def Jacobi(self, b, mat=True):
if isinstance(b, matrix):
b = b.c(range(b.size[0]), 0)
D = self.D
A = self.A.data
B=[[-A_ij/D_i if i!=j \
else 1-A_ij/D_i for j,A_ij in enumerate(A[i])]for i,D_i in enumerate(D)]
if mat == True:
B = matrix(B)
f = [b_i / D[i] for i, b_i in enumerate(b)]
f = matrix(f).T()
return B, f
def GaussEq(A, B):
if isinstance(B, matrix):
B = B.c(range(B.size[0]), 0)
B = copy.deepcopy(B)
size = A.size
data = copy.deepcopy(A.data)
row_arg = range(size[0])
if size[0] != size[1]:
print('error\n')
return ValueError
size = size[0]
for i in row_arg[:-1]:
vec = matrix(data).c(range(i, size), i)
max_index = i + argabsmax(vec)
data[max_index], data[i] = data[i], data[max_index]
B[max_index], B[i] = B[i], B[max_index]
for index in range(i + 1, size):
try:
param = data[index][i] / data[i][i]
data[index] = add_vec(data[index], data[i], -param)
B[index] += B[i] * (-param)
except:
print('error: a singular matrix')
return ValueError
row_arg.reverse()
x = [0 for i in range(size)]
for i in row_arg:
x[i] = (B[i] - sum(dot_vec(x[i:], data[i][i:]))) / data[i][i]
return x
def __init__(self, A):
data = A.data
self.size = A.size
size = A.size[0]
self.A = copy.deepcopy(A)
self.L = [[-data[i][j] for j in range(i)] for i in range(1, size)]
self.D = [data[i][i] for i in range(size)]
self.X = matrix([1 for i in self.D]).T()
def Gauss_Seidel(self, b, mat=True):
if isinstance(b, matrix):
b = b.c(range(b.size[0]), 0)
D = self.D
L = self.L
A = self.A
size = A.size
Dinv = [[-L[i - 1][j] if j < i else 0 for j in xrange(size[1])]
for i in xrange(size[0])]
for i in range(size[0]):
Dinv[i][i] = D[i]
Dinv = matrix(Dinv).inv(mat=True)
B = matrix(size, eye=True) - Dinv * A
if mat == False:
B = B.data
f = Dinv * matrix(b).T()
return B, f
def LU(A):
size = A.size
if size[0] != size[1]:
return ValueError
L = matrix(size, eye=True)
U = matrix(size, eye=False)
size = size[0]
aij = A.data
for i in range(0, size):
rang = range(0, i)
for r in range(i, size):
U.data[i][r] = aij[i][r] - sum(
dot_vec(L.c([i], rang), U.c(rang, [r])))
for r in range(i + 1, size):
L.data[r][i] = (aij[r][i] - sum(
dot_vec(L.c([r], rang), U.c(rang, [i])))) / U.data[i][i]
return L.data, U.data
def _SOR(self, b, w, mat=True):
if isinstance(b, matrix):
b = b.c(range(b.size[0]), 0)
D = self.D
L = self.L
A = self.A
size = A.size
data = A.data
Dinv = [[-w * L[i - 1][j] if j < i else 0 for j in xrange(size[1])]
for i in xrange(size[0])]
LA = [[-w * data[i][j] if j >= i else 0 for j in xrange(size[1])]
for i in xrange(size[0])]
for i in range(size[0]):
Dinv[i][i] = D[i]
LA[i][i] += D[i]
Dinv = matrix(Dinv).inv(mat=True)
LA = matrix(LA)
SORmat = Dinv * LA
b = matrix(b).T()
f = w * Dinv * b
return SORmat, f
def QR_split(B, NumLimit=10e-9, P_is_a_matrix=False):
if isinstance(B, matrix):
size = B.size[0]
A = copy.deepcopy(B.data)
else:
A = copy.deepcopy(B)
size = len(A)
P = functionStack()
for i in xrange(size):
for j in xrange(i + 1, size):
c, s = cg_qr(A, i, j)
kw = {'a': i, 'b': j, 'c': c, 's': s}
P.getin(Givens(A, **kw))
if abs(A[j][i]) < NumLimit:
A[j][i] = 0
if P_is_a_matrix:
e = matrix(size).data
e = P.pullout(e)
del P
P = matrix(e).T()
return P, matrix(A)
def Norm(A, p='inf', epoch_if_norm2=200):
if isinstance(A, matrix):
data = A.data
tdata = A.T().data
else:
data = A
A = matrix(data)
tdata = A.T().data
switch = {
1: lambda A: max([sum(abs(j) for j in i) for i in tdata]),
2: lambda A: (PowIter(A.T() * A).fit(epoch_if_norm2)**(1 / 2)),
'inf': lambda A: max([sum(abs(j) for j in i) for i in data]),
'F':
lambda A: math.sqrt(sum([data[i][i]**2 for i in range(len(data))]))
}
return switch[p](A)
def Chol(A):
size = A.size
if size[0] != size[1]:
return ValueError
if not check_symmetrical(A):
print 'Not symmetrical!\n'
return ValueError
aij = A.data
L = matrix(size, eye=False)
size = size[0]
for j in range(size):
try:
L[j][j] = math.sqrt(aij[j][j] - sum(dot_vec(L[j][:j], L[j][:j])))
except:
print 'error:Not positive!\n'
for i in range(j + 1, size):
try:
L[i][j] = (aij[i][j] -
sum(dot_vec(L[i][:j], L[j][:j]))) / L[j][j]
except:
print 'error:Not positive!\n'
return L
def __init__(self, A):
L, U = LU(A)
self.L = matrix(L)
self.U = matrix(U)
def Cond(A, p=2):
if isinstance(A, matrix):
inv = A.inv()
else:
inv = matrix(A).inv()
return Norm(A, p) * Norm(inv, p)