def parabolic_IDM(expr, x, x0=0, n=5, check=True):
"""
parabolic_IDM(f(x), x) -- parabolic iterate derivative matrix.
Contains in each row the power series for each iterate
where the columns represent the coefficients of the
power series. The IDM is a (n+1)*(n+1) matrix A where:
A_j_k = 1/k! the k-th derivative of the j-th iterate of f
evaluated at x=x0.
"""
# is it a parabolic function?
# the real parabolic-ness check
if check and not is_parabolic(expr, x, x0):
raise ValueError, "x0 == f(x0) must be a parabolic fixed point (f'(x0) == 1)"
# initialize data
ret = []
ser = x
# truncate power series each time,
# because keeping useless coefficients
# makes it so much slower. This is fast!
for i in xrange(n + 1):
# find Taylor series about x=x0
ser = taylor(ser.subs(x=expr), x, x0, n)
cof = get_coeff_list(ser, x, x0, n)
ret.append(cof)
# the coefficients of the identity function
top = [0, 1] + [0 for i in xrange(n - 1)]
return [top] + ret
def parabolic_flow_matrix(expr, t, x, x0=0, n=5):
"""
parabolic_flow_matrix(f(x), t, x)
The flow matrix for a parabolic function,
basically the coefficients of x and t.
This takes the flow coefficients (1-D list)
and 'creates' a dimension (2-D matrix).
"""
# this also checks parabolic-ness
coeffs = parabolic_flow_coeffs(expr, t, x, x0, n)
ret = []
# # extract coeffs from series
# coeffs = ser.coeffs(x)
# coeffs2 = [0 for k in xrange(n + 1)]
# for cof in coeffs:
# try:
# coeffs2[cof[1]] = cof[0]
# except:
# pass
for i in xrange(n):
if (i < 2):
ret.append([coeffs[i]])
else:
ser = coeffs[i]
# cof = coeffs[i].expand().coeffs(t)
# extract coeffs from series
cof = get_coeff_list(ser, t, 0, i-1)
ret.append(cof)
return ret
def Carleman_matrix(expr, x, x0=0, n_row=5, n_col=0, limit=False):
"""
Carleman_matrix(f(x), x) -- Carleman matrix of f(x)
Carleman_matrix(f(x), x, x0) -- Carleman matrix about x=x0
Carleman_matrix(f(x), x, x0, n) -- (n+1)*(n+1) Carleman matrix
The Carleman matrix,
from which the other 2 matrices are derived
"""
if n_col == 0: n_col = n_row
# initialize data
ser = 1
ret = [[1] + [0 for k in xrange(n_col)]]
for j in xrange(1, n_row + 1):
# find Taylor series about x=x0
ser = taylor(ser*expr, x, x0, n_col)
cof = get_coeff_list(ser, x, x0, n_col)
ret.append(cof)
return matrix(ret)