrix is numerically singular')
current = 1/s * absmin(A[k,j])
if current > biggest: # TODO: what if equal?
biggest = current
p[j] = k
# swap rows according to p
swap_row(A, j, p[j])
if absmin(A[j,j]) <= tol:
raise ZeroDivisionError('matrix is numerically singular')
# calculate elimination factors and add rows
for i in xrange(j + 1, n):
A[i,j] /= A[j,j]
for k in xrange(j + 1, n):
A[i,k] -= A[i,j]*A[j,k]
if absmin(A[n - 1,n - 1]) <= tol:
raise ZeroDivisionError('matrix is numerically singular')
# cache decomposition
if not overwrite and isinstance(orig, matrix):
orig._LU = (A, p)
return A, p
