def expms(m, context=FloatContext, order=7):
"""Calculate quickly exp(t*A), for different t.
Takes matrix M and returns function f(t) = exp(t*M).
Each call to that function requires no matrix multiplications and several additions
"""
a = _exp_pade_coeffs(order)
n, n_ = shape_mat(m)
assert (n == n_)
#Prepare list of m powers and Pade coefficients
m_a_list = zip(matrix_powers(m), a)
def calculate_expm(t):
num = zeros(n, n)
den = zeros(n, n)
s = 1
k = context.one # t^n is stored here
for m_i, ai in m_a_list:
aik = ai * k
mat_combine_inplace(num, m_i, aik)
mat_combine_inplace(den, m_i, s * aik)
s = -s
k *= t
return solve(den, num)
return calculate_expm
def qr_inverse(M, context=FloatContext, eps=1e-14):
"""Matrix inverse, using QR transformation"""
n, n_ = shape_mat(M)
assert (n == n_)
IQ, R = qrl_givens(M, context=context, eps=eps, inverse_q=True)
IR = ltri_inverse(R, context=context)
return mmul(IR, IQ)
def expms( m, context=FloatContext, order=7 ):
"""Calculate quickly exp(t*A), for different t.
Takes matrix M and returns function f(t) = exp(t*M).
Each call to that function requires no matrix multiplications and several additions
"""
a = _exp_pade_coeffs( order )
n,n_ = shape_mat( m )
assert( n == n_ )
#Prepare list of m powers and Pade coefficients
m_a_list = zip( matrix_powers(m), a )
def calculate_expm(t):
num = zeros(n,n)
den = zeros(n,n)
s = 1
k = context.one # t^n is stored here
for m_i, ai in m_a_list:
aik = ai*k
mat_combine_inplace( num, m_i, aik )
mat_combine_inplace( den, m_i, s*aik)
s = -s
k *= t
return solve( den, num )
return calculate_expm
def qr_inverse( M, context=FloatContext, eps=1e-14):
"""Matrix inverse, using QR transformation"""
n,n_ = shape_mat(M)
assert(n==n_)
IQ,R = qrl_givens(M, context=context, eps=eps, inverse_q = True)
IR = ltri_inverse(R, context=context)
return mmul( IR, IQ )
def expm(m, context=FloatContext, order=7):
"""Matrix exponent, using Pade approximation of given order
"""
a = _exp_pade_coeffs(order)
n, n_ = shape_mat(m)
assert (n == n_)
num = zeros(n, n)
den = zeros(n, n)
s = 1
for m_i, ai in izip(matrix_powers(m), a):
mat_combine_inplace(num, m_i, ai)
mat_combine_inplace(den, m_i, s * ai)
s = -s
return solve(den, num)
def expm( m, context=FloatContext, order=7 ):
"""Matrix exponent, using Pade approximation of given order
"""
a = _exp_pade_coeffs( order )
n,n_ = shape_mat( m )
assert( n == n_ )
num = zeros(n,n)
den = zeros(n,n)
s = 1
for m_i, ai in izip( matrix_powers(m), a ):
mat_combine_inplace( num, m_i, ai )
mat_combine_inplace( den, m_i, s*ai)
s = -s
return solve( den, num )
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