问需要更好地理解rtol，atol如何在scipy.integrate.odeint中工作

EN

# FROM: "Comparing Numerical Methods for Ordinary Differential Equations"
# T.E. Hull, W.H. Enright, B.M. Fellen and A.E. Sedgwidh
# SIAM J. Numer. Anal. vol 9, no 4, December 1972, pp: 603-637

def deriv_B1(y, x):
    return [2.*(y[0]-y[0]*y[1]), -(y[1]-y[0]*y[1])] # "growth of two conflicting populations"

def deriv_B4(y, x):
    A = 1./np.sqrt(y[0]**2 + y[1]**2)
    return [-y[1] - A*y[0]*y[2],  y[0] - A*y[1]*y[2],  A*y[0]]  # "integral surface of a torus"

def deriv_C1(y, x):
    return [-y[0]] + [y[i]-y[i+1] for i in range(8)] + [y[8]] # a radioactive decay chain

def deriv_D1toD5(y, x):
    A = -(y[0]**2 + y[1]**2)**-1.5
    return [y[2],  y[3],  A*y[0],  A*y[1]] # dimensionless orbit equation

deriv_D1, deriv_D5 = deriv_D1toD5, deriv_D1toD5

def deriv_E1(y, x):
    return [y[1], -(y[1]/(x+1.0) + (1.0 - 0.25/(x+1.0)**2)*y[0])] # derived from Bessel's equation of order 1/2

def deriv_E3(y, x):
    return [y[1], y[0]**3/6.0 - y[0] + 2.0*np.sin(2.78535*x)] # derived from Duffing's equation

import numpy as np
from scipy.integrate import odeint as ODEint
import matplotlib.pyplot as plt
import timeit

y0_B1 = [1.0, 3.0]
y0_B4 = [3.0, 0.0, 0.0]
y0_C1 = [1.0] + [0.0 for i in range(9)]
ep1, ep5 = 0.1, 0.9
y0_D1 = [1.0-ep1, 0.0, 0.0, np.sqrt((1.0+ep1)/(1.0-ep1))]
y0_D5 = [1.0-ep5, 0.0, 0.0, np.sqrt((1.0+ep5)/(1.0-ep5))]
y0_E1 = [0.6713968071418030, 0.09540051444747446] # J(1/2, 1), Jprime(1/2, 1)
y0_E3 = [0.0, 0.0]

x  = np.linspace(0, 20, 51)
xa = np.linspace(0, 20, 2001)

derivs = [deriv_B1, deriv_B4, deriv_C1, deriv_D1, deriv_D5, deriv_E3]
names  = ["deriv_B1", "deriv_B4", "deriv_C1", "deriv_D1", "deriv_D5", "deriv_E3"]
y0s    = [y0_B1, y0_B4, y0_C1, y0_D1, y0_D5, y0_E3]

timeit_dict, answer_dict, info_dict = dict(), dict(), dict()

ntimes = 10
tols   = [10.**-i for i in range(6, 14)]

def F():           # low density of time points, no output for speed test
    ODEint(deriv, y0, x, rtol=tol, atol=tol)
def Fa():           # hight density of time points, full output for plotting
    return ODEint(deriv, y0, xa, rtol=tol, atol=tol, full_output=True)

for deriv, y0, name in zip(derivs, y0s, names):
    timez = [timeit.timeit(F, number=ntimes)/float(ntimes) for tol in tols]
    timeit_dict[name] = timez
    alist, dlist = zip(*[Fa() for tol in tols])
    answer_dict[name] = np.array([a.T for a in alist])
    info_dict[name] = dlist

plt.figure(figsize=[10,6])

for i, name in enumerate(names):
    plt.subplot(2, 3, i+1)
    for thing in answer_dict[name][-1]:
        plt.plot(xa, thing)
    plt.title(name[-2:], fontsize=16)
plt.show()

plt.figure(figsize=[10, 8])
for i, name in enumerate(names):
    plt.subplot(2,3,i+1)
    a = answer_dict[name]
    a13, a10, a8 = a[-1], a[-4], a[-6]
    d10 = np.abs(a10-a13).max(axis=0)
    d8  = np.abs(a8 -a13).max(axis=0)
    plt.plot(xa, d10, label="tol(1E-10)-tol(1E-13)")
    plt.plot(xa, d8,  label="tol(1E-08)-tol(1E-13)")
    plt.yscale('log')
    plt.ylim(1E-11, 1E-03)
    plt.title(name[-2:], fontsize=16)
    if i==3:
        plt.text(3, 1E-10, "1E-10 - 1E-13", fontsize=14)
        plt.text(2, 2E-05, "1E-08 - 1E-13", fontsize=14)
plt.show()

fs = 16
plt.figure(figsize=[12,6])

plt.subplot(1,3,1)
for name in names:
    plt.plot(tols, timeit_dict[name])
plt.title("timing results", fontsize=16)
plt.xscale('log')
plt.yscale('log')
plt.text(1E-09, 5E-02, "D5", fontsize=fs)
plt.text(1E-09, 4.5E-03, "C1", fontsize=fs)

plt.subplot(1,3,2)
for name in names:
    a = answer_dict[name]
    e = a[:-1] - a[-1]
    em = [np.abs(thing).max() for thing in e]
    plt.plot(tols[:-1], em)
plt.title("max difference from smallest tol", fontsize=16)
plt.xscale('log')
plt.yscale('log')
plt.xlim(min(tols), max(tols))
plt.text(1E-09, 3E-03, "D5", fontsize=fs)
plt.text(1E-09, 8E-11, "C1", fontsize=fs)

plt.subplot(1,3,3)
for name in names:
    nsteps = [d['nst'][-1] for d in info_dict[name]]
    plt.plot(tols, nsteps, label=name[-2:])
plt.title("number of steps", fontsize=16)
plt.xscale('log')
plt.yscale('log')
plt.ylim(3E+01, 3E+03)
plt.legend(loc="upper right", shadow=False, fontsize="large")
plt.text(2E-12, 2.3E+03, "D5", fontsize=fs)
plt.text(2E-12, 1.5E+02, "C1", fontsize=fs)

plt.show()