复化求积公式

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一、简介

复化求积公式(composite integration rule )一类重要的求积公式，指将求积区间分为n个子区间，对每个子区间应用同一求积公式，所得到的复合数值积分公式。

详解

二、实现

# -*- coding: utf-8 -*-
"""
Created on Mon Dec 19 10:21:15 2016

    复化梯形、复化Simpson

@author: Administrator
"""

from numpy import *
from scipy import integrate
import matplotlib.pyplot as plt

#x [0,2pi]
def f(x):
    return e**(3*x) * cos(pi * x)

def T(a,b,n=50):
    h = (b - a) / n
    temp = 0
    for i in range(1,n):
        x = a + i * h
        temp += 2 * f(x)
    return (b - a) / (2 * n) * (f(a) + temp + f(b))

#50 100 200 500 1000
def S(a,b,n=50):
    h = (b - a) / n
    temp1 = 0
    temp2 = 0
    for i in range(1,n):
        xk1 = a + h * i
        xk2 = a + h * (i + 1)
        xk12 = (xk1 + xk2) / 2
        temp1 += f(xk1)
        temp2 += f(xk12)
    temp2 += f((a + a + h) / 2)
    return (b - a) / (6 * n) * (f(a) + 4 * temp2 + 2 * temp1 + f(b))

if __name__ == '__main__':
    n =1000 #50 100 200 500 1000
    a = 0
    b = 2 * pi

    print 'Truth-value:',integrate.quad(f,a,b)[0]
    print 'T-Estimated-value:',T(a,b,n)
    print 'S-Estimated-value:',S(a,b,n)

    x = linspace(0,2*pi,100)
    y = f(x)
    fig = plt.figure(figsize=(8,4))
    ax = fig.add_subplot(111)
    ax.set_xlabel('x')
    ax.set_ylabel('y')
    ax.plot(x,y,color='r',linestyle='-',label='f(x)')
    ax.legend(loc='upper left')
    fig.show()
    fig.savefig('a.png')