牛顿插值

牛顿插值

import matplotlib.pyplot as plt
import numpy as np
import pylab as mpl

mpl.rcParams['font.sans-serif'] = ['SimHei']
mpl.rcParams['axes.unicode_minus'] = False

x = [0.4, 0.55, 0.65, 0.80, 0.90, 1.05]
y = [0.41075, 0.57815, 0.69675, 0.88811, 1.02652, 1.25382]

'''计算五次差商的值'''
def five_order_difference_quotient(x, y):
    # i 记录计算差商的次数，这里计算五次差商
    i = 0
    quotient = np.zeros(6)
    while i < 5:
        j = 5
        while j > i:
            if i == 0:
                quotient[j] = (y[j] - y[j - 1]) / (x[j] - x[j - 1])
            else:
                quotient[j] = (quotient[j] - quotient[j - 1]) / (x[j] - x[j - 1 - i])
            j -= 1
        i += 1
    # 返回差商值
    return quotient


def function(data):
    return x[0] + parameters[1] * (data - 0.4) + parameters[2] * (data - 0.4) * (data - 0.55) + \
           parameters[3] * (data - 0.4) * (data - 0.55) * (data - 0.65) \
           + parameters[4] * (data - 0.4) * (data - 0.55) * (data - 0.80)


"""计算插值多项式的值和相应的误差"""
def calculate_data(x, parameters):
    return_data = []
    for data in x:
        return_data.append(function(data))
    return return_data


def draw(newData):
    plt.scatter(x, y, label="离散数据", color="red")
    plt.plot(x, newData, label="牛顿插值拟合曲线", color="black")
    plt.scatter(0.596, function(0.596), label="预测函数点", color="blue")
    plt.title("牛顿插值法")
    plt.legend(loc="upper left")
    plt.show()


parameters = five_order_difference_quotient(x, y)
yuanzu = calculate_data(x, parameters)
draw(yuanzu)