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Black hole quasinormal modes
using the asymptotic iteration method

H.T. Cho, A.S. Cornell, Jason Doukas and Wade Naylor, Class. Quantum Grav. 27 (2010) 155004 (12pp).
https://iopscience.iop.org/article/10.1088/0264-9381/27/15/155004/meta

Equations 33 and 34 in the paper.

$\large \lambda_0(\xi) = -{1\over p} \left[ p' - {2 i\omega \over \kappa_1(\xi - \xi_1)} -2 i \omega \right]$

$\large s_0(\xi) = {1\over p} \left[ \ell(\ell +1) + (1-s^2) \left( 2M\xi - (4-s^2){\Lambda \over 6\xi^2} \right) + {i\omega \over \kappa_1(\xi - \xi_1)^2} \left( {i\omega \over \kappa_1} +1 \right) + (p' - 2i\omega) {i\omega \over \kappa_1(\xi - \xi_1)} \right] $

Import AIM library





In [1]:



    


# Python program to use AIM tools
from asymptotic import *

# a high precision math library: 
# it is used to obtain roots of a function with high precision.
import mpmath

# To draw potentials and its derivatie
import numpy as np

%matplotlib inline
import matplotlib.pyplot as plt

Definitions

Variables





In [2]:



    


# symengine (symbolic) variables for lambda_0 and s_0 
En, r, xi, xi1, k1 = se.symbols(r'En, r, xi, xi_1, kappa_1')
l, s, M, Lambda = se.symbols(r'\ell, s, M, Lambda')

Potential and some other functions





In [3]:



    


def Vf(r):
    """
    Master potential defined in Eq. 2 in the paper.
    """
    return f(r)*(l*(l+1)/r**2 + (1-s**2) * (2*M/r**3 - (4-s**2)*Lambda/6))

def f(r):
    """
    A function defined in Eq 3. with the cosmological constant Lambda .
    """
    return 1 - 2*M/r  - Lambda*r**2/3

def dVf(r0):
    """
    Derivative of the master potential.
    """
    return sym.diff(Vf(r), r).subs(r, r0)

def dVf_mpmath(r0):
    """
    Derivative of the master potential with numerical parameters.
    """
    dV = dVf(r).subs({l:nl, s: ns, Lambda: nLambda, M: nM})
    dVs = sym.lambdify(r, dV, 'mpmath')
    return dVs(r0)
    
def p(xi):
    """
    It is defined with Eq. 26 in the paper.
    """
    return xi**2 - 2*M*xi**3 - Lambda/3

def kappa(xi):
    """
    It is defined with Eq. 31 in the paper.
    """    
    return M*xi**2 - Lambda/(3*xi)



    











In [4]:



    


display(Latex('Master potential and its derivative:'))
symPrintFunc(Vf, (r,))
symPrintFunc(dVf, (r,))
display(Latex(r'Equation 26 in the paper.'))
symPrintFunc(p, (xi,))
display(Latex(r'$\kappa_1$ is the surface gravity at the event horizon $\xi_1$'))
symPrintFunc(kappa, (xi,))



    


















    







Master potential and its derivative:









    







$\displaystyle Vf(r) = \left(\frac{\ell \left(\ell + 1\right)}{r^{2}} + \left(1 - s^{2}\right) \left(- \frac{\Lambda \left(4 - s^{2}\right)}{6} + \frac{2 M}{r^{3}}\right)\right) \left(- \frac{\Lambda r^{2}}{3} - \frac{2 M}{r} + 1\right)$









    







$\displaystyle dVf(r) = \left(- \frac{2 \Lambda r}{3} + \frac{2 M}{r^{2}}\right) \left(\frac{\ell \left(\ell + 1\right)}{r^{2}} + \left(1 - s^{2}\right) \left(- \frac{\Lambda \left(4 - s^{2}\right)}{6} + \frac{2 M}{r^{3}}\right)\right) + \left(- \frac{6 M \left(1 - s^{2}\right)}{r^{4}} - \frac{2 \ell \left(\ell + 1\right)}{r^{3}}\right) \left(- \frac{\Lambda r^{2}}{3} - \frac{2 M}{r} + 1\right)$









    







Equation 26 in the paper.









    







$\displaystyle p(\xi) = - \frac{\Lambda}{3} - 2 M \xi^{3} + \xi^{2}$









    







$\kappa_1$ is the surface gravity at the event horizon $\xi_1$









    







$\displaystyle \kappa(\xi) = - \frac{\Lambda}{3 \xi} + M \xi^{2}$

$\lambda_0$ and $s_0$





In [5]:



    


def lambda_0(xi):
    return (6*(-se.I*En + xi - 3*M*xi**2))/(-3*xi**2 + 6*M*xi**3 + Lambda) + (2*se.I*En)/((xi - xi1)*k1)

def s_0(xi):
    return (2*En**2*xi**2*(-3*xi**2 + 6*M*xi**3 + Lambda) - \
            2*En*xi**2*(6*En*(xi - xi1) - 3*se.I*xi*(2*xi1 + xi*(-1 + 4*M*xi - 6*M*xi1)) + se.I*Lambda)*k1 + \
            (xi - xi1)**2*(-6*xi**2*(l + l**2 - 2*M*(-1 + s**2)*xi) + (4 - 5*s**2 + s**4)*Lambda)*k1**2)/ \
            (2*xi**2*(xi - xi1)**2*(-3*xi**2 + 6*M*xi**3 + Lambda)*k1**2)

Case: $l = 2, s =2$

$\Lambda = 0$

Numerical values for variables





In [6]:



    


# values of variables
nl, ns = 2, 2 # angular mom. and spin
nM = 1 # mass of the black hole

nLambda = o* 0

dprec=500
tol=1e-101
ft.ctx.dps = dprec
mpmath.mp.dps = dprec

sr0 = 3 # initial search point of the root of the der. pot.
nr0 = sym.S(mpmath.findroot(dVf_mpmath, sr0, tol=tol))
nxi0 = o* 1/nr0

solhor = sym.solve(p(xi).subs({Lambda: nLambda, M:nM}), xi)
nxi1 = sym.N(max(map(lambda x: sym.re(x), solhor)), dprec)
#nxi1 = max(map(lambda x: sym.re(x), solhor)) # with infinite precision
nk1 = kappa(nxi1).subs({Lambda: nLambda, M: nM})

display(Math(r"r_0 = %.20f"%sym.N(nr0)))
display(Math(r"\xi_0 = %.20f"%sym.N(nxi0)))
display(Math(r"\xi_1 = %.20f"%sym.N(nxi1)))
display(Math(r"k_1 = %.20f"%sym.N(nk1)))



    


















    







$\displaystyle r_0 = 3.28077640640441492437$









    







$\displaystyle \xi_0 = 0.30480589839889621340$









    







$\displaystyle \xi_1 = 0.50000000000000000000$









    







$\displaystyle k_1 = 0.25000000000000000000$
















In [7]:



    


# pl0 and ps0 to put the numerical values of variables into lambda_0 and s_0. 
pl0 = lambda: {k1: nk1, xi1: nxi1, Lambda: nLambda, M: nM}
ps0 = lambda: {k1: nk1, xi1: nxi1, Lambda: nLambda, M: nM, l: nl, s: ns, }

Drawing of the Master potential and its derivative





In [8]:



    


rspace = np.linspace(1e-10, 30, 1500)

Vf_numpy = sym.lambdify(r, Vf(r).subs({l:nl, s: ns, Lambda: nLambda, M: nM}), "numpy")
dVf_numpy = sym.lambdify(r, dVf_mpmath(r), "numpy")

Vf(r).subs({l:nl, s: ns, Lambda: nLambda, M: nM})


plt.figure(figsize=(12, 8))
plt.plot(rspace, Vf_numpy(rspace), color="red", lw=2, label="Master potential")
plt.plot(rspace, dVf_numpy(rspace), color="blue", lw=2, label="the derivative")
plt.plot(nr0, dVf_numpy(nr0), "go", ms=10, label="an extremum of the pot.")

plt.xlim(0.5, 8)
plt.ylim(-.75, .75)

plt.xlabel('r', fontsize=24)
plt.ylabel('V(r), dV(r)/dr', fontsize=24)

plt.axhline(color="black", zorder=-999)
plt.legend(loc="best")
plt.grid()
plt.show()

Initialize AIM solver





In [9]:



    


%%time
# pass lambda_0, s_0 and variable values to aim class
black_L00l2s2 = aim(lambda_0(xi), s_0(xi), pl0(), ps0())
black_L00l2s2.display_parameters(pprec=7)
black_L00l2s2.display_l0s0(0, pprec=3)
black_L00l2s2.parameters(En, xi, nxi0, nmax=71, nstep=10, dprec=500, tol=1e-101)



    


















    







$\displaystyle \text{Values of the parameters of } \lambda_0:$









    







$\displaystyle M = 1.000000,~~\Lambda = 0,~~\kappa_{1} = 0.2500000,~~\xi_{1} = 0.5000000$









    







$\displaystyle \text{Values of the parameters of } s_0:$









    







$\displaystyle M = 1.000000,~~\Lambda = 0,~~\ell = 2.000000,~~\kappa_{1} = 0.2500000,~~\xi_{1} = 0.5000000,~~s = 2.000000$









    







$\displaystyle            
                \begin{align}
                \lambda_0 &= \left[\frac{2 i En}{\kappa_{1} \left(\xi - \xi_{1}\right)} + \frac{6 \left(- i En - 3 M \xi^{2} + \xi\right)}{\Lambda + 6 M \xi^{3} - 3 \xi^{2}} \right.\\ 
                      s_0 &= \left[\frac{2 En^{2} \xi^{2} \left(\Lambda + 6 M \xi^{3} - 3 \xi^{2}\right) - 2 En \kappa_{1} \xi^{2} \left(6 En \left(\xi - \xi_{1}\right) + i \Lambda - 3 i \xi \left(\xi \left(4 M \xi - 6 M \xi_{1} - 1\right) + 2 \xi_{1}\right)\right) + \kappa_{1}^{2} \left(\xi - \xi_{1}\right)^{2} \left(\Lambda \left(s^{4} - 5 s^{2} + 4\right) - 6 \xi^{2} \left(- 2 M \xi \left(s^{2} - 1\right) + \ell^{2} + \ell\right)\right)}{2 \kappa_{1}^{2} \xi^{2} \left(\xi - \xi_{1}\right)^{2} \left(\Lambda + 6 M \xi^{3} - 3 \xi^{2}\right)} \right.\\ 
                \lambda_0 &= \left[\frac{8.0 i En}{\xi - 0.5} + \frac{6.0 \left(- i En - 3.0 \xi^{2} + \xi\right)}{6.0 \xi^{3} - 3.0 \xi^{2}} \right.\\ 
                      s_0 &= \left[\frac{8.0 \left(2.0 En^{2} \xi^{2} \left(6.0 \xi^{3} - 3.0 \xi^{2}\right) - 0.5 En \xi^{2} \left(6.0 En \left(\xi - 0.5\right) - 3.0 i \xi \left(\xi \left(4.0 \xi - 4.0\right) + 1.0\right)\right) - 0.375 \xi^{2} \left(6.0 - 6.0 \xi\right) \left(\xi - 0.5\right)^{2}\right)}{\xi^{2} \left(\xi - 0.5\right)^{2} \left(6.0 \xi^{3} - 3.0 \xi^{2}\right)} \right.
                \end{align}
                $









    






CPU times: user 91.2 ms, sys: 3.43 ms, total: 94.7 ms
Wall time: 96.4 ms

Calculation necessary coefficients





In [10]:



    


%%time
# create coefficients c0 and d0 for improved AIM
black_L00l2s2.c0()
black_L00l2s2.d0()
black_L00l2s2.cndn()



    


















    






CPU times: user 16.5 s, sys: 93.5 ms, total: 16.6 s
Wall time: 16.6 s

The solution





In [11]:



    


%%time
black_L00l2s2.get_arb_roots()



    


















    






CPU times: user 14.3 s, sys: 32.8 ms, total: 14.4 s
Wall time: 14.3 s

















In [12]:



    


%%time
print("iter" + (3*"{:^24s}").format(*("n=%s"%str(i) for i in range(1,4))))
print("_"*3*24)
black_L00l2s2.printAllRoots(showRoots='+r-i', printFormat="{:10.7f}", col=(0,3))



    


















    






iter          n=1                     n=2                     n=3           
________________________________________________________________________
0001  0.3609169-0.0937215j  0.4144812-0.2973979j
0011  0.3736718-0.0889627j  0.3467437-0.2739338j  0.3011186-0.4770297j
0021  0.3736717-0.0889623j  0.3467118-0.2739145j  0.3010188-0.4783149j
0031  0.3736717-0.0889623j  0.3467110-0.2739149j  0.3010527-0.4782826j
0041  0.3736717-0.0889623j  0.3467110-0.2739149j  0.3010531-0.4782778j
0051  0.3736717-0.0889623j  0.3467110-0.2739149j  0.3010533-0.4782770j
0061  0.3736717-0.0889623j  0.3467110-0.2739149j  0.3010534-0.4782769j
0071  0.3736717-0.0889623j  0.3467110-0.2739149j  0.3010535-0.4782770j
CPU times: user 217 ms, sys: 13 µs, total: 217 ms
Wall time: 215 ms

$\Lambda = 0.02$





In [13]:



    


%%time
# values of variables
nLambda = o* 2/100

sr0 = 3 # initial search point of the root of the der. pot.
nr0 = sym.S(mpmath.findroot(dVf_mpmath, sr0, tol=tol))
nxi0 = o* 1/nr0

solhor = sym.solve(p(xi).subs({Lambda: nLambda, M:nM}), xi)
nxi1 = sym.N(max(map(lambda x: sym.re(x), solhor)), dprec)
nk1 = kappa(nxi1).subs({Lambda: nLambda, M: nM})

display(Math(r"r_0 = %.20f"%sym.N(nr0)))
display(Math(r"\xi_0 = %.20f"%sym.N(nxi0)))
display(Math(r"\xi_1 = %.20f"%sym.N(nxi1)))
display(Math(r"k_1 = %.20f"%sym.N(nk1)))


# pass lambda_0, s_0 and variable values to aim class
black_L02l2s2 = aim(lambda_0(xi), s_0(xi), pl0(), ps0())
black_L02l2s2.display_parameters(pprec=7)
black_L02l2s2.display_l0s0(0, pprec=3)
black_L02l2s2.parameters(En, xi, nxi0, nmax=71, nstep=10, dprec=500, tol=1e-101)

# create coefficients c0 and d0 for improved AIM
black_L02l2s2.c0()
black_L02l2s2.d0()
black_L02l2s2.cndn()

# the solution
print("iter" + (3*"{:^24s}").format(*("n=%s"%str(i) for i in range(1,4))))
print("_"*3*24)
black_L02l2s2.get_arb_roots(showRoots='+r-i', printFormat="  {:10.7f}", col=(0,3))



    


















    







$\displaystyle r_0 = 3.22503324240581257243$









    







$\displaystyle \xi_0 = 0.31007432321969469191$









    







$\displaystyle \xi_1 = 0.48588048315289572576$









    







$\displaystyle k_1 = 0.22235904857377850052$









    







$\displaystyle \text{Values of the parameters of } \lambda_0:$









    







$\displaystyle M = 1.000000,~~\Lambda = 0.02000000,~~\kappa_{1} = 0.2223590,~~\xi_{1} = 0.4858805$









    







$\displaystyle \text{Values of the parameters of } s_0:$









    







$\displaystyle M = 1.000000,~~\Lambda = 0.02000000,~~\ell = 2.000000,~~\kappa_{1} = 0.2223590,~~\xi_{1} = 0.4858805,~~s = 2.000000$









    







$\displaystyle            
                \begin{align}
                \lambda_0 &= \left[\frac{2 i En}{\kappa_{1} \left(\xi - \xi_{1}\right)} + \frac{6 \left(- i En - 3 M \xi^{2} + \xi\right)}{\Lambda + 6 M \xi^{3} - 3 \xi^{2}} \right.\\ 
                      s_0 &= \left[\frac{2 En^{2} \xi^{2} \left(\Lambda + 6 M \xi^{3} - 3 \xi^{2}\right) - 2 En \kappa_{1} \xi^{2} \left(6 En \left(\xi - \xi_{1}\right) + i \Lambda - 3 i \xi \left(\xi \left(4 M \xi - 6 M \xi_{1} - 1\right) + 2 \xi_{1}\right)\right) + \kappa_{1}^{2} \left(\xi - \xi_{1}\right)^{2} \left(\Lambda \left(s^{4} - 5 s^{2} + 4\right) - 6 \xi^{2} \left(- 2 M \xi \left(s^{2} - 1\right) + \ell^{2} + \ell\right)\right)}{2 \kappa_{1}^{2} \xi^{2} \left(\xi - \xi_{1}\right)^{2} \left(\Lambda + 6 M \xi^{3} - 3 \xi^{2}\right)} \right.\\ 
                \lambda_0 &= \left[\frac{8.99 i En}{\xi - 0.486} + \frac{6.0 \left(- i En - 3.0 \xi^{2} + \xi\right)}{6.0 \xi^{3} - 3.0 \xi^{2} + 0.02} \right.\\ 
                      s_0 &= \left[\frac{10.1 \left(2.0 En^{2} \xi^{2} \left(6.0 \xi^{3} - 3.0 \xi^{2} + 0.02\right) - 0.445 En \xi^{2} \left(6.0 En \left(\xi - 0.486\right) - 3.0 i \xi \left(\xi \left(4.0 \xi - 3.92\right) + 0.972\right) + 0.02 i\right) - 0.297 \xi^{2} \left(6.0 - 6.0 \xi\right) \left(\xi - 0.486\right)^{2}\right)}{\xi^{2} \left(\xi - 0.486\right)^{2} \left(6.0 \xi^{3} - 3.0 \xi^{2} + 0.02\right)} \right.
                \end{align}
                $









    






iter          n=1                     n=2                     n=3           
________________________________________________________________________
0001    0.3283549  -0.0856613j    0.3690909  -0.2686128j
0011    0.3383914  -0.0817565j    0.3187579  -0.2492120j    0.2831809  -0.4292619j
0021    0.3383914  -0.0817564j    0.3187586  -0.2491967j    0.2827376  -0.4294769j
0031    0.3383914  -0.0817564j    0.3187587  -0.2491966j    0.2827324  -0.4294843j
0041    0.3383914  -0.0817564j    0.3187587  -0.2491966j    0.2827322  -0.4294841j
0051    0.3383914  -0.0817564j    0.3187587  -0.2491966j    0.2827322  -0.4294841j
0061    0.3383914  -0.0817564j    0.3187587  -0.2491966j    0.2827322  -0.4294841j
0071    0.3383914  -0.0817564j    0.3187587  -0.2491966j    0.2827322  -0.4294841j
CPU times: user 52.4 s, sys: 308 ms, total: 52.7 s
Wall time: 52.6 s

$\Lambda = 0.04$





In [14]:



    


%%time
# values of variables
nLambda = o* 4/100

sr0 = 3 # initial search point of the root of the der. pot.
nr0 = sym.S(mpmath.findroot(dVf_mpmath, sr0, tol=tol))
nxi0 = o* 1/nr0

solhor = sym.solve(p(xi).subs({Lambda: nLambda, M:nM}), xi)
nxi1 = sym.N(max(map(lambda x: sym.re(x), solhor)), dprec)
nk1 = kappa(nxi1).subs({Lambda: nLambda, M: nM})

display(Math(r"r_0 = %.20f"%sym.N(nr0)))
display(Math(r"\xi_0 = %.20f"%sym.N(nxi0)))
display(Math(r"\xi_1 = %.20f"%sym.N(nxi1)))
display(Math(r"k_1 = %.20f"%sym.N(nk1)))


# pass lambda_0, s_0 and variable values to aim class
black_L04l2s2 = aim(lambda_0(xi), s_0(xi), pl0(), ps0())
black_L04l2s2.display_parameters(pprec=7)
black_L04l2s2.display_l0s0(0, pprec=3)
black_L04l2s2.parameters(En, xi, nxi0, nmax=71, nstep=10, dprec=500, tol=1e-101)

# create coefficients c0 and d0 for improved AIM
black_L04l2s2.c0()
black_L04l2s2.d0()
black_L04l2s2.cndn()

# the solution
print("iter" + (3*"{:^24s}").format(*("n=%s"%str(i) for i in range(1,4))))
print("_"*3*24)
black_L04l2s2.get_arb_roots(showRoots='+r-i', printFormat="  {:10.7f}", col=(0,3))



    


















    







$\displaystyle r_0 = 3.17182740602516677342$









    







$\displaystyle \xi_0 = 0.31527566667102113751$









    







$\displaystyle \xi_1 = 0.46979395281592567502$









    







$\displaystyle k_1 = 0.19232512149131092416$









    







$\displaystyle \text{Values of the parameters of } \lambda_0:$









    







$\displaystyle M = 1.000000,~~\Lambda = 0.04000000,~~\kappa_{1} = 0.1923251,~~\xi_{1} = 0.4697940$









    







$\displaystyle \text{Values of the parameters of } s_0:$









    







$\displaystyle M = 1.000000,~~\Lambda = 0.04000000,~~\ell = 2.000000,~~\kappa_{1} = 0.1923251,~~\xi_{1} = 0.4697940,~~s = 2.000000$









    







$\displaystyle            
                \begin{align}
                \lambda_0 &= \left[\frac{2 i En}{\kappa_{1} \left(\xi - \xi_{1}\right)} + \frac{6 \left(- i En - 3 M \xi^{2} + \xi\right)}{\Lambda + 6 M \xi^{3} - 3 \xi^{2}} \right.\\ 
                      s_0 &= \left[\frac{2 En^{2} \xi^{2} \left(\Lambda + 6 M \xi^{3} - 3 \xi^{2}\right) - 2 En \kappa_{1} \xi^{2} \left(6 En \left(\xi - \xi_{1}\right) + i \Lambda - 3 i \xi \left(\xi \left(4 M \xi - 6 M \xi_{1} - 1\right) + 2 \xi_{1}\right)\right) + \kappa_{1}^{2} \left(\xi - \xi_{1}\right)^{2} \left(\Lambda \left(s^{4} - 5 s^{2} + 4\right) - 6 \xi^{2} \left(- 2 M \xi \left(s^{2} - 1\right) + \ell^{2} + \ell\right)\right)}{2 \kappa_{1}^{2} \xi^{2} \left(\xi - \xi_{1}\right)^{2} \left(\Lambda + 6 M \xi^{3} - 3 \xi^{2}\right)} \right.\\ 
                \lambda_0 &= \left[\frac{10.4 i En}{\xi - 0.47} + \frac{6.0 \left(- i En - 3.0 \xi^{2} + \xi\right)}{6.0 \xi^{3} - 3.0 \xi^{2} + 0.04} \right.\\ 
                      s_0 &= \left[\frac{13.5 \left(2.0 En^{2} \xi^{2} \left(6.0 \xi^{3} - 3.0 \xi^{2} + 0.04\right) - 0.385 En \xi^{2} \left(6.0 En \left(\xi - 0.47\right) - 3.0 i \xi \left(\xi \left(4.0 \xi - 3.82\right) + 0.94\right) + 0.04 i\right) - 0.222 \xi^{2} \left(6.0 - 6.0 \xi\right) \left(\xi - 0.47\right)^{2}\right)}{\xi^{2} \left(\xi - 0.47\right)^{2} \left(6.0 \xi^{3} - 3.0 \xi^{2} + 0.04\right)} \right.
                \end{align}
                $









    






iter          n=1                     n=2                     n=3           
________________________________________________________________________
0001    0.2915300  -0.0762555j    0.3204505  -0.2365696j
0011    0.2988947  -0.0732967j    0.2858378  -0.2217255j    0.2600628  -0.3771886j
0021    0.2988947  -0.0732967j    0.2858409  -0.2217241j    0.2599916  -0.3770924j
0031    0.2988947  -0.0732967j    0.2858409  -0.2217241j    0.2599919  -0.3770922j
0041    0.2988947  -0.0732967j    0.2858409  -0.2217241j    0.2599919  -0.3770922j
0051    0.2988947  -0.0732967j    0.2858409  -0.2217241j    0.2599919  -0.3770922j
0061    0.2988947  -0.0732967j    0.2858409  -0.2217241j    0.2599919  -0.3770922j
0071    0.2988947  -0.0732967j    0.2858409  -0.2217241j    0.2599919  -0.3770922j
CPU times: user 5min 12s, sys: 1.79 s, total: 5min 14s
Wall time: 5min 13s

$\Lambda = 0.06$





In [15]:



    


%%time
# values of variables
nLambda = o* 6/100

sr0 = 3 # initial search point of the root of the der. pot.
nr0 = sym.S(mpmath.findroot(dVf_mpmath, sr0, tol=tol))
nxi0 = o* 1/nr0

solhor = sym.solve(p(xi).subs({Lambda: nLambda, M:nM}), xi)
nxi1 = sym.N(max(map(lambda x: sym.re(x), solhor)), dprec)
nk1 = kappa(nxi1).subs({Lambda: nLambda, M: nM})

display(Math(r"r_0 = %.20f"%sym.N(nr0)))
display(Math(r"\xi_0 = %.20f"%sym.N(nxi0)))
display(Math(r"\xi_1 = %.20f"%sym.N(nxi1)))
display(Math(r"k_1 = %.20f"%sym.N(nk1)))


# pass lambda_0, s_0 and variable values to aim class
black_L06l2s2 = aim(lambda_0(xi), s_0(xi), pl0(), ps0())
black_L06l2s2.display_parameters(pprec=7)
black_L06l2s2.display_l0s0(0, pprec=3)
black_L06l2s2.parameters(En, xi, nxi0, nmax=71, nstep=10, dprec=500, tol=1e-101)

# create coefficients c0 and d0 for improved AIM
black_L06l2s2.c0()
black_L06l2s2.d0()
black_L06l2s2.cndn()

# the solution
print("iter" + (3*"{:^24s}").format(*("n=%s"%str(i) for i in range(1,4))))
print("_"*3*24)
black_L06l2s2.get_arb_roots(showRoots='+r-i', printFormat="  {:10.7f}", col=(0,3))



    


















    







$\displaystyle r_0 = 3.12092802581297501519$









    







$\displaystyle \xi_0 = 0.32041751419099406828$









    







$\displaystyle \xi_1 = 0.45079027713765573715$









    







$\displaystyle k_1 = 0.15884534474787764813$









    







$\displaystyle \text{Values of the parameters of } \lambda_0:$









    







$\displaystyle M = 1.000000,~~\Lambda = 0.06000000,~~\kappa_{1} = 0.1588453,~~\xi_{1} = 0.4507903$









    







$\displaystyle \text{Values of the parameters of } s_0:$









    







$\displaystyle M = 1.000000,~~\Lambda = 0.06000000,~~\ell = 2.000000,~~\kappa_{1} = 0.1588453,~~\xi_{1} = 0.4507903,~~s = 2.000000$









    







$\displaystyle            
                \begin{align}
                \lambda_0 &= \left[\frac{2 i En}{\kappa_{1} \left(\xi - \xi_{1}\right)} + \frac{6 \left(- i En - 3 M \xi^{2} + \xi\right)}{\Lambda + 6 M \xi^{3} - 3 \xi^{2}} \right.\\ 
                      s_0 &= \left[\frac{2 En^{2} \xi^{2} \left(\Lambda + 6 M \xi^{3} - 3 \xi^{2}\right) - 2 En \kappa_{1} \xi^{2} \left(6 En \left(\xi - \xi_{1}\right) + i \Lambda - 3 i \xi \left(\xi \left(4 M \xi - 6 M \xi_{1} - 1\right) + 2 \xi_{1}\right)\right) + \kappa_{1}^{2} \left(\xi - \xi_{1}\right)^{2} \left(\Lambda \left(s^{4} - 5 s^{2} + 4\right) - 6 \xi^{2} \left(- 2 M \xi \left(s^{2} - 1\right) + \ell^{2} + \ell\right)\right)}{2 \kappa_{1}^{2} \xi^{2} \left(\xi - \xi_{1}\right)^{2} \left(\Lambda + 6 M \xi^{3} - 3 \xi^{2}\right)} \right.\\ 
                \lambda_0 &= \left[\frac{12.6 i En}{\xi - 0.451} + \frac{6.0 \left(- i En - 3.0 \xi^{2} + \xi\right)}{6.0 \xi^{3} - 3.0 \xi^{2} + 0.06} \right.\\ 
                      s_0 &= \left[\frac{19.8 \left(2.0 En^{2} \xi^{2} \left(6.0 \xi^{3} - 3.0 \xi^{2} + 0.06\right) - 0.318 En \xi^{2} \left(6.0 En \left(\xi - 0.451\right) - 3.0 i \xi \left(\xi \left(4.0 \xi - 3.7\right) + 0.902\right) + 0.06 i\right) - 0.151 \xi^{2} \left(6.0 - 6.0 \xi\right) \left(\xi - 0.451\right)^{2}\right)}{\xi^{2} \left(\xi - 0.451\right)^{2} \left(6.0 \xi^{3} - 3.0 \xi^{2} + 0.06\right)} \right.
                \end{align}
                $









    






iter          n=1                     n=2                     n=3           
________________________________________________________________________
0001    0.2484765  -0.0650221j    0.2667728  -0.1997838j
0011    0.2532892  -0.0630425j    0.2457417  -0.1897908j    0.2300666  -0.3191627j
0021    0.2532892  -0.0630425j    0.2457420  -0.1897910j    0.2300764  -0.3191573j
0031    0.2532892  -0.0630425j    0.2457420  -0.1897910j    0.2300764  -0.3191573j
0041    0.2532892  -0.0630425j    0.2457420  -0.1897910j    0.2300764  -0.3191573j
0051    0.2532892  -0.0630425j    0.2457420  -0.1897910j    0.2300764  -0.3191573j
0061    0.2532892  -0.0630425j    0.2457420  -0.1897910j    0.2300764  -0.3191573j
0071    0.2532892  -0.0630425j    0.2457420  -0.1897910j    0.2300764  -0.3191573j
CPU times: user 6min 29s, sys: 2.22 s, total: 6min 31s
Wall time: 6min 31s

$\Lambda = 0.08$





In [16]:



    


%%time
# values of variables
nLambda = o* 8/100

sr0 = 3 # initial search point of the root of the der. pot.
nr0 = sym.S(mpmath.findroot(dVf_mpmath, sr0, tol=tol))
nxi0 = o* 1/nr0

solhor = sym.solve(p(xi).subs({Lambda: nLambda, M:nM}), xi)
nxi1 = sym.N(max(map(lambda x: sym.re(x), solhor)), dprec)
nk1 = kappa(nxi1).subs({Lambda: nLambda, M: nM})

display(Math(r"r_0 = %.20f"%sym.N(nr0)))
display(Math(r"\xi_0 = %.20f"%sym.N(nxi0)))
display(Math(r"\xi_1 = %.20f"%sym.N(nxi1)))
display(Math(r"k_1 = %.20f"%sym.N(nk1)))


# pass lambda_0, s_0 and variable values to aim class
black_L08l2s2 = aim(lambda_0(xi), s_0(xi), pl0(), ps0())
black_L08l2s2.display_parameters(pprec=7)
black_L08l2s2.display_l0s0(0, pprec=3)
black_L08l2s2.parameters(En, xi, nxi0, nmax=71, nstep=10, dprec=500, tol=1e-101)

# create coefficients c0 and d0 for improved AIM
black_L08l2s2.c0()
black_L08l2s2.d0()
black_L08l2s2.cndn()

# the solution
print("iter" + (3*"{:^24s}").format(*("n=%s"%str(i) for i in range(1,4))))
print("_"*3*24)
black_L08l2s2.get_arb_roots(showRoots='+r-i', printFormat="  {:10.7f}", col=(0,3))



    


















    







$\displaystyle r_0 = 3.07213303248536773182$









    







$\displaystyle \xi_0 = 0.32550673731436557956$









    







$\displaystyle \xi_1 = 0.42680542981557051396$









    







$\displaystyle k_1 = 0.11968319494459116192$









    







$\displaystyle \text{Values of the parameters of } \lambda_0:$









    







$\displaystyle M = 1.000000,~~\Lambda = 0.08000000,~~\kappa_{1} = 0.1196832,~~\xi_{1} = 0.4268054$









    







$\displaystyle \text{Values of the parameters of } s_0:$









    







$\displaystyle M = 1.000000,~~\Lambda = 0.08000000,~~\ell = 2.000000,~~\kappa_{1} = 0.1196832,~~\xi_{1} = 0.4268054,~~s = 2.000000$









    







$\displaystyle            
                \begin{align}
                \lambda_0 &= \left[\frac{2 i En}{\kappa_{1} \left(\xi - \xi_{1}\right)} + \frac{6 \left(- i En - 3 M \xi^{2} + \xi\right)}{\Lambda + 6 M \xi^{3} - 3 \xi^{2}} \right.\\ 
                      s_0 &= \left[\frac{2 En^{2} \xi^{2} \left(\Lambda + 6 M \xi^{3} - 3 \xi^{2}\right) - 2 En \kappa_{1} \xi^{2} \left(6 En \left(\xi - \xi_{1}\right) + i \Lambda - 3 i \xi \left(\xi \left(4 M \xi - 6 M \xi_{1} - 1\right) + 2 \xi_{1}\right)\right) + \kappa_{1}^{2} \left(\xi - \xi_{1}\right)^{2} \left(\Lambda \left(s^{4} - 5 s^{2} + 4\right) - 6 \xi^{2} \left(- 2 M \xi \left(s^{2} - 1\right) + \ell^{2} + \ell\right)\right)}{2 \kappa_{1}^{2} \xi^{2} \left(\xi - \xi_{1}\right)^{2} \left(\Lambda + 6 M \xi^{3} - 3 \xi^{2}\right)} \right.\\ 
                \lambda_0 &= \left[\frac{16.7 i En}{\xi - 0.427} + \frac{6.0 \left(- i En - 3.0 \xi^{2} + \xi\right)}{6.0 \xi^{3} - 3.0 \xi^{2} + 0.08} \right.\\ 
                      s_0 &= \left[\frac{34.9 \left(2.0 En^{2} \xi^{2} \left(6.0 \xi^{3} - 3.0 \xi^{2} + 0.08\right) - 0.239 En \xi^{2} \left(6.0 En \left(\xi - 0.427\right) - 3.0 i \xi \left(\xi \left(4.0 \xi - 3.56\right) + 0.854\right) + 0.08 i\right) - 0.0859 \xi^{2} \left(6.0 - 6.0 \xi\right) \left(\xi - 0.427\right)^{2}\right)}{\xi^{2} \left(\xi - 0.427\right)^{2} \left(6.0 \xi^{3} - 3.0 \xi^{2} + 0.08\right)} \right.
                \end{align}
                $









    






iter          n=1                     n=2                     n=3           
________________________________________________________________________
0001    0.1949902  -0.0509151j    0.2041500  -0.1550700j
0011    0.1974823  -0.0498773j    0.1941148  -0.1497866j    0.1871197  -0.2502566j
0021    0.1974823  -0.0498773j    0.1941148  -0.1497866j    0.1871198  -0.2502570j
0031    0.1974823  -0.0498773j    0.1941148  -0.1497866j    0.1871198  -0.2502570j
0041    0.1974823  -0.0498773j    0.1941148  -0.1497866j    0.1871198  -0.2502570j
0051    0.1974823  -0.0498773j    0.1941148  -0.1497866j    0.1871198  -0.2502570j
0061    0.1974823  -0.0498773j    0.1941148  -0.1497866j    0.1871198  -0.2502570j
0071    0.1974823  -0.0498773j    0.1941148  -0.1497866j    0.1871198  -0.2502570j
CPU times: user 51.2 s, sys: 296 ms, total: 51.5 s
Wall time: 51.4 s

$\Lambda = 0.09$





In [17]:



    


%%time
# values of variables
nLambda = o* 9/100

sr0 = 3 # initial search point of the root of the der. pot.
nr0 = sym.S(mpmath.findroot(dVf_mpmath, sr0, tol=tol))
nxi0 = o* 1/nr0

solhor = sym.solve(p(xi).subs({Lambda: nLambda, M:nM}), xi)
nxi1 = sym.N(max(map(lambda x: sym.re(x), solhor)), dprec)
nk1 = kappa(nxi1).subs({Lambda: nLambda, M: nM})

display(Math(r"r_0 = %.20f"%sym.N(nr0)))
display(Math(r"\xi_0 = %.20f"%sym.N(nxi0)))
display(Math(r"\xi_1 = %.20f"%sym.N(nxi1)))
display(Math(r"k_1 = %.20f"%sym.N(nk1)))


# pass lambda_0, s_0 and variable values to aim class
black_L09l2s2 = aim(lambda_0(xi), s_0(xi), pl0(), ps0())
black_L09l2s2.display_parameters(pprec=7)
black_L09l2s2.display_l0s0(0, pprec=3)
black_L09l2s2.parameters(En, xi, nxi0, nmax=71, nstep=10, dprec=500, tol=1e-101)

# create coefficients c0 and d0 for improved AIM
black_L09l2s2.c0()
black_L09l2s2.d0()
black_L09l2s2.cndn()

# the solution
print("iter" + (3*"{:^24s}").format(*("n=%s"%str(i) for i in range(1,4))))
print("_"*3*24)
black_L09l2s2.get_arb_roots(showRoots='+r-i', printFormat="  {:10.7f}", col=(0,3))



    


















    







$\displaystyle r_0 = 3.04846823336026906404$









    







$\displaystyle \xi_0 = 0.32803359702315770807$









    







$\displaystyle \xi_1 = 0.41135376108773658022$









    







$\displaystyle k_1 = 0.09628198919534343425$









    







$\displaystyle \text{Values of the parameters of } \lambda_0:$









    







$\displaystyle M = 1.000000,~~\Lambda = 0.09000000,~~\kappa_{1} = 0.09628199,~~\xi_{1} = 0.4113538$









    







$\displaystyle \text{Values of the parameters of } s_0:$









    







$\displaystyle M = 1.000000,~~\Lambda = 0.09000000,~~\ell = 2.000000,~~\kappa_{1} = 0.09628199,~~\xi_{1} = 0.4113538,~~s = 2.000000$









    







$\displaystyle            
                \begin{align}
                \lambda_0 &= \left[\frac{2 i En}{\kappa_{1} \left(\xi - \xi_{1}\right)} + \frac{6 \left(- i En - 3 M \xi^{2} + \xi\right)}{\Lambda + 6 M \xi^{3} - 3 \xi^{2}} \right.\\ 
                      s_0 &= \left[\frac{2 En^{2} \xi^{2} \left(\Lambda + 6 M \xi^{3} - 3 \xi^{2}\right) - 2 En \kappa_{1} \xi^{2} \left(6 En \left(\xi - \xi_{1}\right) + i \Lambda - 3 i \xi \left(\xi \left(4 M \xi - 6 M \xi_{1} - 1\right) + 2 \xi_{1}\right)\right) + \kappa_{1}^{2} \left(\xi - \xi_{1}\right)^{2} \left(\Lambda \left(s^{4} - 5 s^{2} + 4\right) - 6 \xi^{2} \left(- 2 M \xi \left(s^{2} - 1\right) + \ell^{2} + \ell\right)\right)}{2 \kappa_{1}^{2} \xi^{2} \left(\xi - \xi_{1}\right)^{2} \left(\Lambda + 6 M \xi^{3} - 3 \xi^{2}\right)} \right.\\ 
                \lambda_0 &= \left[\frac{20.8 i En}{\xi - 0.411} + \frac{6.0 \left(- i En - 3.0 \xi^{2} + \xi\right)}{6.0 \xi^{3} - 3.0 \xi^{2} + 0.09} \right.\\ 
                      s_0 &= \left[\frac{53.9 \left(2.0 En^{2} \xi^{2} \left(6.0 \xi^{3} - 3.0 \xi^{2} + 0.09\right) - 0.193 En \xi^{2} \left(6.0 En \left(\xi - 0.411\right) - 3.0 i \xi \left(\xi \left(4.0 \xi - 3.47\right) + 0.823\right) + 0.09 i\right) - 0.0556 \xi^{2} \left(6.0 - 6.0 \xi\right) \left(\xi - 0.411\right)^{2}\right)}{\xi^{2} \left(\xi - 0.411\right)^{2} \left(6.0 \xi^{3} - 3.0 \xi^{2} + 0.09\right)} \right.
                \end{align}
                $









    






iter          n=1                     n=2                     n=3           
________________________________________________________________________
0001    0.1611362  -0.0419793j    0.1664529  -0.1273112j
0011    0.1626104  -0.0413665j    0.1607886  -0.1241522j    0.1570423  -0.2071172j
0021    0.1626104  -0.0413665j    0.1607886  -0.1241522j    0.1570423  -0.2071172j
0031    0.1626104  -0.0413665j    0.1607886  -0.1241522j    0.1570423  -0.2071172j
0041    0.1626104  -0.0413665j    0.1607886  -0.1241522j    0.1570423  -0.2071172j
0051    0.1626104  -0.0413665j    0.1607886  -0.1241522j    0.1570423  -0.2071172j
0061    0.1626104  -0.0413665j    0.1607886  -0.1241522j    0.1570423  -0.2071172j
0071    0.1626104  -0.0413665j    0.1607886  -0.1241522j    0.1570423  -0.2071172j
CPU times: user 6min 46s, sys: 2.64 s, total: 6min 49s
Wall time: 7min

$\Lambda = 0.10$





In [18]:



    


%%time
# values of variables
nLambda = o* 10/100

sr0 = 3 # initial search point of the root of the der. pot.
nr0 = sym.S(mpmath.findroot(dVf_mpmath, sr0, tol=tol))
nxi0 = o* 1/nr0

solhor = sym.solve(p(xi).subs({Lambda: nLambda, M:nM}), xi)
nxi1 = sym.N(max(map(lambda x: sym.re(x), solhor)), dprec)
nk1 = kappa(nxi1).subs({Lambda: nLambda, M: nM})

display(Math(r"r_0 = %.20f"%sym.N(nr0)))
display(Math(r"\xi_0 = %.20f"%sym.N(nxi0)))
display(Math(r"\xi_1 = %.20f"%sym.N(nxi1)))
display(Math(r"k_1 = %.20f"%sym.N(nk1)))


# pass lambda_0, s_0 and variable values to aim class
black_L10l2s2 = aim(lambda_0(xi), s_0(xi), pl0(), ps0())
black_L10l2s2.display_parameters(pprec=7)
black_L10l2s2.display_l0s0(0, pprec=3)
black_L10l2s2.parameters(En, xi, nxi0, nmax=71, nstep=10, dprec=500, tol=1e-101)

# create coefficients c0 and d0 for improved AIM
black_L10l2s2.c0()
black_L10l2s2.d0()
black_L10l2s2.cndn()

# the solution
print("iter" + (3*"{:^24s}").format(*("n=%s"%str(i) for i in range(1,4))))
print("_"*3*24)
black_L10l2s2.get_arb_roots(showRoots='+r-i', printFormat="  {:10.7f}", col=(0,3))



    


















    







$\displaystyle r_0 = 3.02526445868100379855$









    







$\displaystyle \xi_0 = 0.33054961430908874620$









    







$\displaystyle \xi_1 = 0.39096099092712277567$









    







$\displaystyle k_1 = 0.06759049835303057585$









    







$\displaystyle \text{Values of the parameters of } \lambda_0:$









    







$\displaystyle M = 1.000000,~~\Lambda = 0.1000000,~~\kappa_{1} = 0.06759050,~~\xi_{1} = 0.3909610$









    







$\displaystyle \text{Values of the parameters of } s_0:$









    







$\displaystyle M = 1.000000,~~\Lambda = 0.1000000,~~\ell = 2.000000,~~\kappa_{1} = 0.06759050,~~\xi_{1} = 0.3909610,~~s = 2.000000$









    







$\displaystyle            
                \begin{align}
                \lambda_0 &= \left[\frac{2 i En}{\kappa_{1} \left(\xi - \xi_{1}\right)} + \frac{6 \left(- i En - 3 M \xi^{2} + \xi\right)}{\Lambda + 6 M \xi^{3} - 3 \xi^{2}} \right.\\ 
                      s_0 &= \left[\frac{2 En^{2} \xi^{2} \left(\Lambda + 6 M \xi^{3} - 3 \xi^{2}\right) - 2 En \kappa_{1} \xi^{2} \left(6 En \left(\xi - \xi_{1}\right) + i \Lambda - 3 i \xi \left(\xi \left(4 M \xi - 6 M \xi_{1} - 1\right) + 2 \xi_{1}\right)\right) + \kappa_{1}^{2} \left(\xi - \xi_{1}\right)^{2} \left(\Lambda \left(s^{4} - 5 s^{2} + 4\right) - 6 \xi^{2} \left(- 2 M \xi \left(s^{2} - 1\right) + \ell^{2} + \ell\right)\right)}{2 \kappa_{1}^{2} \xi^{2} \left(\xi - \xi_{1}\right)^{2} \left(\Lambda + 6 M \xi^{3} - 3 \xi^{2}\right)} \right.\\ 
                \lambda_0 &= \left[\frac{29.6 i En}{\xi - 0.391} + \frac{6.0 \left(- i En - 3.0 \xi^{2} + \xi\right)}{6.0 \xi^{3} - 3.0 \xi^{2} + 0.1} \right.\\ 
                      s_0 &= \left[\frac{109.0 \left(2.0 En^{2} \xi^{2} \left(6.0 \xi^{3} - 3.0 \xi^{2} + 0.1\right) - 0.135 En \xi^{2} \left(6.0 En \left(\xi - 0.391\right) - 3.0 i \xi \left(\xi \left(4.0 \xi - 3.35\right) + 0.782\right) + 0.1 i\right) - 0.0274 \xi^{2} \left(6.0 - 6.0 \xi\right) \left(\xi - 0.391\right)^{2}\right)}{\xi^{2} \left(\xi - 0.391\right)^{2} \left(6.0 \xi^{3} - 3.0 \xi^{2} + 0.1\right)} \right.
                \end{align}
                $









    






iter          n=1                     n=2                     n=3           
________________________________________________________________________
0001    0.1173080  -0.0304594j    0.1194521  -0.0919654j
0011    0.1179164  -0.0302105j    0.1172432  -0.0906409j    0.1158764  -0.1511018j
0021    0.1179164  -0.0302105j    0.1172432  -0.0906409j    0.1158764  -0.1511018j
0031    0.1179164  -0.0302105j    0.1172432  -0.0906409j    0.1158764  -0.1511018j
0041    0.1179164  -0.0302105j    0.1172432  -0.0906409j    0.1158764  -0.1511018j
0051    0.1179164  -0.0302105j    0.1172432  -0.0906409j    0.1158764  -0.1511018j
0061    0.1179164  -0.0302105j    0.1172432  -0.0906409j    0.1158764  -0.1511018j
0071    0.1179164  -0.0302105j    0.1172432  -0.0906409j    0.1158764  -0.1511018j
CPU times: user 5min 45s, sys: 2.24 s, total: 5min 47s
Wall time: 5min 47s

$\Lambda = 0.11$





In [19]:



    


%%time
# values of variables
nLambda = o* 11/100

sr0 = 3 # initial search point of the root of the der. pot.
nr0 = sym.S(mpmath.findroot(dVf_mpmath, sr0, tol=tol))
nxi0 = o* 1/nr0

solhor = sym.solve(p(xi).subs({Lambda: nLambda, M:nM}), xi)
nxi1 = sym.N(max(map(lambda x: sym.re(x), solhor)), dprec)
nk1 = kappa(nxi1).subs({Lambda: nLambda, M: nM})

display(Math(r"r_0 = %.20f"%sym.N(nr0)))
display(Math(r"\xi_0 = %.20f"%sym.N(nxi0)))
display(Math(r"\xi_1 = %.20f"%sym.N(nxi1)))
display(Math(r"k_1 = %.20f"%sym.N(nk1)))


# pass lambda_0, s_0 and variable values to aim class
black_L11l2s2 = aim(lambda_0(xi), s_0(xi), pl0(), ps0())
black_L11l2s2.display_parameters(pprec=7)
black_L11l2s2.display_l0s0(0, pprec=3)
black_L11l2s2.parameters(En, xi, nxi0, nmax=71, nstep=10, dprec=500, tol=1e-101)

# create coefficients c0 and d0 for improved AIM
black_L11l2s2.c0()
black_L11l2s2.d0()
black_L11l2s2.cndn()

# the solution
print("iter" + (3*"{:^24s}").format(*("n=%s"%str(i) for i in range(1,4))))
print("_"*3*24)
black_L11l2s2.get_arb_roots(showRoots='+r-i', printFormat="  {:10.7f}", col=(0,3))



    


















    







$\displaystyle r_0 = 3.00250260815149738747$









    







$\displaystyle \xi_0 = 0.33305549753232488452$









    







$\displaystyle \xi_1 = 0.35222476494479487696$









    







$\displaystyle k_1 = 0.01996209017645310391$









    







$\displaystyle \text{Values of the parameters of } \lambda_0:$









    







$\displaystyle M = 1.000000,~~\Lambda = 0.1100000,~~\kappa_{1} = 0.01996209,~~\xi_{1} = 0.3522248$









    







$\displaystyle \text{Values of the parameters of } s_0:$









    







$\displaystyle M = 1.000000,~~\Lambda = 0.1100000,~~\ell = 2.000000,~~\kappa_{1} = 0.01996209,~~\xi_{1} = 0.3522248,~~s = 2.000000$









    







$\displaystyle            
                \begin{align}
                \lambda_0 &= \left[\frac{2 i En}{\kappa_{1} \left(\xi - \xi_{1}\right)} + \frac{6 \left(- i En - 3 M \xi^{2} + \xi\right)}{\Lambda + 6 M \xi^{3} - 3 \xi^{2}} \right.\\ 
                      s_0 &= \left[\frac{2 En^{2} \xi^{2} \left(\Lambda + 6 M \xi^{3} - 3 \xi^{2}\right) - 2 En \kappa_{1} \xi^{2} \left(6 En \left(\xi - \xi_{1}\right) + i \Lambda - 3 i \xi \left(\xi \left(4 M \xi - 6 M \xi_{1} - 1\right) + 2 \xi_{1}\right)\right) + \kappa_{1}^{2} \left(\xi - \xi_{1}\right)^{2} \left(\Lambda \left(s^{4} - 5 s^{2} + 4\right) - 6 \xi^{2} \left(- 2 M \xi \left(s^{2} - 1\right) + \ell^{2} + \ell\right)\right)}{2 \kappa_{1}^{2} \xi^{2} \left(\xi - \xi_{1}\right)^{2} \left(\Lambda + 6 M \xi^{3} - 3 \xi^{2}\right)} \right.\\ 
                \lambda_0 &= \left[\frac{100.0 i En}{\xi - 0.352} + \frac{6.0 \left(- i En - 3.0 \xi^{2} + \xi\right)}{6.0 \xi^{3} - 3.0 \xi^{2} + 0.11} \right.\\ 
                      s_0 &= \left[\frac{1.25 \cdot 10^{3} \left(2.0 En^{2} \xi^{2} \left(6.0 \xi^{3} - 3.0 \xi^{2} + 0.11\right) - 0.0399 En \xi^{2} \left(6.0 En \left(\xi - 0.352\right) - 3.0 i \xi \left(\xi \left(4.0 \xi - 3.11\right) + 0.704\right) + 0.11 i\right) - 0.00239 \xi^{2} \left(6.0 - 6.0 \xi\right) \left(\xi - 0.352\right)^{2}\right)}{\xi^{2} \left(\xi - 0.352\right)^{2} \left(6.0 \xi^{3} - 3.0 \xi^{2} + 0.11\right)} \right.
                \end{align}
                $









    






iter          n=1                     n=2                     n=3           
________________________________________________________________________
0001    0.0372473  -0.0096242j    0.0373238  -0.0288985j
0011    0.0372699  -0.0096157j    0.0372493  -0.0288470j    0.0372081  -0.0480784j
0021    0.0372699  -0.0096157j    0.0372493  -0.0288470j    0.0372081  -0.0480784j
0031    0.0372699  -0.0096157j    0.0372493  -0.0288470j    0.0372081  -0.0480784j
0041    0.0372699  -0.0096157j    0.0372493  -0.0288470j    0.0372081  -0.0480784j
0051    0.0372699  -0.0096157j    0.0372493  -0.0288470j    0.0372081  -0.0480784j
0061    0.0372699  -0.0096157j    0.0372493  -0.0288470j    0.0372081  -0.0480784j
0071    0.0372699  -0.0096157j    0.0372493  -0.0288470j    0.0372081  -0.0480784j
CPU times: user 6min 7s, sys: 2.22 s, total: 6min 9s
Wall time: 6min 9s

Case: $l = 3, s =2$

$\Lambda = 0$

Numerical values for variables





In [20]:



    


# values of variables
nl, ns = 3, 2 # angular mom. and spin
nM = 1 # mass of the black hole

nLambda = o* 0

dprec=500
tol=1e-101
ft.ctx.dps = dprec
mpmath.mp.dps = dprec

sr0 = 3 # initial search point of the root of the der. pot.
nr0 = sym.S(mpmath.findroot(dVf_mpmath, sr0, tol=tol))
nxi0 = o* 1/nr0

solhor = sym.solve(p(xi).subs({Lambda: nLambda, M:nM}), xi)
nxi1 = sym.N(max(map(lambda x: sym.re(x), solhor)), dprec)
#nxi1 = max(map(lambda x: sym.re(x), solhor)) # with infinite precision
nk1 = kappa(nxi1).subs({Lambda: nLambda, M: nM})

display(Math(r"r_0 = %.20f"%sym.N(nr0)))
display(Math(r"\xi_0 = %.20f"%sym.N(nxi0)))
display(Math(r"\xi_1 = %.20f"%sym.N(nxi1)))
display(Math(r"k_1 = %.20f"%sym.N(nk1)))



    


















    







$\displaystyle r_0 = 3.10610722522451299099$









    







$\displaystyle \xi_0 = 0.32194638738774344899$









    







$\displaystyle \xi_1 = 0.50000000000000000000$









    







$\displaystyle k_1 = 0.25000000000000000000$
















In [21]:



    


# pl0 and ps0 to put the numerical values of variables into lambda_0 and s_0. 
pl0 = lambda: {k1: nk1, xi1: nxi1, Lambda: nLambda, M: nM}
ps0 = lambda: {k1: nk1, xi1: nxi1, Lambda: nLambda, M: nM, l: nl, s: ns, }

Drawing of the Master potential and its derivative





In [22]:



    


rspace = np.linspace(1e-10, 30, 1500)

Vf_numpy = sym.lambdify(r, Vf(r).subs({l:nl, s: ns, Lambda: nLambda, M: nM}), "numpy")
dVf_numpy = sym.lambdify(r, dVf_mpmath(r), "numpy")

Vf(r).subs({l:nl, s: ns, Lambda: nLambda, M: nM})


plt.figure(figsize=(12, 8))
plt.plot(rspace, Vf_numpy(rspace), color="red", lw=2, label="Master potential")
plt.plot(rspace, dVf_numpy(rspace), color="blue", lw=2, label="the derivative")
plt.plot(nr0, dVf_numpy(nr0), "go", ms=10, label="an extremum of the pot.")

plt.xlim(0.5, 8)
plt.ylim(-.75, .75)

plt.xlabel('r', fontsize=24)
plt.ylabel('V(r), dV(r)/dr', fontsize=24)

plt.axhline(color="black", zorder=-999)
plt.legend(loc="best")
plt.grid()
plt.show()

Initialize AIM solver





In [23]:



    


%%time
# pass lambda_0, s_0 and variable values to aim class
black_L00l3s2 = aim(lambda_0(xi), s_0(xi), pl0(), ps0())
black_L00l3s2.display_parameters(pprec=7)
black_L00l3s2.display_l0s0(0, pprec=3)
black_L00l3s2.parameters(En, xi, nxi0, nmax=111, nstep=10, dprec=500, tol=1e-101)



    


















    







$\displaystyle \text{Values of the parameters of } \lambda_0:$









    







$\displaystyle M = 1.000000,~~\Lambda = 0,~~\kappa_{1} = 0.2500000,~~\xi_{1} = 0.5000000$









    







$\displaystyle \text{Values of the parameters of } s_0:$









    







$\displaystyle M = 1.000000,~~\Lambda = 0,~~\ell = 3.000000,~~\kappa_{1} = 0.2500000,~~\xi_{1} = 0.5000000,~~s = 2.000000$









    







$\displaystyle            
                \begin{align}
                \lambda_0 &= \left[\frac{2 i En}{\kappa_{1} \left(\xi - \xi_{1}\right)} + \frac{6 \left(- i En - 3 M \xi^{2} + \xi\right)}{\Lambda + 6 M \xi^{3} - 3 \xi^{2}} \right.\\ 
                      s_0 &= \left[\frac{2 En^{2} \xi^{2} \left(\Lambda + 6 M \xi^{3} - 3 \xi^{2}\right) - 2 En \kappa_{1} \xi^{2} \left(6 En \left(\xi - \xi_{1}\right) + i \Lambda - 3 i \xi \left(\xi \left(4 M \xi - 6 M \xi_{1} - 1\right) + 2 \xi_{1}\right)\right) + \kappa_{1}^{2} \left(\xi - \xi_{1}\right)^{2} \left(\Lambda \left(s^{4} - 5 s^{2} + 4\right) - 6 \xi^{2} \left(- 2 M \xi \left(s^{2} - 1\right) + \ell^{2} + \ell\right)\right)}{2 \kappa_{1}^{2} \xi^{2} \left(\xi - \xi_{1}\right)^{2} \left(\Lambda + 6 M \xi^{3} - 3 \xi^{2}\right)} \right.\\ 
                \lambda_0 &= \left[\frac{8.0 i En}{\xi - 0.5} + \frac{6.0 \left(- i En - 3.0 \xi^{2} + \xi\right)}{6.0 \xi^{3} - 3.0 \xi^{2}} \right.\\ 
                      s_0 &= \left[\frac{8.0 \left(2.0 En^{2} \xi^{2} \left(6.0 \xi^{3} - 3.0 \xi^{2}\right) - 0.5 En \xi^{2} \left(6.0 En \left(\xi - 0.5\right) - 3.0 i \xi \left(\xi \left(4.0 \xi - 4.0\right) + 1.0\right)\right) - 0.375 \xi^{2} \left(12.0 - 6.0 \xi\right) \left(\xi - 0.5\right)^{2}\right)}{\xi^{2} \left(\xi - 0.5\right)^{2} \left(6.0 \xi^{3} - 3.0 \xi^{2}\right)} \right.
                \end{align}
                $









    






CPU times: user 80.4 ms, sys: 60 µs, total: 80.5 ms
Wall time: 81.7 ms

Calculation necessary coefficients





In [24]:



    


%%time
# create coefficients c0 and d0 for improved AIM
black_L00l3s2.c0()
black_L00l3s2.d0()
black_L00l3s2.cndn()



    


















    






CPU times: user 2min 20s, sys: 768 ms, total: 2min 21s
Wall time: 2min 21s

The solution





In [25]:



    


%%time
black_L00l3s2.get_arb_roots()



    


















    






CPU times: user 1min 31s, sys: 217 ms, total: 1min 31s
Wall time: 1min 31s

















In [26]:



    


%%time
print("iter" + (3*"{:^24s}").format(*("n=%s"%str(i) for i in range(1,4))))
print("_"*3*24)
black_L00l3s2.printAllRoots(showRoots='+r-i', printFormat="  {:10.7f}", col=(0,3))



    


















    






iter          n=1                     n=2                     n=3           
________________________________________________________________________
0001    0.5911685  -0.0980648j    0.6261972  -0.3015236j
0011    0.5994433  -0.0927031j    0.5826454  -0.2813016j    0.5517855  -0.4790778j
0021    0.5994433  -0.0927030j    0.5826438  -0.2812981j    0.5516845  -0.4790922j
0031    0.5994433  -0.0927030j    0.5826438  -0.2812981j    0.5516849  -0.4790927j
0041    0.5994433  -0.0927030j    0.5826438  -0.2812981j    0.5516849  -0.4790928j
0051    0.5994433  -0.0927030j    0.5826438  -0.2812981j    0.5516849  -0.4790928j
0061    0.5994433  -0.0927030j    0.5826438  -0.2812981j    0.5516849  -0.4790928j
0071    0.5994433  -0.0927030j    0.5826438  -0.2812981j    0.5516849  -0.4790928j
0081    0.5994433  -0.0927030j    0.5826438  -0.2812981j    0.5516849  -0.4790928j
0091    0.5994433  -0.0927030j    0.5826438  -0.2812981j    0.5516849  -0.4790928j
0101    0.5994433  -0.0927030j    0.5826438  -0.2812981j    0.5516849  -0.4790928j
0111    0.5994433  -0.0927030j    0.5826438  -0.2812981j    0.5516849  -0.4790928j
CPU times: user 545 ms, sys: 12 ms, total: 557 ms
Wall time: 552 ms

$\Lambda = 0.02$





In [27]:



    


%%time
# values of variables
nLambda = o* 2/100

sr0 = 3 # initial search point of the root of the der. pot.
nr0 = sym.S(mpmath.findroot(dVf_mpmath, sr0, tol=tol))
nxi0 = o* 1/nr0

solhor = sym.solve(p(xi).subs({Lambda: nLambda, M:nM}), xi)
nxi1 = sym.N(max(map(lambda x: sym.re(x), solhor)), dprec)
nk1 = kappa(nxi1).subs({Lambda: nLambda, M: nM})

display(Math(r"r_0 = %.20f"%sym.N(nr0)))
display(Math(r"\xi_0 = %.20f"%sym.N(nxi0)))
display(Math(r"\xi_1 = %.20f"%sym.N(nxi1)))
display(Math(r"k_1 = %.20f"%sym.N(nk1)))


# pass lambda_0, s_0 and variable values to aim class
black_L02l3s2 = aim(lambda_0(xi), s_0(xi), pl0(), ps0())
black_L02l3s2.display_parameters(pprec=7)
black_L02l3s2.display_l0s0(0, pprec=3)
black_L02l3s2.parameters(En, xi, nxi0, nmax=71, nstep=10, dprec=500, tol=1e-101)

# create coefficients c0 and d0 for improved AIM
black_L02l3s2.c0()
black_L02l3s2.d0()
black_L02l3s2.cndn()

# the solution
print("iter" + (3*"{:^24s}").format(*("n=%s"%str(i) for i in range(1,4))))
print("_"*3*24)
black_L02l3s2.get_arb_roots(showRoots='+r-i', printFormat="  {:10.7f}", col=(0,3))



    


















    







$\displaystyle r_0 = 3.08604939933184230227$









    







$\displaystyle \xi_0 = 0.32403888292148175232$









    







$\displaystyle \xi_1 = 0.48588048315289572576$









    







$\displaystyle k_1 = 0.22235904857377850052$









    







$\displaystyle \text{Values of the parameters of } \lambda_0:$









    







$\displaystyle M = 1.000000,~~\Lambda = 0.02000000,~~\kappa_{1} = 0.2223590,~~\xi_{1} = 0.4858805$









    







$\displaystyle \text{Values of the parameters of } s_0:$









    







$\displaystyle M = 1.000000,~~\Lambda = 0.02000000,~~\ell = 3.000000,~~\kappa_{1} = 0.2223590,~~\xi_{1} = 0.4858805,~~s = 2.000000$









    







$\displaystyle            
                \begin{align}
                \lambda_0 &= \left[\frac{2 i En}{\kappa_{1} \left(\xi - \xi_{1}\right)} + \frac{6 \left(- i En - 3 M \xi^{2} + \xi\right)}{\Lambda + 6 M \xi^{3} - 3 \xi^{2}} \right.\\ 
                      s_0 &= \left[\frac{2 En^{2} \xi^{2} \left(\Lambda + 6 M \xi^{3} - 3 \xi^{2}\right) - 2 En \kappa_{1} \xi^{2} \left(6 En \left(\xi - \xi_{1}\right) + i \Lambda - 3 i \xi \left(\xi \left(4 M \xi - 6 M \xi_{1} - 1\right) + 2 \xi_{1}\right)\right) + \kappa_{1}^{2} \left(\xi - \xi_{1}\right)^{2} \left(\Lambda \left(s^{4} - 5 s^{2} + 4\right) - 6 \xi^{2} \left(- 2 M \xi \left(s^{2} - 1\right) + \ell^{2} + \ell\right)\right)}{2 \kappa_{1}^{2} \xi^{2} \left(\xi - \xi_{1}\right)^{2} \left(\Lambda + 6 M \xi^{3} - 3 \xi^{2}\right)} \right.\\ 
                \lambda_0 &= \left[\frac{8.99 i En}{\xi - 0.486} + \frac{6.0 \left(- i En - 3.0 \xi^{2} + \xi\right)}{6.0 \xi^{3} - 3.0 \xi^{2} + 0.02} \right.\\ 
                      s_0 &= \left[\frac{10.1 \left(2.0 En^{2} \xi^{2} \left(6.0 \xi^{3} - 3.0 \xi^{2} + 0.02\right) - 0.445 En \xi^{2} \left(6.0 En \left(\xi - 0.486\right) - 3.0 i \xi \left(\xi \left(4.0 \xi - 3.92\right) + 0.972\right) + 0.02 i\right) - 0.297 \xi^{2} \left(12.0 - 6.0 \xi\right) \left(\xi - 0.486\right)^{2}\right)}{\xi^{2} \left(\xi - 0.486\right)^{2} \left(6.0 \xi^{3} - 3.0 \xi^{2} + 0.02\right)} \right.
                \end{align}
                $









    






iter          n=1                     n=2                     n=3           
________________________________________________________________________
0001    0.5365207  -0.0887419j    0.5637064  -0.2717185j
0011    0.5431149  -0.0844957j    0.5307438  -0.2553644j    0.5070445  -0.4320831j
0021    0.5431149  -0.0844957j    0.5307443  -0.2553631j    0.5070154  -0.4320588j
0031    0.5431149  -0.0844957j    0.5307443  -0.2553631j    0.5070153  -0.4320588j
0041    0.5431149  -0.0844957j    0.5307443  -0.2553631j    0.5070153  -0.4320588j
0051    0.5431149  -0.0844957j    0.5307443  -0.2553631j    0.5070153  -0.4320588j
0061    0.5431149  -0.0844957j    0.5307443  -0.2553631j    0.5070153  -0.4320588j
0071    0.5431149  -0.0844957j    0.5307443  -0.2553631j    0.5070153  -0.4320588j
CPU times: user 31.5 s, sys: 264 ms, total: 31.8 s
Wall time: 31.7 s

$\Lambda = 0.04$





In [28]:



    


%%time
# values of variables
nLambda = o* 4/100

sr0 = 3 # initial search point of the root of the der. pot.
nr0 = sym.S(mpmath.findroot(dVf_mpmath, sr0, tol=tol))
nxi0 = o* 1/nr0

solhor = sym.solve(p(xi).subs({Lambda: nLambda, M:nM}), xi)
nxi1 = sym.N(max(map(lambda x: sym.re(x), solhor)), dprec)
nk1 = kappa(nxi1).subs({Lambda: nLambda, M: nM})

display(Math(r"r_0 = %.20f"%sym.N(nr0)))
display(Math(r"\xi_0 = %.20f"%sym.N(nxi0)))
display(Math(r"\xi_1 = %.20f"%sym.N(nxi1)))
display(Math(r"k_1 = %.20f"%sym.N(nk1)))


# pass lambda_0, s_0 and variable values to aim class
black_L04l3s2 = aim(lambda_0(xi), s_0(xi), pl0(), ps0())
black_L04l3s2.display_parameters(pprec=7)
black_L04l3s2.display_l0s0(0, pprec=3)
black_L04l3s2.parameters(En, xi, nxi0, nmax=71, nstep=10, dprec=500, tol=1e-101)

# create coefficients c0 and d0 for improved AIM
black_L04l3s2.c0()
black_L04l3s2.d0()
black_L04l3s2.cndn()

# the solution
print("iter" + (3*"{:^24s}").format(*("n=%s"%str(i) for i in range(1,4))))
print("_"*3*24)
black_L04l3s2.get_arb_roots(showRoots='+r-i', printFormat="  {:10.7f}", col=(0,3))



    


















    







$\displaystyle r_0 = 3.06643302378009963149$









    







$\displaystyle \xi_0 = 0.32611180229440162037$









    







$\displaystyle \xi_1 = 0.46979395281592567502$









    







$\displaystyle k_1 = 0.19232512149131092416$









    







$\displaystyle \text{Values of the parameters of } \lambda_0:$









    







$\displaystyle M = 1.000000,~~\Lambda = 0.04000000,~~\kappa_{1} = 0.1923251,~~\xi_{1} = 0.4697940$









    







$\displaystyle \text{Values of the parameters of } s_0:$









    







$\displaystyle M = 1.000000,~~\Lambda = 0.04000000,~~\ell = 3.000000,~~\kappa_{1} = 0.1923251,~~\xi_{1} = 0.4697940,~~s = 2.000000$









    







$\displaystyle            
                \begin{align}
                \lambda_0 &= \left[\frac{2 i En}{\kappa_{1} \left(\xi - \xi_{1}\right)} + \frac{6 \left(- i En - 3 M \xi^{2} + \xi\right)}{\Lambda + 6 M \xi^{3} - 3 \xi^{2}} \right.\\ 
                      s_0 &= \left[\frac{2 En^{2} \xi^{2} \left(\Lambda + 6 M \xi^{3} - 3 \xi^{2}\right) - 2 En \kappa_{1} \xi^{2} \left(6 En \left(\xi - \xi_{1}\right) + i \Lambda - 3 i \xi \left(\xi \left(4 M \xi - 6 M \xi_{1} - 1\right) + 2 \xi_{1}\right)\right) + \kappa_{1}^{2} \left(\xi - \xi_{1}\right)^{2} \left(\Lambda \left(s^{4} - 5 s^{2} + 4\right) - 6 \xi^{2} \left(- 2 M \xi \left(s^{2} - 1\right) + \ell^{2} + \ell\right)\right)}{2 \kappa_{1}^{2} \xi^{2} \left(\xi - \xi_{1}\right)^{2} \left(\Lambda + 6 M \xi^{3} - 3 \xi^{2}\right)} \right.\\ 
                \lambda_0 &= \left[\frac{10.4 i En}{\xi - 0.47} + \frac{6.0 \left(- i En - 3.0 \xi^{2} + \xi\right)}{6.0 \xi^{3} - 3.0 \xi^{2} + 0.04} \right.\\ 
                      s_0 &= \left[\frac{13.5 \left(2.0 En^{2} \xi^{2} \left(6.0 \xi^{3} - 3.0 \xi^{2} + 0.04\right) - 0.385 En \xi^{2} \left(6.0 En \left(\xi - 0.47\right) - 3.0 i \xi \left(\xi \left(4.0 \xi - 3.82\right) + 0.94\right) + 0.04 i\right) - 0.222 \xi^{2} \left(12.0 - 6.0 \xi\right) \left(\xi - 0.47\right)^{2}\right)}{\xi^{2} \left(\xi - 0.47\right)^{2} \left(6.0 \xi^{3} - 3.0 \xi^{2} + 0.04\right)} \right.
                \end{align}
                $









    






iter          n=1                     n=2                     n=3           
________________________________________________________________________
0001    0.4751298  -0.0782844j    0.4949005  -0.2387318j
0011    0.4800575  -0.0751464j    0.4716580  -0.2263950j    0.4550094  -0.3807826j
0021    0.4800575  -0.0751464j    0.4716583  -0.2263948j    0.4550106  -0.3807731j
0031    0.4800575  -0.0751464j    0.4716583  -0.2263948j    0.4550106  -0.3807731j
0041    0.4800575  -0.0751464j    0.4716583  -0.2263948j    0.4550106  -0.3807731j
0051    0.4800575  -0.0751464j    0.4716583  -0.2263948j    0.4550106  -0.3807731j
0061    0.4800575  -0.0751464j    0.4716583  -0.2263948j    0.4550106  -0.3807731j
0071    0.4800575  -0.0751464j    0.4716583  -0.2263948j    0.4550106  -0.3807731j
CPU times: user 53.1 s, sys: 376 ms, total: 53.5 s
Wall time: 53.4 s

$\Lambda = 0.06$





In [29]:



    


%%time
# values of variables
nLambda = o* 6/100

sr0 = 3 # initial search point of the root of the der. pot.
nr0 = sym.S(mpmath.findroot(dVf_mpmath, sr0, tol=tol))
nxi0 = o* 1/nr0

solhor = sym.solve(p(xi).subs({Lambda: nLambda, M:nM}), xi)
nxi1 = sym.N(max(map(lambda x: sym.re(x), solhor)), dprec)
nk1 = kappa(nxi1).subs({Lambda: nLambda, M: nM})

display(Math(r"r_0 = %.20f"%sym.N(nr0)))
display(Math(r"\xi_0 = %.20f"%sym.N(nxi0)))
display(Math(r"\xi_1 = %.20f"%sym.N(nxi1)))
display(Math(r"k_1 = %.20f"%sym.N(nk1)))


# pass lambda_0, s_0 and variable values to aim class
black_L06l3s2 = aim(lambda_0(xi), s_0(xi), pl0(), ps0())
black_L06l3s2.display_parameters(pprec=7)
black_L06l3s2.display_l0s0(0, pprec=3)
black_L06l3s2.parameters(En, xi, nxi0, nmax=71, nstep=10, dprec=500, tol=1e-101)

# create coefficients c0 and d0 for improved AIM
black_L06l3s2.c0()
black_L06l3s2.d0()
black_L06l3s2.cndn()

# the solution
print("iter" + (3*"{:^24s}").format(*("n=%s"%str(i) for i in range(1,4))))
print("_"*3*24)
black_L06l3s2.get_arb_roots(showRoots='+r-i', printFormat="  {:10.7f}", col=(0,3))



    


















    







$\displaystyle r_0 = 3.04724001308494241158$









    







$\displaystyle \xi_0 = 0.32816581421416407549$









    







$\displaystyle \xi_1 = 0.45079027713765573715$









    







$\displaystyle k_1 = 0.15884534474787764813$









    







$\displaystyle \text{Values of the parameters of } \lambda_0:$









    







$\displaystyle M = 1.000000,~~\Lambda = 0.06000000,~~\kappa_{1} = 0.1588453,~~\xi_{1} = 0.4507903$









    







$\displaystyle \text{Values of the parameters of } s_0:$









    







$\displaystyle M = 1.000000,~~\Lambda = 0.06000000,~~\ell = 3.000000,~~\kappa_{1} = 0.1588453,~~\xi_{1} = 0.4507903,~~s = 2.000000$









    







$\displaystyle            
                \begin{align}
                \lambda_0 &= \left[\frac{2 i En}{\kappa_{1} \left(\xi - \xi_{1}\right)} + \frac{6 \left(- i En - 3 M \xi^{2} + \xi\right)}{\Lambda + 6 M \xi^{3} - 3 \xi^{2}} \right.\\ 
                      s_0 &= \left[\frac{2 En^{2} \xi^{2} \left(\Lambda + 6 M \xi^{3} - 3 \xi^{2}\right) - 2 En \kappa_{1} \xi^{2} \left(6 En \left(\xi - \xi_{1}\right) + i \Lambda - 3 i \xi \left(\xi \left(4 M \xi - 6 M \xi_{1} - 1\right) + 2 \xi_{1}\right)\right) + \kappa_{1}^{2} \left(\xi - \xi_{1}\right)^{2} \left(\Lambda \left(s^{4} - 5 s^{2} + 4\right) - 6 \xi^{2} \left(- 2 M \xi \left(s^{2} - 1\right) + \ell^{2} + \ell\right)\right)}{2 \kappa_{1}^{2} \xi^{2} \left(\xi - \xi_{1}\right)^{2} \left(\Lambda + 6 M \xi^{3} - 3 \xi^{2}\right)} \right.\\ 
                \lambda_0 &= \left[\frac{12.6 i En}{\xi - 0.451} + \frac{6.0 \left(- i En - 3.0 \xi^{2} + \xi\right)}{6.0 \xi^{3} - 3.0 \xi^{2} + 0.06} \right.\\ 
                      s_0 &= \left[\frac{19.8 \left(2.0 En^{2} \xi^{2} \left(6.0 \xi^{3} - 3.0 \xi^{2} + 0.06\right) - 0.318 En \xi^{2} \left(6.0 En \left(\xi - 0.451\right) - 3.0 i \xi \left(\xi \left(4.0 \xi - 3.7\right) + 0.902\right) + 0.06 i\right) - 0.151 \xi^{2} \left(12.0 - 6.0 \xi\right) \left(\xi - 0.451\right)^{2}\right)}{\xi^{2} \left(\xi - 0.451\right)^{2} \left(6.0 \xi^{3} - 3.0 \xi^{2} + 0.06\right)} \right.
                \end{align}
                $









    






iter          n=1                     n=2                     n=3           
________________________________________________________________________
0001    0.4038715  -0.0662061j    0.4167558  -0.2011021j
0011    0.4071752  -0.0641396j    0.4021705  -0.1928074j    0.3920515  -0.3227696j
0021    0.4071752  -0.0641396j    0.4021706  -0.1928074j    0.3920528  -0.3227693j
0031    0.4071752  -0.0641396j    0.4021706  -0.1928074j    0.3920528  -0.3227693j
0041    0.4071752  -0.0641396j    0.4021706  -0.1928074j    0.3920528  -0.3227693j
0051    0.4071752  -0.0641396j    0.4021706  -0.1928074j    0.3920528  -0.3227693j
0061    0.4071752  -0.0641396j    0.4021706  -0.1928074j    0.3920528  -0.3227693j
0071    0.4071752  -0.0641396j    0.4021706  -0.1928074j    0.3920528  -0.3227693j
CPU times: user 51.2 s, sys: 360 ms, total: 51.6 s
Wall time: 51.4 s

$\Lambda = 0.08$





In [30]:



    


%%time
# values of variables
nLambda = o* 8/100

sr0 = 3 # initial search point of the root of the der. pot.
nr0 = sym.S(mpmath.findroot(dVf_mpmath, sr0, tol=tol))
nxi0 = o* 1/nr0

solhor = sym.solve(p(xi).subs({Lambda: nLambda, M:nM}), xi)
nxi1 = sym.N(max(map(lambda x: sym.re(x), solhor)), dprec)
nk1 = kappa(nxi1).subs({Lambda: nLambda, M: nM})

display(Math(r"r_0 = %.20f"%sym.N(nr0)))
display(Math(r"\xi_0 = %.20f"%sym.N(nxi0)))
display(Math(r"\xi_1 = %.20f"%sym.N(nxi1)))
display(Math(r"k_1 = %.20f"%sym.N(nk1)))


# pass lambda_0, s_0 and variable values to aim class
black_L08l3s2 = aim(lambda_0(xi), s_0(xi), pl0(), ps0())
black_L08l3s2.display_parameters(pprec=7)
black_L08l3s2.display_l0s0(0, pprec=3)
black_L08l3s2.parameters(En, xi, nxi0, nmax=71, nstep=10, dprec=500, tol=1e-101)

# create coefficients c0 and d0 for improved AIM
black_L08l3s2.c0()
black_L08l3s2.d0()
black_L08l3s2.cndn()

# the solution
print("iter" + (3*"{:^24s}").format(*("n=%s"%str(i) for i in range(1,4))))
print("_"*3*24)
black_L08l3s2.get_arb_roots(showRoots='+r-i', printFormat="  {:10.7f}", col=(0,3))



    


















    







$\displaystyle r_0 = 3.02845336494387673554$









    







$\displaystyle \xi_0 = 0.33020155158259534289$









    







$\displaystyle \xi_1 = 0.42680542981557051396$









    







$\displaystyle k_1 = 0.11968319494459116192$









    







$\displaystyle \text{Values of the parameters of } \lambda_0:$









    







$\displaystyle M = 1.000000,~~\Lambda = 0.08000000,~~\kappa_{1} = 0.1196832,~~\xi_{1} = 0.4268054$









    







$\displaystyle \text{Values of the parameters of } s_0:$









    







$\displaystyle M = 1.000000,~~\Lambda = 0.08000000,~~\ell = 3.000000,~~\kappa_{1} = 0.1196832,~~\xi_{1} = 0.4268054,~~s = 2.000000$









    







$\displaystyle            
                \begin{align}
                \lambda_0 &= \left[\frac{2 i En}{\kappa_{1} \left(\xi - \xi_{1}\right)} + \frac{6 \left(- i En - 3 M \xi^{2} + \xi\right)}{\Lambda + 6 M \xi^{3} - 3 \xi^{2}} \right.\\ 
                      s_0 &= \left[\frac{2 En^{2} \xi^{2} \left(\Lambda + 6 M \xi^{3} - 3 \xi^{2}\right) - 2 En \kappa_{1} \xi^{2} \left(6 En \left(\xi - \xi_{1}\right) + i \Lambda - 3 i \xi \left(\xi \left(4 M \xi - 6 M \xi_{1} - 1\right) + 2 \xi_{1}\right)\right) + \kappa_{1}^{2} \left(\xi - \xi_{1}\right)^{2} \left(\Lambda \left(s^{4} - 5 s^{2} + 4\right) - 6 \xi^{2} \left(- 2 M \xi \left(s^{2} - 1\right) + \ell^{2} + \ell\right)\right)}{2 \kappa_{1}^{2} \xi^{2} \left(\xi - \xi_{1}\right)^{2} \left(\Lambda + 6 M \xi^{3} - 3 \xi^{2}\right)} \right.\\ 
                \lambda_0 &= \left[\frac{16.7 i En}{\xi - 0.427} + \frac{6.0 \left(- i En - 3.0 \xi^{2} + \xi\right)}{6.0 \xi^{3} - 3.0 \xi^{2} + 0.08} \right.\\ 
                      s_0 &= \left[\frac{34.9 \left(2.0 En^{2} \xi^{2} \left(6.0 \xi^{3} - 3.0 \xi^{2} + 0.08\right) - 0.239 En \xi^{2} \left(6.0 En \left(\xi - 0.427\right) - 3.0 i \xi \left(\xi \left(4.0 \xi - 3.56\right) + 0.854\right) + 0.08 i\right) - 0.0859 \xi^{2} \left(12.0 - 6.0 \xi\right) \left(\xi - 0.427\right)^{2}\right)}{\xi^{2} \left(\xi - 0.427\right)^{2} \left(6.0 \xi^{3} - 3.0 \xi^{2} + 0.08\right)} \right.
                \end{align}
                $









    






iter          n=1                     n=2                     n=3           
________________________________________________________________________
0001    0.3160308  -0.0514590j    0.3227388  -0.1556829j
0011    0.3178048  -0.0503821j    0.3154946  -0.1512490j    0.3108032  -0.2524504j
0021    0.3178048  -0.0503821j    0.3154946  -0.1512490j    0.3108033  -0.2524505j
0031    0.3178048  -0.0503821j    0.3154946  -0.1512490j    0.3108033  -0.2524505j
0041    0.3178048  -0.0503821j    0.3154946  -0.1512490j    0.3108033  -0.2524505j
0051    0.3178048  -0.0503821j    0.3154946  -0.1512490j    0.3108033  -0.2524505j
0061    0.3178048  -0.0503821j    0.3154946  -0.1512490j    0.3108033  -0.2524505j
0071    0.3178048  -0.0503821j    0.3154946  -0.1512490j    0.3108033  -0.2524505j
CPU times: user 6min 26s, sys: 2.29 s, total: 6min 29s
Wall time: 6min 28s

$\Lambda = 0.09$





In [31]:



    


%%time
# values of variables
nLambda = o* 9/100

sr0 = 3 # initial search point of the root of the der. pot.
nr0 = sym.S(mpmath.findroot(dVf_mpmath, sr0, tol=tol))
nxi0 = o* 1/nr0

solhor = sym.solve(p(xi).subs({Lambda: nLambda, M:nM}), xi)
nxi1 = sym.N(max(map(lambda x: sym.re(x), solhor)), dprec)
nk1 = kappa(nxi1).subs({Lambda: nLambda, M: nM})

display(Math(r"r_0 = %.20f"%sym.N(nr0)))
display(Math(r"\xi_0 = %.20f"%sym.N(nxi0)))
display(Math(r"\xi_1 = %.20f"%sym.N(nxi1)))
display(Math(r"k_1 = %.20f"%sym.N(nk1)))


# pass lambda_0, s_0 and variable values to aim class
black_L09l3s2 = aim(lambda_0(xi), s_0(xi), pl0(), ps0())
black_L09l3s2.display_parameters(pprec=7)
black_L09l3s2.display_l0s0(0, pprec=3)
black_L09l3s2.parameters(En, xi, nxi0, nmax=71, nstep=10, dprec=500, tol=1e-101)

# create coefficients c0 and d0 for improved AIM
black_L09l3s2.c0()
black_L09l3s2.d0()
black_L09l3s2.cndn()

# the solution
print("iter" + (3*"{:^24s}").format(*("n=%s"%str(i) for i in range(1,4))))
print("_"*3*24)
black_L09l3s2.get_arb_roots(showRoots='+r-i', printFormat="  {:10.7f}", col=(0,3))



    


















    







$\displaystyle r_0 = 3.01920738867269466965$









    







$\displaystyle \xi_0 = 0.33121275595434351846$









    







$\displaystyle \xi_1 = 0.41135376108773658022$









    







$\displaystyle k_1 = 0.09628198919534343425$









    







$\displaystyle \text{Values of the parameters of } \lambda_0:$









    







$\displaystyle M = 1.000000,~~\Lambda = 0.09000000,~~\kappa_{1} = 0.09628199,~~\xi_{1} = 0.4113538$









    







$\displaystyle \text{Values of the parameters of } s_0:$









    







$\displaystyle M = 1.000000,~~\Lambda = 0.09000000,~~\ell = 3.000000,~~\kappa_{1} = 0.09628199,~~\xi_{1} = 0.4113538,~~s = 2.000000$









    







$\displaystyle            
                \begin{align}
                \lambda_0 &= \left[\frac{2 i En}{\kappa_{1} \left(\xi - \xi_{1}\right)} + \frac{6 \left(- i En - 3 M \xi^{2} + \xi\right)}{\Lambda + 6 M \xi^{3} - 3 \xi^{2}} \right.\\ 
                      s_0 &= \left[\frac{2 En^{2} \xi^{2} \left(\Lambda + 6 M \xi^{3} - 3 \xi^{2}\right) - 2 En \kappa_{1} \xi^{2} \left(6 En \left(\xi - \xi_{1}\right) + i \Lambda - 3 i \xi \left(\xi \left(4 M \xi - 6 M \xi_{1} - 1\right) + 2 \xi_{1}\right)\right) + \kappa_{1}^{2} \left(\xi - \xi_{1}\right)^{2} \left(\Lambda \left(s^{4} - 5 s^{2} + 4\right) - 6 \xi^{2} \left(- 2 M \xi \left(s^{2} - 1\right) + \ell^{2} + \ell\right)\right)}{2 \kappa_{1}^{2} \xi^{2} \left(\xi - \xi_{1}\right)^{2} \left(\Lambda + 6 M \xi^{3} - 3 \xi^{2}\right)} \right.\\ 
                \lambda_0 &= \left[\frac{20.8 i En}{\xi - 0.411} + \frac{6.0 \left(- i En - 3.0 \xi^{2} + \xi\right)}{6.0 \xi^{3} - 3.0 \xi^{2} + 0.09} \right.\\ 
                      s_0 &= \left[\frac{53.9 \left(2.0 En^{2} \xi^{2} \left(6.0 \xi^{3} - 3.0 \xi^{2} + 0.09\right) - 0.193 En \xi^{2} \left(6.0 En \left(\xi - 0.411\right) - 3.0 i \xi \left(\xi \left(4.0 \xi - 3.47\right) + 0.823\right) + 0.09 i\right) - 0.0556 \xi^{2} \left(12.0 - 6.0 \xi\right) \left(\xi - 0.411\right)^{2}\right)}{\xi^{2} \left(\xi - 0.411\right)^{2} \left(6.0 \xi^{3} - 3.0 \xi^{2} + 0.09\right)} \right.
                \end{align}
                $









    






iter          n=1                     n=2                     n=3           
________________________________________________________________________
0001    0.2607670  -0.0422812j    0.2647636  -0.1276435j
0011    0.2618425  -0.0416439j    0.2605716  -0.1249688j    0.2579976  -0.2084119j
0021    0.2618425  -0.0416439j    0.2605716  -0.1249688j    0.2579976  -0.2084119j
0031    0.2618425  -0.0416439j    0.2605716  -0.1249688j    0.2579976  -0.2084119j
0041    0.2618425  -0.0416439j    0.2605716  -0.1249688j    0.2579976  -0.2084119j
0051    0.2618425  -0.0416439j    0.2605716  -0.1249688j    0.2579976  -0.2084119j
0061    0.2618425  -0.0416439j    0.2605716  -0.1249688j    0.2579976  -0.2084119j
0071    0.2618425  -0.0416439j    0.2605716  -0.1249688j    0.2579976  -0.2084119j
CPU times: user 5min 22s, sys: 2.02 s, total: 5min 24s
Wall time: 5min 24s

$\Lambda = 0.10$





In [32]:



    


%%time
# values of variables
nLambda = o* 10/100

sr0 = 3 # initial search point of the root of the der. pot.
nr0 = sym.S(mpmath.findroot(dVf_mpmath, sr0, tol=tol))
nxi0 = o* 1/nr0

solhor = sym.solve(p(xi).subs({Lambda: nLambda, M:nM}), xi)
nxi1 = sym.N(max(map(lambda x: sym.re(x), solhor)), dprec)
nk1 = kappa(nxi1).subs({Lambda: nLambda, M: nM})

display(Math(r"r_0 = %.20f"%sym.N(nr0)))
display(Math(r"\xi_0 = %.20f"%sym.N(nxi0)))
display(Math(r"\xi_1 = %.20f"%sym.N(nxi1)))
display(Math(r"k_1 = %.20f"%sym.N(nk1)))


# pass lambda_0, s_0 and variable values to aim class
black_L10l3s2 = aim(lambda_0(xi), s_0(xi), pl0(), ps0())
black_L10l3s2.display_parameters(pprec=7)
black_L10l3s2.display_l0s0(0, pprec=3)
black_L10l3s2.parameters(En, xi, nxi0, nmax=71, nstep=10, dprec=500, tol=1e-101)

# create coefficients c0 and d0 for improved AIM
black_L10l3s2.c0()
black_L10l3s2.d0()
black_L10l3s2.cndn()

# the solution
print("iter" + (3*"{:^24s}").format(*("n=%s"%str(i) for i in range(1,4))))
print("_"*3*24)
black_L10l3s2.get_arb_roots(showRoots='+r-i', printFormat="  {:10.7f}", col=(0,3))



    


















    







$\displaystyle r_0 = 3.01005707511366971119$









    







$\displaystyle \xi_0 = 0.33221961412882400344$









    







$\displaystyle \xi_1 = 0.39096099092712277567$









    







$\displaystyle k_1 = 0.06759049835303057585$









    







$\displaystyle \text{Values of the parameters of } \lambda_0:$









    







$\displaystyle M = 1.000000,~~\Lambda = 0.1000000,~~\kappa_{1} = 0.06759050,~~\xi_{1} = 0.3909610$









    







$\displaystyle \text{Values of the parameters of } s_0:$









    







$\displaystyle M = 1.000000,~~\Lambda = 0.1000000,~~\ell = 3.000000,~~\kappa_{1} = 0.06759050,~~\xi_{1} = 0.3909610,~~s = 2.000000$









    







$\displaystyle            
                \begin{align}
                \lambda_0 &= \left[\frac{2 i En}{\kappa_{1} \left(\xi - \xi_{1}\right)} + \frac{6 \left(- i En - 3 M \xi^{2} + \xi\right)}{\Lambda + 6 M \xi^{3} - 3 \xi^{2}} \right.\\ 
                      s_0 &= \left[\frac{2 En^{2} \xi^{2} \left(\Lambda + 6 M \xi^{3} - 3 \xi^{2}\right) - 2 En \kappa_{1} \xi^{2} \left(6 En \left(\xi - \xi_{1}\right) + i \Lambda - 3 i \xi \left(\xi \left(4 M \xi - 6 M \xi_{1} - 1\right) + 2 \xi_{1}\right)\right) + \kappa_{1}^{2} \left(\xi - \xi_{1}\right)^{2} \left(\Lambda \left(s^{4} - 5 s^{2} + 4\right) - 6 \xi^{2} \left(- 2 M \xi \left(s^{2} - 1\right) + \ell^{2} + \ell\right)\right)}{2 \kappa_{1}^{2} \xi^{2} \left(\xi - \xi_{1}\right)^{2} \left(\Lambda + 6 M \xi^{3} - 3 \xi^{2}\right)} \right.\\ 
                \lambda_0 &= \left[\frac{29.6 i En}{\xi - 0.391} + \frac{6.0 \left(- i En - 3.0 \xi^{2} + \xi\right)}{6.0 \xi^{3} - 3.0 \xi^{2} + 0.1} \right.\\ 
                      s_0 &= \left[\frac{109.0 \left(2.0 En^{2} \xi^{2} \left(6.0 \xi^{3} - 3.0 \xi^{2} + 0.1\right) - 0.135 En \xi^{2} \left(6.0 En \left(\xi - 0.391\right) - 3.0 i \xi \left(\xi \left(4.0 \xi - 3.35\right) + 0.782\right) + 0.1 i\right) - 0.0274 \xi^{2} \left(12.0 - 6.0 \xi\right) \left(\xi - 0.391\right)^{2}\right)}{\xi^{2} \left(\xi - 0.391\right)^{2} \left(6.0 \xi^{3} - 3.0 \xi^{2} + 0.1\right)} \right.
                \end{align}
                $









    






iter          n=1                     n=2                     n=3           
________________________________________________________________________
0001    0.1895354  -0.0305759j    0.1912054  -0.0920840j
0011    0.1899943  -0.0303145j    0.1895170  -0.0909507j    0.1885554  -0.1516089j
0021    0.1899943  -0.0303145j    0.1895170  -0.0909507j    0.1885554  -0.1516089j
0031    0.1899943  -0.0303145j    0.1895170  -0.0909507j    0.1885554  -0.1516089j
0041    0.1899943  -0.0303145j    0.1895170  -0.0909507j    0.1885554  -0.1516089j
0051    0.1899943  -0.0303145j    0.1895170  -0.0909507j    0.1885554  -0.1516089j
0061    0.1899943  -0.0303145j    0.1895170  -0.0909507j    0.1885554  -0.1516089j
0071    0.1899943  -0.0303145j    0.1895170  -0.0909507j    0.1885554  -0.1516089j
CPU times: user 6min 18s, sys: 2.34 s, total: 6min 20s
Wall time: 6min 20s

$\Lambda = 0.11$





In [33]:



    


%%time
# values of variables
nLambda = o* 11/100

sr0 = 3 # initial search point of the root of the der. pot.
nr0 = sym.S(mpmath.findroot(dVf_mpmath, sr0, tol=tol))
nxi0 = o* 1/nr0

solhor = sym.solve(p(xi).subs({Lambda: nLambda, M:nM}), xi)
nxi1 = sym.N(max(map(lambda x: sym.re(x), solhor)), dprec)
nk1 = kappa(nxi1).subs({Lambda: nLambda, M: nM})

display(Math(r"r_0 = %.20f"%sym.N(nr0)))
display(Math(r"\xi_0 = %.20f"%sym.N(nxi0)))
display(Math(r"\xi_1 = %.20f"%sym.N(nxi1)))
display(Math(r"k_1 = %.20f"%sym.N(nk1)))


# pass lambda_0, s_0 and variable values to aim class
black_L11l2s3 = aim(lambda_0(xi), s_0(xi), pl0(), ps0())
black_L11l2s3.display_parameters(pprec=7)
black_L11l2s3.display_l0s0(0, pprec=3)
black_L11l2s3.parameters(En, xi, nxi0, nmax=71, nstep=10, dprec=500, tol=1e-101)

# create coefficients c0 and d0 for improved AIM
black_L11l2s3.c0()
black_L11l2s3.d0()
black_L11l2s3.cndn()

# the solution
print("iter" + (3*"{:^24s}").format(*("n=%s"%str(i) for i in range(1,4))))
print("_"*3*24)
black_L11l2s3.get_arb_roots(showRoots='+r-i', printFormat="  {:10.7f}", col=(0,3))



    


















    







$\displaystyle r_0 = 3.00100056707217532903$









    







$\displaystyle \xi_0 = 0.33322219628089444488$









    







$\displaystyle \xi_1 = 0.35222476494479487696$









    







$\displaystyle k_1 = 0.01996209017645310391$









    







$\displaystyle \text{Values of the parameters of } \lambda_0:$









    







$\displaystyle M = 1.000000,~~\Lambda = 0.1100000,~~\kappa_{1} = 0.01996209,~~\xi_{1} = 0.3522248$









    







$\displaystyle \text{Values of the parameters of } s_0:$









    







$\displaystyle M = 1.000000,~~\Lambda = 0.1100000,~~\ell = 3.000000,~~\kappa_{1} = 0.01996209,~~\xi_{1} = 0.3522248,~~s = 2.000000$









    







$\displaystyle            
                \begin{align}
                \lambda_0 &= \left[\frac{2 i En}{\kappa_{1} \left(\xi - \xi_{1}\right)} + \frac{6 \left(- i En - 3 M \xi^{2} + \xi\right)}{\Lambda + 6 M \xi^{3} - 3 \xi^{2}} \right.\\ 
                      s_0 &= \left[\frac{2 En^{2} \xi^{2} \left(\Lambda + 6 M \xi^{3} - 3 \xi^{2}\right) - 2 En \kappa_{1} \xi^{2} \left(6 En \left(\xi - \xi_{1}\right) + i \Lambda - 3 i \xi \left(\xi \left(4 M \xi - 6 M \xi_{1} - 1\right) + 2 \xi_{1}\right)\right) + \kappa_{1}^{2} \left(\xi - \xi_{1}\right)^{2} \left(\Lambda \left(s^{4} - 5 s^{2} + 4\right) - 6 \xi^{2} \left(- 2 M \xi \left(s^{2} - 1\right) + \ell^{2} + \ell\right)\right)}{2 \kappa_{1}^{2} \xi^{2} \left(\xi - \xi_{1}\right)^{2} \left(\Lambda + 6 M \xi^{3} - 3 \xi^{2}\right)} \right.\\ 
                \lambda_0 &= \left[\frac{100.0 i En}{\xi - 0.352} + \frac{6.0 \left(- i En - 3.0 \xi^{2} + \xi\right)}{6.0 \xi^{3} - 3.0 \xi^{2} + 0.11} \right.\\ 
                      s_0 &= \left[\frac{1.25 \cdot 10^{3} \left(2.0 En^{2} \xi^{2} \left(6.0 \xi^{3} - 3.0 \xi^{2} + 0.11\right) - 0.0399 En \xi^{2} \left(6.0 En \left(\xi - 0.352\right) - 3.0 i \xi \left(\xi \left(4.0 \xi - 3.11\right) + 0.704\right) + 0.11 i\right) - 0.00239 \xi^{2} \left(12.0 - 6.0 \xi\right) \left(\xi - 0.352\right)^{2}\right)}{\xi^{2} \left(\xi - 0.352\right)^{2} \left(6.0 \xi^{3} - 3.0 \xi^{2} + 0.11\right)} \right.
                \end{align}
                $









    






iter          n=1                     n=2                     n=3           
________________________________________________________________________
0001    0.0600733  -0.0096282j    0.0601373  -0.0289013j
0011    0.0600915  -0.0096189j    0.0600766  -0.0288567j    0.0600469  -0.0480945j
0021    0.0600915  -0.0096189j    0.0600766  -0.0288567j    0.0600469  -0.0480945j
0031    0.0600915  -0.0096189j    0.0600766  -0.0288567j    0.0600469  -0.0480945j
0041    0.0600915  -0.0096189j    0.0600766  -0.0288567j    0.0600469  -0.0480945j
0051    0.0600915  -0.0096189j    0.0600766  -0.0288567j    0.0600469  -0.0480945j
0061    0.0600915  -0.0096189j    0.0600766  -0.0288567j    0.0600469  -0.0480945j
0071    0.0600915  -0.0096189j    0.0600766  -0.0288567j    0.0600469  -0.0480945j
CPU times: user 5min 53s, sys: 2.25 s, total: 5min 55s
Wall time: 5min 55s

















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Content source: mkarakoc/aim

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