<SYSTEM_TASK:>
Given the following text description, write Python code to implement the functionality described below step by step
<END_TASK>
<USER_TASK:>
Description:
 <p>
<img src="http
Step1:  Catalan array $\mathcal{C}$
Step2:  power function
Step3:  inverse function
Step4:  sqrt function
Step5:  expt function
Step6:  log function
Step7:  sin function
Step8:  cos function
<ASSISTANT_TASK:>
Python Code:
from sympy import *
from sympy.abc import n, i, N, x, lamda, phi, z, j, r, k, a, alpha
from commons import *
from matrix_functions import *
from sequences import *
import functions_catalog
init_printing()
Explanation: <p>
<img src="http://www.cerm.unifi.it/chianti/images/logo%20unifi_positivo.jpg" 
        alt="UniFI logo" style="float: left; width: 20%; height: 20%;">
<div align="right">
Massimo Nocentini<br>
<small>
<br>April 20, 2018: cleanup, `sin` and `cos` $g$ poly
<br>November 16 and 18, 2016: `expt` and `log` $g$ poly
<br>November 9, 2016: splitting from "big" notebook
</small>
</div>
</p>
<br>
<br>
<div align="center">
<b>Abstract</b><br>
Theory of matrix functions, with applications to Catalan array $\mathcal{C}$.
</div>
End of explanation
m=8
C = define(let=Symbol(r'\mathcal{{C}}_{{ {} }}'.format(m)), 
           be=Matrix(m, m, lambda n,k: (k+1)*binomial(2*n-k, n-k)/(n+1) if n > 0 else int(not k)))
C
eigendata = spectrum(C)
eigendata
data, eigenvals, multiplicities = eigendata.rhs
Phi_poly = Phi_poly_ctor(deg=m-1)
Phi_poly
Phi_polynomials = component_polynomials(eigendata, early_eigenvals_subs=True)
Phi_polynomials
cmatrices = component_matrices(C, Phi_polynomials)
list(cmatrices.values())
Explanation: Catalan array $\mathcal{C}$
End of explanation
f_power, g_power, G_power = functions_catalog.power(eigendata, Phi_polynomials)
C_power = G_power(C)
C_power
define(C_power.lhs, C_power.rhs.applyfunc(factor)) # factored
assert C_power.rhs == (C.rhs**r).applyfunc(simplify)
inspect(C_power.rhs)
production_matrix(C_power.rhs).applyfunc(factor)
Explanation: power function
End of explanation
f_inverse, g_inverse, G_inverse = functions_catalog.inverse(eigendata, Phi_polynomials)
C_inverse = G_inverse(C)
C_inverse, G_inverse(C_inverse)
inspect(C_inverse.rhs)
production_matrix(C_inverse.rhs)
assert C_inverse.rhs*C.rhs == Matrix(m, m, identity_matrix())
assert C_inverse.rhs == C_power.rhs.subs({r:-1})
Explanation: inverse function
End of explanation
f_sqrt, g_sqrt, G_sqrt = functions_catalog.square_root(eigendata, Phi_polynomials)
C_sqrt = G_sqrt(C)
C_sqrt
inspect(C_sqrt.rhs)
production_matrix(C_sqrt.rhs)
assert C.rhs**(S(1)/2) == C_sqrt.rhs
assert C_sqrt.rhs == C_power.rhs.subs({r:S(1)/2})
Explanation: sqrt function
End of explanation
f_exp, g_exp, G_exp = functions_catalog.exp(eigendata, Phi_polynomials)
C_exp = G_exp(C)
C_exp
define(C_exp.lhs, C_exp.rhs.applyfunc(factor))
C_exp1 = define(let=Subs(C_exp.lhs, alpha, 1), be=C_exp.rhs.subs({alpha:1}))
C_exp1
inspect(C_exp.rhs)
inspect(C_exp1.rhs)
eigendata_Cexpt = spectrum(C_exp1)
eigendata_Cexpt
Phi_polynomials_Cexpt = component_polynomials(eigendata_Cexpt, early_eigenvals_subs=True)
Phi_polynomials_Cexpt
Explanation: expt function
End of explanation
f_log, g_log, G_log = functions_catalog.log(eigendata, Phi_polynomials)
C_log = G_log(C)
C_log
inspect(C_log.rhs[1:,:-1])
production_matrix(C_log.rhs[1:,:-1])
g_log_Cexpt = Hermite_interpolation_polynomial(f_log, eigendata_Cexpt, Phi_polynomials_Cexpt)
g_log_Cexpt
g_log_Cexpt = g_log_Cexpt.subs(eigendata_Cexpt.rhs[1])
g_log_Cexpt
with lift_to_matrix_function(g_log_Cexpt) as G_log_Cexpt:
    CC = G_log_Cexpt(C_exp1)
CC
Explanation: log function
End of explanation
f_sin, g_sin, G_sin = functions_catalog.sin(eigendata, Phi_polynomials)
C_sin = G_sin(C)    
C_sin
inspect(C_sin.rhs) # takes long to evaluate
Explanation: sin function
End of explanation
f_cos, g_cos, G_cos = functions_catalog.cos(eigendata, Phi_polynomials)
C_cos = G_cos(C)    
C_cos
inspect(C_sin.rhs) # takes long to evaluate
assert (C_sin.rhs**2 + C_cos.rhs**2).applyfunc(trigsimp) == Matrix(m,m, identity_matrix())
Explanation: cos function
End of explanation
<END_TASK>