scipy.odeint奇怪的行为

Question

Here is my code to solve differential equation dy / dt = 2 / sqrt(pi) * exp(-x * x) to plot erf(x).

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


def euler(df, f0, x):
    h = x[1] - x[0]
    y = [f0]
    for i in xrange(len(x) - 1):
        y.append(y[i] + h * df(y[i], x[i]))
    return y


def i(df, f0, x):
    h = x[1] - x[0]
    y = [f0]
    y.append(y[0] + h * df(y[0], x[0]))
    for i in xrange(1, len(x) - 1):
        fn = df(y[i], x[i])
        fn1 = df(y[i - 1], x[i - 1])
        y.append(y[i] + (3 * fn - fn1) * h / 2)
    return y


if __name__ == "__main__":
    df = lambda y, x: 2.0 / math.sqrt(math.pi) * math.exp(-x * x)
    f0 = 0.0
    x = np.linspace(-10.0, 10.0, 10000)

    y1 = euler(df, f0, x)
    y2 = i(df, f0, x)
    y3 = odeint(df, f0, x)

    plt.plot(x, y1, x, y2, x, y3)
    plt.legend(["euler", "modified", "odeint"], loc='best')
    plt.grid(True)
    plt.show()

And here is a plot:

Am I using odeint in a wrong way or it's a bug?

Answer 1

Notice that if you change x to x = np.linspace(-5.0, 5.0, 10000) , then your code works. Therefore, I suspect the problem has something to do with exp(-x*x) being too small when x is very small or very large. [Total speculation: Perhaps the odeint (lsoda) algorithm adapts its stepsize based on values sampled around x = -10 and increases the stepsize in such a way that values around x = 0 are missed?]

The code can be fixed by using the tcrit parameter, which tells odeint to pay special attention around certain critical points.

So, by setting

y3 = integrate.odeint(df, f0, x, tcrit = [0])

we tell odeint to sample more carefully around 0.

import matplotlib.pyplot as plt
import scipy.integrate as integrate
import numpy as np
import math


def euler(df, f0, x):
    h = x[1] - x[0]
    y = [f0]
    for i in xrange(len(x) - 1):
        y.append(y[i] + h * df(y[i], x[i]))
    return y


def i(df, f0, x):
    h = x[1] - x[0]
    y = [f0]
    y.append(y[0] + h * df(y[0], x[0]))
    for i in xrange(1, len(x) - 1):
        fn = df(y[i], x[i])
        fn1 = df(y[i - 1], x[i - 1])
        y.append(y[i] + (3 * fn - fn1) * h / 2)
    return y

def df(y, x):
   return 2.0 / np.sqrt(np.pi) * np.exp(-x * x)

if __name__ == "__main__":
    f0 = 0.0
    x = np.linspace(-10.0, 10.0, 10000)

    y1 = euler(df, f0, x)
    y2 = i(df, f0, x)
    y3 = integrate.odeint(df, f0, x, tcrit = [0])

    plt.plot(x, y1)
    plt.plot(x, y2)
    plt.plot(x, y3)
    plt.legend(["euler", "modified", "odeint"], loc='best')
    plt.grid(True)
    plt.show()