def matrix_powers(m):
"""Generates infinite sequence of matrix powers, starting from m^0"""
yield eye(len(m)) # m^0
yield m # m^1
m_i = mmul(m, m)
while True:
yield m_i
m_i = mmul(m_i, m)
def matrix_powers(m):
"""Generates infinite sequence of matrix powers, starting from m^0"""
yield eye(len(m)) # m^0
yield m # m^1
m_i = mmul( m, m )
while True:
yield m_i
m_i = mmul( m_i, m )
def ltri_inverse(M, context=FloatContext):
"""Inverse of the left-triangular matrix"""
R = eye(len(M), context=context)
for i in xrange(len(M)):
m_ii = M[i][i] #diagonal element
r_i = R[i]
#normalize
r_i[:i + 1] = (x / m_ii for x in r_i[:i + 1])
#now subtract it from other rows
for j in xrange(i + 1, len(M)):
r_j = R[j]
m_ji = M[j][i]
r_j[:i + 1] = (x - m_ji * y
for x, y in izip(r_j[:i + 1], r_i[:i + 1]))
return R
def ltri_inverse( M, context=FloatContext):
"""Inverse of the left-triangular matrix"""
R = eye(len(M),context=context)
for i in xrange(len(M)):
m_ii = M[i][i] #diagonal element
r_i = R[i]
#normalize
r_i[:i+1] = ( x / m_ii for x in r_i[:i+1])
#now subtract it from other rows
for j in xrange(i+1,len(M)):
r_j = R[j]
m_ji = M[j][i]
r_j[:i+1] = ( x - m_ji * y
for x,y in izip( r_j[:i+1], r_i[:i+1] ) )
return R
def qrl_givens(M, eps=1e-14, context=FloatContext, inverse_q=False):
"""QR decomposition of a rectangular matrix. R is right-triangular"""
n, m = shape_mat(M)
R = copy_mat(M) #will be used for inplace computations
Q = eye(n, context=context)
for j in xrange(min(n, m) - 1, 0, -1): #from last column to the first
for i in xrange(j):
k_sin = R[i][j]
k_cos = R[i + 1][j]
k2 = k_sin**2 + k_cos**2
if k2 < eps: continue
k = context.sqrt(k2)
s = k_sin / k
c = k_cos / k
givens_inplace(Q, i, i + 1, -s, c)
givens_inplace(R, i, i + 1, -s, c)
if inverse_q: return Q, R
else: return transpose(Q), R
def qrl_givens(M, eps=1e-14, context=FloatContext, inverse_q=False ):
"""QR decomposition of a rectangular matrix. R is right-triangular"""
n,m = shape_mat(M)
R = copy_mat(M) #will be used for inplace computations
Q = eye( n, context=context )
for j in xrange(min(n,m)-1,0,-1): #from last column to the first
for i in xrange(j):
k_sin = R[i ][j]
k_cos = R[i+1][j]
k2 = k_sin**2 + k_cos**2
if k2 < eps: continue
k = context.sqrt(k2)
s = k_sin / k
c = k_cos / k
givens_inplace( Q, i, i+1, -s, c )
givens_inplace( R, i, i+1, -s, c )
if inverse_q: return Q,R
else: return transpose(Q), R