数值计算方法三次样条插值法三种边界条件求插值函数并画出图像的python代码

数值计算方法三次样条插值法求插值函数并画出图像的python代码

第一类插值条件

#三次样条插值法
import matplotlib.pyplot as plt
import numpy as np
from sympy import *

#第一类插值条件函数
def one(x, y, a, b, n):
    h = np.zeros(n)
    u = np.zeros(n-1)
    r = np.zeros(n-1)
    g = np.zeros(n+1)
    A = np.zeros(shape=(n+1, n+1))
    q = np.zeros(n)
    w = np.zeros(n)
    e = np.zeros(n)
    t = np.zeros(n)
    #计算和，u，r，g
    for i in range(0, n):
        h[i] = x[i+1] - x[i]
    print("h=", h)
    for i in range(0, n-1):
        u[i] = h[i] / (h[i]+h[i+1])
    print("u=", u)
    for i in range(0, n-1):
        r[i] = 1 - u[1]
    print("r=", r)
    for i in range(1, n):
        g[i] = (6/(h[i-1]+h[i]))*((y[i+1]-y[i])/h[i]-(y[i]-y[i-1])/h[i-1])
    g[0] = 6/h[0]*((y[1]-y[0])/h[0]-a)
    g[n] = (6/h[n-1])*(b-(y[n]-y[n-1])/h[n-1])
#对线性方程组系数矩阵进行赋值
    print("g=", g)
    for i in range(0, n+1):
        A[i][i] = 2
    A[0][1] = 1
    A[n][n-1] = 1
    for i in range(0, n-1):
        A[i+1][i] = u[i]
    for i in range(2, n+1):
        A[i-1][i] = r[i-2]
#求解线性方程组
    M = np.linalg.solve(A, g)
    print("A=", A)
    print("M=", M)
    for i in range(0, n):
        q[i] = (M[i+1]-M[i])/6*h[i]
        w[i] = (M[i]*x[i+1]-M[i+1]*x[i])/2*h[i]
        e[i] = (3*M[i+1]*pow(x[i], 2)-3*M[i]*pow(x[i+1], 2)-6*y[i]+M[i]
                *pow(h[i], 2)+6*y[i+1]-M[i+1]*pow(h[i], 2))/6*h[i]
        t[i] = (M[i]*pow(x[i+1], 3)-M[i+1]*pow(x[i], 3)+6*y[i]*x[i+1]-M[i]
                *pow(h[i], 2)*x[i+1]-6*y[i+1]*x[i]+M[i+1]*pow(h[i], 2)*x[i])/6*h[i]
    print("q=", q)
    print("w=", w)
    print("e=", e)
    print("t=", t)
    
#主函数部分
x = [0, 1, 2, 3, 4, 5, 6]
y = [1, 0, 0, 1, 2, 2, 1]
n = len(x) - 1
one(x, y, -0.6, -1.8, n)

#对区间进行划分
x1 = np.linspace(0, 1, 100)
x2 = np.linspace(1, 2, 100)
x3 = np.linspace(2, 3, 100)
x4 = np.linspace(3, 4, 100)
x5 = np.linspace(4, 5, 100)
x6 = np.linspace(5, 6, 100)

#分段函数定义
y1 = (0.64*pow(x1, 3)+(-1.04)*pow(x1, 2)+(-0.6)*x1+1)
y2 = (-0.12*pow(x2, 3)+1.24*pow(x2, 2)+(-2.88)*x2+1.76)
y3 = (-0.16*pow(x3, 3)+1.48*pow(x3, 2)+(-3.36)*x3+2.08)
y4 = (-0.24*pow(x4, 3)+2.2*pow(x4, 2)+(-5.52)*x4+4.24)
y5 = (0.12*pow(x5, 3)+(-2.12)*pow(x5, 2)+11.76*x5-18.8)
y6 = (-0.24*pow(x6, 3)+3.28*pow(x6, 2)+(-15.24)*x6+26.2)

#画出图像
plt.plot(x1, y1)
plt.plot(x2, y2)
plt.plot(x3, y3)
plt.plot(x4, y4)
plt.plot(x5, y5)
plt.plot(x6, y6)
plt.show()

第二类插值条件和自然插值条件

#三次样条插值法
import matplotlib.pyplot as plt
import numpy as np
from sympy import *

#第二类插值条件函数与自然边界条件
def one(x, y, a, b, n):
    h = np.zeros(n)
    u = np.zeros(n-1)
    r = np.zeros(n-1)
    g = np.zeros(n+1)
    A = np.zeros(shape=(n-1, n-1))
    z = np.zeros(n-1)
    q = np.zeros(n)
    w = np.zeros(n)
    e = np.zeros(n)
    t = np.zeros(n)
    K = np.zeros(n+1)
    #计算和，u，r，g
    for i in range(0, n):
        h[i] = x[i+1] - x[i]
    print("h=", h)
    for i in range(0, n-1):
        u[i] = h[i] / (h[i]+h[i+1])
    print("u=", u)
    for i in range(0, n-1):
        r[i] = 1 - u[1]
    print("r=", r)
    for i in range(1, n):
        g[i] = (6/(h[i-1]+h[i]))*((y[i+1]-y[i])/h[i]-(y[i]-y[i-1])/h[i-1])
    g[1] = g[1] - u[0]*a
    g[n-1] = g[n-1]-r[n-2]
    g[0] = a
    g[n] = b
    print("g=", g)
    for i in range(0, n-1):
        z[i] = g[i+1]
    print("z=", z)
#对线性方程组系数矩阵进行赋值
    for i in range(0, n-1):
        A[i][i] = 2
    for i in range(0, n-2):
        A[i+1][i] = u[i+1]
    for i in range(1, n-1):
        A[i-1][i] = r[i-1]
    print("A=", A)
#求解线性方程组
    M = np.linalg.solve(A, z)
    for i in range(1, n):
        K[i] = M[i-1]
    K[0] = a
    K[n] = b
    print("K=", K)
    for i in range(0, n):
        q[i] = (K[i+1]-K[i])/6*h[i]
        w[i] = (K[i]*x[i+1]-K[i+1]*x[i])/2*h[i]
        e[i] = (3*K[i+1]*pow(x[i], 2)-3*K[i]*pow(x[i+1], 2)-6*y[i]+K[i]
                *pow(h[i], 2)+6*y[i+1]-K[i+1]*pow(h[i], 2))/6*h[i]
        t[i] = (K[i]*pow(x[i+1], 3)-K[i+1]*pow(x[i], 3)+6*y[i]*x[i+1]-K[i]
                *pow(h[i], 2)*x[i+1]-6*y[i+1]*x[i]+K[i+1]*pow(h[i], 2)*x[i])/6*h[i]
    print("q=", q)
    print("w=", w)
    print("e=", e)
    print("t=", t)

    # 对区间进行划分
    x1 = np.linspace(0, 1, 100)
    x2 = np.linspace(1, 2, 100)
    x3 = np.linspace(2, 3, 100)
    x4 = np.linspace(3, 4, 100)
    x5 = np.linspace(4, 5, 100)
    x6 = np.linspace(5, 6, 100)

    # 分段函数定义
    y1 = (q[0] * pow(x1, 3) + w[0] * pow(x1, 2) + e[0] * x1 + t[0])
    y2 = (q[1] * pow(x2, 3) + w[1] * pow(x2, 2) + e[1] * x2 + t[1])
    y3 = (q[2] * pow(x3, 3) + w[2] * pow(x3, 2) + e[2] * x3 + t[2])
    y4 = (q[3] * pow(x4, 3) + w[3] * pow(x4, 2) + e[3] * x4 + t[3])
    y5 = (q[4] * pow(x5, 3) + w[4] * pow(x5, 2) + e[4] * x5 + t[4])
    y6 = (q[5] * pow(x6, 3) + w[5] * pow(x6, 2) + e[5] * x6 + t[5])

    # 画出图像
    plt.plot(x1, y1)
    plt.plot(x2, y2)
    plt.plot(x3, y3)
    plt.plot(x4, y4)
    plt.plot(x5, y5)
    plt.plot(x6, y6)
    plt.show()

#主函数部分
x = [0, 1, 2, 3, 4, 5, 6]
y = [1, 0, 0, 1, 2, 2, 1]
n = len(x) - 1
one(x, y, 1, -1, n)

来源：CSDN

作者：狗儿胖

链接：https://blog.csdn.net/weixin_43652753/article/details/103246328