scipy.odeint奇怪的行为

Question

这是我的代码，用于求解微分方程dy / dt = 2 / sqrt（pi）* exp（-x * x）以绘制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()

这是一个情节：

我是以错误的方式使用odeint还是错误？

Answer 1

请注意，如果您将x更改为x = np.linspace(-5.0, 5.0, 10000)，那么您的代码就可以运行。因此，我怀疑当exp(-x*x)非常小或非常大时，问题与x太小有关。 [总推测：也许odeint（lsoda）算法根据x = -10周围的值调整其步长，并以x = 0周围的值丢失的方式增加步长？]

可以使用tcrit参数修复代码，该参数告诉odeint围绕某些关键点要特别注意。

所以，通过设置

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

我们告诉odeint在0附近更仔细地采样。

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()