三次样条插值的编程

题目背景：对y=1/（1+x^2）在[-1，1]区间以Xn=-1+0.1*（n-1），n=1 (21)为插值点做三次样条插值求解思路简析：以插值为四段三次函数为例进行说明（题干为插值20段三次函数），可看出方程组为q*x=d，其中q为方程组系数矩阵，x为所求三次函数的系数矩阵，其中方程组系数矩阵和d均呈规律性变化（边界点除外，首位两个点特殊堪虑）function qiujieyangtiao %%定义求解函数q=zeros(80); %%方程组的系数矩阵，赋初值为0n=-1:0.1:1; %%插值点的横坐标nd=zeros(80,1); %%插值点q*x=d中的dy=zeros(21,1); %%插值点的纵坐标向量a=1;for i=-1:0.1:1y(a)=1/(1+i^2);a=a+1; %%给插值点的纵坐标y通过原函数赋值endq(1,3)=2;q(1,4)=6*n(1);q(2,1)=1;q(2,2)=n(1);q(2,3)=n(1)^2;q(2,4)=n(1)^3;d(2)=y(1); %%给左端边界点的两个方程组系数赋值j=2;for i=3:4:75q(i,i-1)=1;q(i,i)=2*n(j);q(i,i+1)=3*n(j)^2;q(i,i+3)=-1;q(i,i+4)=-2*n(j);q(i,i+5)=-3*n(j)^2;d(i)=0;q(i+1,i)=2;q(i+1,i+1)=6*n(j);q(i+1,i+4)=-2;q(i+1,i+5)=-6*n(j);d(i+1)=0;q(i+2,i-2)=1;q(i+2,i-1)=n(j);q(i+2,i)=n(j)^2;q(i+2,i+1)=n(j)^3;d(i+2)=y(j);q(i+3,i+2)=1;q(i+3,i+3)=n(j);q(i+3,i+4)=n(j)^2;q(i+3,i+5)=n(j)^3;d(i+3)=y(j);j=j+1;end %%给系数矩阵赋值q(79,79)=2;q(79,80)=6*n(21);d(79)=0;q(80,77)=1;q(80,78)=n(21);q(80,79)=n(21)^2;q(80,80)=n(21)^3;d(80)=y(21); %%给右端边界点的两个方程组系数赋值result=q\d; %%求解系数矩阵function A=fun(x)if x>=-1&&x<-0.9A=result(1)+result(2)*x+result(3)*x*x+result(4)*x*x*x;elseif x>=-0.9&x<-0.8A=result(5)+result(6)*x+result(7)*x*x+result(8)*x*x*x;elseif x>=-0.8&x<-0.7A=result(9)+result(10)*x+result(11)*x*x+result(12)*x*x*x; elseif x>=-0.7&x<-0.6A=result(13)+result(14)*x+result(15)*x*x+result(16)*x*x*x; elseif x>=-0.6&x<-0.5A=result(17)+result(18)*x+result(19)*x*x+result(20)*x*x*x; elseif x>=-0.5&x<-0.4A=result(21)+result(22)*x+result(23)*x*x+result(24)*x*x*x; elseif x>=-0.4&x<-0.3A=result(25)+result(26)*x+result(27)*x*x+result(28)*x*x*x; elseif x>=-0.3&x<-0.2A=result(29)+result(30)*x+result(31)*x*x+result(32)*x*x*x; elseif x>=-0.2&x<-0.1A=result(33)+result(34)*x+result(35)*x*x+result(36)*x*x*x; elseif x>=-0.1&x<0A=result(37)+result(38)*x+result(39)*x*x+result(40)*x*x*x; elseif x>=0&x<0.1A=result(41)+result(42)*x+result(43)*x*x+result(44)*x*x*x; elseif x>=0.1&x<0.2A=result(45)+result(46)*x+result(47)*x*x+result(48)*x*x*x; elseif x>=0.2&x<0.3A=result(49)+result(50)*x+result(51)*x*x+result(52)*x*x*x; elseif x>=0.3&x<0.4A=result(53)+result(54)*x+result(55)*x*x+result(56)*x*x*x; elseif x>=0.4&x<0.5A=result(57)+result(58)*x+result(59)*x*x+result(60)*x*x*x; elseif x>=0.5&x<0.6A=result(61)+result(62)*x+result(63)*x*x+result(64)*x*x*x; elseif x>=0.6&x<0.7A=result(65)+result(66)*x+result(67)*x*x+result(68)*x*x*x; elseif x>=0.7&x<0.8A=result(69)+result(70)*x+result(71)*x*x+result(72)*x*x*x; elseif x>=0.8&x<0.9A=result(73)+result(74)*x+result(75)*x*x+result(76)*x*x*x; elseA=result(77)+result(78)*x+result(79)*x*x+result(80)*x*x*x; endend %%插值函数用子函数表达，方便调用x=linspace(-1,1);for i=1:length(x)A(i)=fun(x(i));endY=1./(1+x.^2);plot(x,Y,'--',x,A,':')legend('primitive','fitting') %%将原函数与该插值函数画在同一图上进行比较grid ontitle('三次样条插值')for m=1:20fprintf("S%d=%.3f+%.3f*x+%.3f*x.^2+%.3f*x.^3\n",m,result(4*m-3,1),result(4*m-2,1),result(4*m-1,1),result(4*m,1)) %%输出结果endend输出结果：S1=2.049+3.619*x+3.104*x.^2+1.035*x.^3S2=1.010+0.156*x+-0.743*x.^2+-0.390*x.^3S3=1.137+0.632*x+-0.149*x.^2+-0.143*x.^3S4=1.054+0.273*x+-0.660*x.^2+-0.386*x.^3S5=1.023+0.120*x+-0.916*x.^2+-0.528*x.^3S6=1.003+-0.002*x+-1.160*x.^2+-0.691*x.^3S7=0.997+-0.044*x+-1.265*x.^2+-0.779*x.^3S8=0.998+-0.034*x+-1.233*x.^2+-0.743*x.^3S9=1.000+-0.010*x+-1.113*x.^2+-0.543*x.^3S10=1.000+-0.000*x+-1.010*x.^2+-0.200*x.^3S11=1.000+-0.000*x+-1.010*x.^2+0.200*x.^3S12=1.000+0.010*x+-1.113*x.^2+0.543*x.^3S13=0.998+0.034*x+-1.233*x.^2+0.743*x.^3S14=0.997+0.044*x+-1.265*x.^2+0.779*x.^3S15=1.003+0.002*x+-1.160*x.^2+0.691*x.^3S16=1.023+-0.120*x+-0.916*x.^2+0.528*x.^3S17=1.054+-0.273*x+-0.660*x.^2+0.386*x.^3S18=1.137+-0.632*x+-0.149*x.^2+0.143*x.^3S19=1.010+-0.156*x+-0.743*x.^2+0.390*x.^3S20=2.049+-3.619*x+3.104*x.^2+-1.035*x.^3对比图。

2 三次样条插值程序 三次样条插值利用方案二（求解固支样条或压紧样条）按照要求要起点和终点的一阶导数值已知， 可得关于01,,.....,n M M M 的严格对角占优势的三对角方程组然后利用三对角法（追赶法）解此线性方程组。

（1）编写M 文件，并保存文件名scfit.m% x,y 分别为n 个节点的横坐标和纵坐标值组成的向量% dx0和dxn 分别为S 的导数在x0和xn 处的值，即m 0和m nn=length(x)-1;h=diff(x);d=diff(y)./h;a=h(2:n-1);b=2*(h(1:n-1)+h(2:n));c=h(2:n);u=6*diff(d);b(1)=b(1)-h(1)/2;u(1)=u(1)-3*(d(1)-dx0);b(n-1)=b(n-1)-h(n)/2;u(n-1)=u(n-1)-3*(dxn-d(n));%追赶法部分for k=2:n-1temp=a(k-1)/b(k-1);b(k)=b(k)-temp*c(k-1);u(k)=u(k)-temp*u(k-1);endm(n)=u(n-1)/b(n-1);for k=n-2:-1:1m(k+1)=(u(k)-c(k)*m(k+2))/b(k);end%求S K1，S K2，S K3，S K4m(1)=3*(d(1)-dx0)/h(1)-m(2)/2;m(n+1)=3*(dxn-d(n))/h(n)-m(n)/2;for k=0:n-100()S x m '=()n n S x m '=0011111111212212n n n n n n M d M d M d M d μλμλ----⎡⎤⎡⎤⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥=⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦⎣⎦S(k+1,1)=(m(k+2)-m(k+1))/(6*h(k+1));S(k+1,2)=m(k+1)/2;S(k+1,3)=d(k+1)-h(k+1)*(2*m(k+1)+m(k+2))/6;S(k+1,4)=y(k+1);EndP=mkpp(x,S);ppval(P,x);plot(x,y ,'-*')（或fnplt(P)）（2）在Command window 内输入已知条件（课本数值试验一后例子，令x0和xn 处一阶导数为0）>> x=[1 2 3 4 5 6 7 8 9 10];>> y=[3.56 2.56 1.54 0.53 0.26 0.90 1.81 2.12 1.53 0.56];>> dx0=0;dxn=0;调用编写的M 文件>> S=scfit(x,y ,dx0,dxn)回车计算结果：S =0.7390 -1.7390 0 3.5600-0.2369 0.4780 -1.2610 2.56000.2387 -0.2328 -1.0159 1.54000.0120 0.4834 -0.7654 0.5300-0.1167 0.5193 0.2373 0.2600-0.1853 0.1693 0.9260 0.9000-0.0122 -0.3865 0.7088 1.8100-0.0658 -0.4232 -0.1010 2.12000.7953 -0.6205 -1.1447 1.5300即插值结果为.().()().,[,].().().().,[,]().........................().().().,[,]32232322073901173901013561202369204780212610225623079539062059114479153910x x x x x x x x s x x x x x ⎧---+-+∈⎪--+---+∈⎪=⎨⎪⎪-----+∈⎩第一个图是利用ppval(P,x);plot(x,y)显示的图形，图中*为数据点 第二个图是利用fnplt(P)显示的图形。

三次样条插值算法C++实现三次样条插值算法1 总体说明三次样条插值算法是⼀种计算量和效果都⽐较理想的插值算法。

关于三次样条插值算法的原理这⾥不做过多的解释，下⾯的代码是我在⽹上收集了两种C++实现版本的基础上⾃⼰整合的⼀个版本。

由于本⼈刚接触C++不久，⽔平有限。

没有使⽤模板机制将代码做的更通⽤。

关于算法实现有下⾯⼏点说明。

1. 所有有关的类都被包含到SplineSpace命名空间中。

2. SplineSpace中⼀个有三个类分别是异常类(SplineFailure),接⼝类(SplineInterface)和实现类(Spline)。

有⼀个枚举类型说明边界条件(BoundaryCondition),取值为：GivenFirstOrder和GivenSecondOrder。

分别对应I型边界条件和II型边界条件。

3. 接⼝类定义了Spline在实现的过程中必须要有的三个⽅法：单点插值、多点插值和⾃动⽣成插值序列。

4. 异常类是可能被实现类抛出的类，如果在实现类的运⾏过程中出现了已知数据过少构造失败、使⽤了外插值、设定输出点数过少等⾏为会抛出该类。

因此应该将插值的过程⽤try...catch(SplineFailure sf)包裹起来。

如：double x0[2]={1,2};double y0[2]={3,4};try{SplineInterface* sp = new Spline(x0,y0,2);//...}catch(SplineFailure sf){cout<<sf.GetMessage()<<endl;}上⾯代码就会抛出异常并显⽰“构造失败,已知点数过少”。

2 插值⽅法调⽤2.1单点插值调⽤⽅法如下：#include <iostream>#include "Spline.h"using namespace std;using namespace SplineSpace;int main(void){//单点插值测试double x0[5]={1,2,4,5,6}; //已知的数据点double y0[5]={1,3,4,2,5};try{//Spline sp(x0,y0,5,GivenSecondOrder,0,0);SplineInterface* sp = new Spline(x0,y0,5); //使⽤接⼝，且使⽤默认边界条件double x=4.5;double y;sp->SinglePointInterp(x,y); //求x的插值结果ycout<<"x="<<x<<"时的插值结果为:"<<y<<endl;}catch(SplineFailure sf){cout<<sf.GetMessage()<<endl;}getchar(); //程序暂停}此时屏幕会输出"x=4.5时的插值结果为2.71107"。

文章标题：深度解析Matlab三次样条插值1. 前言在数学和工程领域中，插值是一种常见的数值分析技术，它可以用来估计不连续数据点之间的值。

而三次样条插值作为一种常用的插值方法，在Matlab中有着广泛的应用。

本文将从简单到复杂，由浅入深地解析Matlab中的三次样条插值方法，以便读者更深入地理解这一技术。

2. 三次样条插值概述三次样条插值是一种利用分段三次多项式对数据点进行插值的方法。

在Matlab中，可以使用spline函数来进行三次样条插值。

该函数需要输入数据点的x和y坐标，然后可以根据需要进行插值操作。

3. 三次样条插值的基本原理在进行三次样条插值时，首先需要对数据点进行分段处理，然后在每个分段上构造出一个三次多项式函数。

这些多项式函数需要满足一定的插值条件，如在数据点处函数值相等、一阶导数相等等。

通过这些条件，可以得到一个关于数据点的插值函数。

4. Matlab中的三次样条插值实现在Matlab中，可以使用spline函数来进行三次样条插值。

通过传入数据点的x和y坐标，可以得到一个关于x的插值函数。

spline函数也支持在已知插值函数上进行插值点的求值，这为用户提供了极大的灵活性。

5. 三次样条插值的适用范围和局限性虽然三次样条插值在许多情况下都能够得到较好的插值效果，但也存在一些局限性。

在数据点分布不均匀或有较大噪音的情况下，三次样条插值可能会出现较大的误差。

在实际应用中，需要根据具体情况选择合适的插值方法。

6. 个人观点和总结通过对Matlab中三次样条插值的深度解析，我深刻地理解了这一插值方法的原理和实现方式。

在实际工程应用中，我会根据数据点的情况选择合适的插值方法，以确保得到准确且可靠的结果。

我也意识到插值方法的局限性，这为我在实际工作中的决策提供了重要的参考。

通过以上深度解析，相信读者已经对Matlab中的三次样条插值有了更加全面、深刻和灵活的理解。

在实际应用中，希望读者能够根据具体情况选择合适的插值方法，以提高工作效率和准确性。

/* 三次样条插值计算算法*/#include "math.h "#include "stdio.h "#include "stdlib.h "/*N：已知节点数N+1R：欲求插值点数R+1x，y为给定函数f(x)的节点值{x(i)} (x(i) <x(i+1)) ,以及相应的函数值{f(i)} 0 <=i <=NP0=f(x0)的二阶导数；Pn=f(xn)的二阶导数u:存插值点{u(i)} 0 <=i <=R求得的结果s(ui)放入s[R+1] 0 <=i <=R返回0表示成功，1表示失败*/int SPL(int N,int R,double x[],double y[],double P0,double Pn,double u[],double s[]){/*声明局部变量*/double *h; /*存放步长:{hi} 0 <=i <=N-1 */double *a; /*存放系数矩阵{ai} 1 <=i <=N ；分量0没有利用*/ double *c; /*先存放系数矩阵{ci} 后存放{Bi} 0 <=i <=N-1 */double *g; /*先存放方程组右端项{gi} 后存放求解中间结果{yi} 0 <=i <=N */double *af; /*存放系数矩阵{a(f)i} 1 <=i <=N ；*/double *ba; /*存放中间结果0 <=i <=N-1*/double *m; /*存放方程组的解{m(i)} 0 <=i <=N ；*/int i,k;double p1,p2,p3,p4;/*分配空间*/if(!(h=(double*)malloc(N*sizeof(double)))) exit(1);if(!(a=(double*)malloc((N+1)*sizeof(double)))) exit(1);if(!(c=(double*)malloc(N*sizeof(double)))) exit(1);if(!(g=(double*)malloc((N+1)*sizeof(double)))) exit(1);if(!(af=(double*)malloc((N+1)*sizeof(double)))) exit(1);if(!(ba=(double*)malloc((N)*sizeof(double)))) exit(1);if(!(m=(double*)malloc((N+1)*sizeof(double)))) exit(1);/*第一步：计算方程组的系数*/for(k=0;k <N;k++)h[k]=x[k+1]-x[k];for(k=1;k <N;k++)a[k]=h[k]/(h[k]+h[k-1]);for(k=1;k <N;k++)c[k]=1-a[k];for(k=1;k <N;k++)g[k]=3*(c[k]*(y[k+1]-y[k])/h[k]+a[k]*(y[k]-y[k-1])/h[k-1]); c[0]=a[N]=1;g[0]=3*(y[1]-y[0])/h[0]-P0*h[0]/2;g[N]=3*(y[N]-y[N-1])/h[N-1]+Pn*h[N-1]/2;/*第二步：用追赶法解方程组求{m(i)} */ba[0]=c[0]/2;g[0]=g[0]/2;for(i=1;i <N;i++){af[i]=2-a[i]*ba[i-1];g[i]=(g[i]-a[i]*g[i-1])/af[i];ba[i]=c[i]/af[i];}af[N]=2-a[N]*ba[N-1];g[N]=(g[N]-a[N]*g[N-1])/af[N];m[N]=g[N]; /*P110 公式：6.32*/ for(i=N-1;i> =0;i--)m[i]=g[i]-ba[i]*m[i+1];/*第三步：求值*/for(i=0;i <=R;i++){/*判断u(i)属于哪一个子区间，即确定k */if(u[i] <x[0] || u[i]> x[N]){/*释放空间*/free(h);free(a);free(c);free(g);free(af);free(ba);free(m);return 1;}k=0;while(u[i]> x[k+1])k++;//p1=(h[k]+2*(u[i]-x[k])*pow((u[i]-x[k+1]),2)*y[k])/pow(h[k],3); //p2=(h[k]-2*(u[i]-x[k+1])*pow((u[i]-x[k]),2)*y[k+1])/pow(h[k],3);p1=(h[k]+2*(u[i]-x[k]))*pow((u[i]-x[k+1]),2)*y[k]/pow(h[k],3);p2=(h[k]-2*(u[i]-x[k+1]))*pow((u[i]-x[k]),2)*y[k+1]/pow(h[k],3); p3=(u[i]-x[k])*pow((u[i]-x[k+1]),2)*m[k]/pow(h[k],2);p4=(u[i]-x[k+1])*pow((u[i]-x[k]),2)*m[k+1]/pow(h[k],2);s[i]=p1+p2+p3+p4;}/*释放空间*/free(h);free(a);free(c);free(g);free(af);free(ba);free(m);return 0;}void main(){int N,R;double *x,*y,*u,*s;double P0,Pn;int i;/*验证算法：*/N=7;R=6;/*分配空间*/if(!(x=(double*)malloc((N+1)*sizeof(double)))){printf( "malloc error!\n ");exit(1);}if(!(y=(double*)malloc((N+1)*sizeof(double)))){printf( "malloc error!\n ");exit(1);}if(!(u=(double*)malloc((R+1)*sizeof(double)))){printf( "malloc error!\n ");exit(1);}if(!(s=(double*)malloc((R+1)*sizeof(double)))){printf( "malloc error!\n ");exit(1);}x[0]=0.5;x[1]=0.7;x[2]=0.9;x[3]=1.1;x[4]=1.3;x[5]=1.5;x[6]=1.7;x[7]=1.9;y[0]=0.4794;y[1]=0.6442;y[2]=0.7833;y[3]=0.8912;y[4]=0.9636;y[5]=0.9975;y[6]=0.9917;y[7]=0.9 463;u[0]=0.6;u[1]=0.8;u[2]=1.0;u[3]=1.2;u[4]=1.4;u[5]=1.6;u[6]=1.8;P0=-0.4794;Pn=-0.9463;if(!SPL( N, R, x, y, P0, Pn, u, s)){/*打印结果*/printf( "\nx= ");for(i=0;i <=N;i++)printf( "%8.1f ",x[i]);printf( "\ny= ");for(i=0;i <=N;i++)printf( "%8.4f ",y[i]);printf( "\n\nu= ");for(i=0;i <=R;i++)printf( "%9.2f ",u[i]);printf( "\ns= ");for(i=0;i <=R;i++)printf( "%9.5f ",s[i]);printf( "\nsin= ");for(i=0;i <=R;i++)printf( "%9.5f ",sin(u[i]));}/*释放空间*/free(x);free(y);free(u);free(s);}/* 测试数据来自课本55页例5 《数值分析》清华大学出版社第四版*/ //输入327.7 4.128 4.329 4.130 3.013.0 -4.0//输出输出三次样条插值函数：1: [27.7 , 28]13.07*(x - 28)^3 + 0.22*(x - 27.7)^3+ 14.84*(28 - x) + 14.31*(x - 27.7)2: [28 , 29]0.066*(29 - x)^3 + 0.1383*(x - 28)^3+ 4.234*(29 - x) + 3.962*(x - 28)3: [29 , 30]0.1383*(30 - x)^3 - 1.519*(x - 29)^3+ 3.962*(30 - x) + 4.519*(x - 29)//三次样条插值函数#include<iostream>#include<iomanip>using namespace std;const int MAX = 50;float x[MAX], y[MAX], h[MAX];float c[MAX], a[MAX], fxym[MAX];float f(int x1, int x2, int x3){float a = (y[x3] - y[x2]) / (x[x3] - x[x2]);float b = (y[x2] - y[x1]) / (x[x2] - x[x1]);return (a - b)/(x[x3] - x[x1]);} //求差分void cal_m(int n){ //用追赶法求解出弯矩向量M……float B[MAX];B[0] = c[0] / 2;for(int i = 1; i < n; i++)B[i] = c[i] / (2 - a[i]*B[i-1]);fxym[0] = fxym[0] / 2;for(i = 1; i <= n; i++)fxym[i] = (fxym[i] - a[i]*fxym[i-1]) / (2 - a[i]*B[i-1]);for(i = n-1; i >= 0; i--)fxym[i] = fxym[i] - B[i]*fxym[i+1];}void printout(int n);int main(){int n,i; char ch;do{cout<<"Please put in the number of the dots:";cin>>n;for(i = 0; i <= n; i++){cout<<"Please put in X"<<i<<':';cin>>x[i]; //cout<<endl;cout<<"Please put in Y"<<i<<':';cin>>y[i]; //cout<<endl;}for(i = 0; i < n; i++) //求步长h[i] = x[i+1] - x[i];cout<<"Please 输入边界条件\n 1: 已知两端的一阶导数\n 2:两端的二阶导数已知\n 默认:自然边界条件\n";int t;float f0, f1;cin>>t;switch(t){case 1:cout<<"Please put in Y0\' Y"<<n<<"\'\n";cin>>f0>>f1;c[0] = 1; a[n] = 1;fxym[0] = 6*((y[1] - y[0]) / (x[1] - x[0]) - f0) / h[0];fxym[n] = 6*(f1 - (y[n] - y[n-1]) / (x[n] - x[n-1])) / h[n-1];break;case 2:cout<<"Please put in Y0\" Y"<<n<<"\"\n";cin>>f0>>f1;c[0] = a[n] = 0;fxym[0] = 2*f0; fxym[n] = 2*f1;break;default:cout<<"不可用\n";//待定};//switchfor(i = 1; i < n; i++)fxym[i] = 6 * f(i-1, i, i+1);for(i = 1; i < n; i++){a[i] = h[i-1] / (h[i] + h[i-1]);c[i] = 1 - a[i];}a[n] = h[n-1] / (h[n-1] + h[n]);cal_m(n);cout<<"\n输出三次样条插值函数：\n";printout(n);cout<<"Do you to have anther try ? y/n :";cin>>ch;}while(ch == 'y' || ch == 'Y');return 0;}void printout(int n){cout<<setprecision(6);for(int i = 0; i < n; i++){cout<<i+1<<": ["<<x[i]<<" , "<<x[i+1]<<"]\n"<<"\t";/*cout<<fxym[i]/(6*h[i])<<" * ("<<x[i+1]<<" - x)^3 + "<<<<" * (x - "<<x[i]<<")^3 + "<<(y[i] - fxym[i]*h[i]*h[i]/6)/h[i]<<" * ("<<x[i+1]<<" - x) + "<<(y[i+1] - fxym[i+1]*h[i]*h[i]/6)/h[i]<<"(x - "<<x[i]<<")\n";cout<<endl;*/float t = fxym[i]/(6*h[i]);if(t > 0)cout<<t<<"*("<<x[i+1]<<" - x)^3";else cout<<-t<<"*(x - "<<x[i+1]<<")^3";t = fxym[i+1]/(6*h[i]);if(t > 0)cout<<" + "<<t<<"*(x - "<<x[i]<<")^3";else cout<<" - "<<-t<<"*(x - "<<x[i]<<")^3";cout<<"\n\t";t = (y[i] - fxym[i]*h[i]*h[i]/6)/h[i];if(t > 0)cout<<"+ "<<t<<"*("<<x[i+1]<<" - x)";else cout<<"- "<<-t<<"*("<<x[i+1]<<" - x)";t = (y[i+1] - fxym[i+1]*h[i]*h[i]/6)/h[i];if(t > 0)cout<<" + "<<t<<"*(x - "<<x[i]<<")";else cout<<" - "<<-t<<"*(x - "<<x[i]<<")";cout<<endl<<endl;}cout<<endl;}。

// Čý´Î×ÔČťŃůĚő˛ĺÖľąŕłĚ.cpp : Defines the entry point for the console application.////#include "stdafx.h"#include "stdio.h"void GetValueSpline(int n,double x[],double y[],double dy[],double ddy[],int m,double t[],double z[],double dz[],double ddz[]){int i,j;double h0,h1,alpha,beta,g,*s;h0=x[1]-x[0];s[0]=3.0*(y[1]-y[0])/2.0*h0-ddy[0]*h0/4.0;for(j=1;j<=n-2;j++){h1=x[j+1]-x[j];alpha=h0/(h0+h1);beta=(1.0-alpha)*(y[j]-y[j-1])/h0;beta=3.0*(beta+alpha*(y[j+1]-y[j]))/h1;dy[j]=-alpha/(2.0+(1.0-alpha))*dy[j-1];s[j]=s[j-1]/(2.0+(1.0-alpha))*dy[j-1];h0=h1;}dy[n-1]=(3.0*(y[n-1]-y[n-2])/h1+ddy[n-1]*h1/2.0-s[n-2])/(2.0+dy[n-2]);for(j=n-2;j>=0;j--)dy[j]=dy[j]*dy[j+1]+s[j];for(j=0;j<=n-2;j++)s[j]=x[j+1]-x[j];for(j=0;j<=n-2;j++){h1=s[j]*s[j];ddy[j]=6.0*(y[j+1]-y[j])/h1-2.0*(2.0*dy[j]+dy[j+1])/s[j];}h1=s[n-2]*s[n-2];ddy[n-1]=6.0*(y[n-2]-y[n-1])/h1+2.0*(2.0*dy[n-1]+dy[n-2])/s[n-2]; g=0.0;for(i=0;i<=n-2;i++){h1=0.5*s[i]*(y[i]+y[i+1]);h1=h1-s[i]*s[i]*s[i]*(ddy[i]+ddy[i+1])/24.0;g=g+h1;}for(j=0;j<=m-1;j++){if(t[j]>=x[n-1])i=n-2;else{i=0;while(t[j]>x[i+1])i=i+1;}h1=(x[i+1]-t[j])/s[i];h0=h1*h1;z[j]=(3.0*h0-2.0*h0*h1)*y[i];z[j]=z[j]+s[i]*(h0-h0*h1)*dy[i];dz[j]=6.0*(h0-h1)*y[i]/s[i];dz[j]=dz[j]+(3.0*h0-2.0*h1)*dy[i];ddz[j]=(6.0-12.0*h1)*y[i]/(s[i]*s[i]);ddz[j]=ddz[j]+(2.0-6.0*h1)*dy[i]/s[i];h1=(t[j]-x[i])/s[i];h0=h1*h1;z[j]=z[j]+(3.0*h0-2.0*h0*h1)*y[i+1];z[j]=z[j]-s[i]*(h0-h0*h1)*dy[i+1];dz[j]=dz[j]-6.0*(h0-h1)*y[i+1]/s[i];dz[j]=dz[j]+(3.0*h0-2.0*h1)*dy[i+1];ddz[j]=ddz[j]+(6.0-12.0*h1)*y[i+1]/(s[i]*s[i]);ddz[j]=ddz[j]-(2.0-6.0*h1)*dy[i+1]/s[i];for(i=0;i<4;i++);{printf("Ö¸ś¨˛ĺÖľľă Čý´Î×ÔČťŃůĚő˛ĺÖľ Ňť˝×ľźĘýÖľ śţ˝×ľźĘýÖľ\n ");printf(" t(i) z(i) dz(i) ddz(i) \n"); printf(" %f %f %f %f\n",t[i],z[i],dz[i],ddz[i]);}//delete[] s;//return (g);}}void main(){int i;double x[5],y[5],dy[5],ddy[5];double t[4],z[4],dz[4],ddz[4]; printf("ÇëĘäČëÎĺ×éÔ­ĘźĘýžÝ:\n");for(i=0;i<5;i++){printf("x(%d)=",i);scanf("%f ",&x[i]);printf(" y(%d)=",i);scanf("%f\n",&y[i]);}printf("ÇëĘäČë˛ĺÖľľăĘýžÝ:\n");for(i=0;i<4;i++){printf("t(%d)=",i);scanf("%f\n",&t[i]);}GetValueSpline(5,x[5],y[5],dy[5],ddy[5],4,t[4],z[4],dz[4],ddz[4]);//for(i=0;i<4;i++);//{printf("Ö¸ś¨˛ĺÖľľă Čý´Î×ÔČťŃůĚő˛ĺÖľ Ňť˝×ľźĘýÖľ śţ˝×ľźĘýÖľ\n ");//printf(" t(i) z(i) dz(i) ddz(i)\n");//for(i=0;i<4;i++)//printf(" %f %f %f %f\n",t[i],z[i],dz[i],ddz[i]);//}}/*************************************************************************************************************************/// aa.cpp : Defines the entry point for the console application.////#include "stdafx.h"#include"stdio.h"void main(){int i,j;double h0,h1,alpha,beta,*s;double x[5],y[5],dy[5],ddy[5]; double t[4],z[4],dz[4],ddz[4];int m,n;m=5;n=4;x[0]=-3.0;x[1]=-1.0;x[2]=2.0;x[3]=4.0;x[4]=7.0;y[0]=3.0;y[1]=6.0;y[2]=8.0;y[3]=2.0;y[4]=5.0;t[0]=-2.5;t[1]=0.0;t[2]=2.5;t[3]=5.0;dy[0]=-0.5;ddy[0]=0;ddy[4]=0;h0=x[1]-x[0];s[0]=3.0*(y[1]-y[0])/2.0*h0-ddy[0]*h0/4.0;for(j=1;j<=n-2;j++){h1=x[j+1]-x[j];alpha=h0/(h0+h1);beta=(1.0-alpha)*(y[j]-y[j-1])/h0;beta=3.0*(beta+alpha*(y[j+1]-y[j]))/h1;dy[j]=-alpha/(2.0+(1.0-alpha))*dy[j-1];s[j]=s[j-1]/(2.0+(1.0-alpha))*dy[j-1];h0=h1;}dy[n-1]=(3.0*(y[n-1]-y[n-2])/h1+ddy[n-1]*h1/2.0-s[n-2])/(2.0+dy[n-2]);for(j=n-2;j>=0;j--)dy[j]=dy[j]*dy[j+1]+s[j];for(j=0;j<=n-2;j++)s[j]=x[j+1]-x[j];for(j=1;j<=n-2;j++){h1=s[j]*s[j];ddy[j]=6.0*(y[j+1]-y[j])/h1-2.0*(2.0*dy[j]+dy[j+1])/s[j];}for(j=0;j<=m-1;j++){if(t[j]>=x[n-1])i=n-2;else{i=0;while(t[j]>x[i+1])i=i+1;}h1=(x[i+1]-t[j])/s[i];h0=h1*h1;z[j]=(3.0*h0-2.0*h0*h1)*y[i];z[j]=z[j]+s[i]*(h0-h0*h1)*dy[i];dz[j]=6.0*(h0-h1)*y[i]/s[i];dz[j]=dz[j]+(3.0*h0-2.0*h1)*dy[i];ddz[j]=(6.0-12.0*h1)*y[i]/(s[i]*s[i]);ddz[j]=ddz[j]+(2.0-6.0*h1)*dy[i]/s[i];h1=(t[j]-x[i])/s[i];h0=h1*h1;z[j]=z[j]+(3.0*h0-2.0*h0*h1)*y[i+1];z[j]=z[j]-s[i]*(h0-h0*h1)*dy[i+1];dz[j]=dz[j]-6.0*(h0-h1)*y[i+1]/s[i];dz[j]=dz[j]+(3.0*h0-2.0*h1)*dy[i+1];ddz[j]=ddz[j]+(6.0-12.0*h1)*y[i+1]/(s[i]*s[i]);ddz[j]=ddz[j]-(2.0-6.0*h1)*dy[i+1]/s[i];}printf("Ö¸ś¨˛ĺÖľľă Čý´Î×ÔČťŃůĚő˛ĺÖľ Ňť˝×ľźĘýÖľ śţ˝×ľźĘýÖľ\n ");printf(" t(i) z(i) dz(i) ddz(i) \n");for(i=0;i<4;i++)printf(" %f %f %f %f\n",t[i],z[i],dz[i],ddz[i]);。

三次样条插值（CubicSplineInterpolation）及代码实现（C语⾔）样条插值是⼀种⼯业设计中常⽤的、得到平滑曲线的⼀种插值⽅法，三次样条⼜是其中⽤的较为⼴泛的⼀种。

本篇介绍⼒求⽤容易理解的⽅式，介绍⼀下三次样条插值的原理，并附C语⾔的实现代码。

1. 三次样条曲线原理假设有以下节点1.1 定义样条曲线是⼀个分段定义的公式。

给定n+1个数据点，共有n个区间，三次样条⽅程满⾜以下条件：a. 在每个分段区间（i = 0, 1, …, n-1，x递增），都是⼀个三次多项式。

b. 满⾜（i = 0, 1, …, n ）c. ，导数，⼆阶导数在[a, b]区间都是连续的，即曲线是光滑的。

所以n个三次多项式分段可以写作：，i = 0, 1, …, n-1其中ai, bi, ci, di代表4n个未知系数。

1.2 求解已知:a. n+1个数据点[xi, yi], i = 0, 1, …, nb. 每⼀分段都是三次多项式函数曲线c. 节点达到⼆阶连续d. 左右两端点处特性（⾃然边界，固定边界，⾮节点边界）根据定点，求出每段样条曲线⽅程中的系数，即可得到每段曲线的具体表达式。

插值和连续性：, 其中 i = 0, 1, …, n-1微分连续性：, 其中 i = 0, 1, …, n-2样条曲线的微分式：将步长带⼊样条曲线的条件：a. 由 (i = 0, 1, …, n-1)推出b. 由 (i = 0, 1, …, n-1)推出c. 由 (i = 0, 1, …, n-2)推出由此可得：d. 由 (i = 0, 1, …, n-2)推出设，则a. 可写为：，推出b. 将ci, di带⼊可得：c. 将bi, ci, di带⼊ (i = 0, 1, …, n-2)可得：端点条件由i的取值范围可知，共有n-1个公式，但却有n+1个未知量m 。

要想求解该⽅程组，还需另外两个式⼦。

所以需要对两端点x0和xn的微分加些限制。

一、引言在计算机编程和数据处理领域，插值是一种常见的数值分析方法，用于在已知数据点之间估算未知点的数值。

而三次样条插值是插值方法中的一种重要技术，它可以在使用较少插值节点的情况下，实现更为平滑和精确的插值结果。

本文将着重探讨三次样条插值的原理和C++代码实现，并给出详细的注释和解释。

二、三次样条插值的原理三次样条插值是一种分段插值方法，它将整个插值区间分割为若干个小区间，每个小区间内采用三次多项式进行插值。

这样做的好处是可以在每个小区间内实现更为细致和精确的插值，从而提高插值的准确性和平滑性。

而三次样条插值的核心在于确定每个小区间内的三次多项式的系数，一般采用自然边界条件进行求解。

在具体实现中，我们需要先对给定的插值节点进行排序，并求解出每个小区间内的三次多项式系数。

最终将这些系数整合起来，就可以得到整个插值区间的三次样条插值函数。

三、C++代码实现及注释接下来，我们将给出使用C++语言实现三次样条插值的代码，并对每个关键步骤进行详细注释和解释。

```cpp// include necessary libraries#include <iostream>#include <vector>using namespace std;// define the function for cubic spline interpolationvector<double> cubicSplineInterpolation(vector<double> x, vector<double> y) {// initialize necessary variables and containersint n = x.size();vector<double> h(n-1), alpha(n), l(n), mu(n), z(n), c(n), b(n), d(n);vector<double> interpolatedValues;// step 1: calculate the differences between x valuesfor (int i = 0; i < n-1; i++) {h[i] = x[i+1] - x[i];}// step 2: calculate alpha valuesfor (int i = 1; i < n-1; i++) {alpha[i] = (3/h[i]) * (y[i+1] - y[i]) - (3/h[i-1]) * (y[i] - y[i-1]); }// step 3: calculate l, mu, and z valuesl[0] = 1;mu[0] = 0;z[0] = 0;for (int i = 1; i < n-1; i++) {l[i] = 2*(x[i+1] - x[i-1]) - h[i-1]*mu[i-1];mu[i] = h[i]/l[i];z[i] = (alpha[i] - h[i-1]*z[i-1])/l[i];}l[n-1] = 1;z[n-1] = 0;c[n-1] = 0;// step 4: calculate coefficients for the cubic polynomials for (int j = n-2; j >= 0; j--) {c[j] = z[j] - mu[j]*c[j+1];b[j] = (y[j+1] - y[j])/h[j] - h[j]*(c[j+1] + 2*c[j])/3;d[j] = (c[j+1] - c[j])/(3*h[j]);}// step 5: interpolate values using the cubic polynomials for (int i = 0; i < n-1; i++) {double xi = x[i];while (xi < x[i+1]) {double dx = xi - x[i];double interpolatedValue = y[i] + b[i]*dx + c[i]*dx*dx + d[i]*dx*dx*dx;interpolatedValues.push_back(interpolatedValue);xi += 0.1; // adjust the step size for finer interpolation }}return interpolatedValues;}// main function for testing the cubic spline interpolation int main() {vector<double> x = {1, 2, 3, 4, 5};vector<double> y = {3, 6, 8, 10, 15};vector<double> interpolatedValues = cubicSplineInterpolation(x, y);for (int i = 0; i < interpolatedValues.size(); i++) {cout << "Interpolated value " << i << " : " << interpolatedValues[i] << endl;}return 0;}```四、总结与展望通过本文的学习，我们了解了三次样条插值的原理和C++代码实现。

用vb实现利用三次样条插值函数进行编程网友 cz5360 于提问vb三次样条插值函数绘图Dim X(1000) As Single, Y(1000) As SingleDim u1(0 To 80000) As Single, v1(0 To 80000) As SingleDim num As LongDim t As IntegerPrivate Declare Sub Sleep Lib "kernel32" (ByVal dwMilliseconds As Long)Dim de As IntegerDim ToInit As BooleanDim DownX As Single, DownY As SingleSub Drawposi(Index As Integer)Me.Picture1.ForeColor = 0Me.Picture1.Line (0, Y(Index))-(1024, Y(Index))Me.Picture1.Line (X(Index), 0)-(X(Index), 770)End SubFunction hypot(ByVal X As Single, ByVal Y As Single)hypot = Sqr(X ^ 2 + Y ^ 2)End FunctionSub MovePic(Index As Integer)Dim i As IntegerX(Index) = Picture2(Index).Left + 4Y(Index) = Picture2(Index).Top + 4lblX.Caption = "X: " + CStr(CInt(X(Index)))lblY.Caption = "Y: " + CStr(CInt(Y(Index)))lblX.RefreshlblY.RefreshMe.Picture1.ClsMe.Picture1.ForeColor = QBColor(10)For i = 0 To t - 1Me.Picture1.CurrentX = X(i) + 4Me.Picture1.CurrentY = Y(i) + 4Me.Picture1.Print iNext iEnd SubPrivate Sub Command1_Click()Dim i As Long'Picture1.Scale (0, 0)-(640, 550)DrawWidth = 3Picture1.Cls'If Check1.Value Then Command2_Click'X(0) = 1'Y(0) = 1'X(t - 1) = 638'Y(t - 1) = 548Picture1.ForeColor = QBColor(10)For i = 0 To t - 1Picture1.Line (X(i) - 1, Y(i) - 1)-(X(i) + 1, Y(i) + 1), QBColor(10), B Picture1.Print iNext iPicture1.ForeColor = QBColor(12)DrawWidth = 1tspLine t - 1, 2, 0, 0, 0, 0Picture1.PSet (u1(0), v1(0))For i = 1 To num - 1Picture1.Line -(u1(i), v1(i))'For de = 1 To 12000: Next de 'Sleep 1Next iPicture1.ForeColor = QBColor(10)For i = 0 To t - 1Picture1.Line (X(i) - 1, Y(i) - 1)-(X(i) + 1, Y(i) + 1), QBColor(10), B Picture1.Print iNext iEnd SubPrivate Sub Command2_Click()Dim i As IntegerRandomize TimerToInit = Not ToInitIf ToInit Thenmand1.Enabled = Falsemand2.Caption = "结束初始化"Me.ClsFor i = 1 To t - 1Load Me.Picture2(i)Next iFor i = 0 To t - 1Picture2(i).Left = X(i) - 4Picture2(i).Top = Y(i) - 4Picture2(i).Visible = TrueNext iPicture1.ClsMe.Picture1.ForeColor = QBColor(10)For i = 0 To t - 1Picture1.Line (X(i) - 1, Y(i) - 1)-(X(i) + 1, Y(i) + 1), QBColor(10), BPicture1.Print iNext iElsemand1.Enabled = Truemand2.Caption = "开始初始化"For i = 1 To t - 1Unload Me.Picture2(i)Next iMe.Picture2(0).Visible = FalseEnd IfExit SubFor i = 0 To tX(i) = Rnd(1) * 500 + Rnd(1) * 50 + 12Y(i) = Rnd(1) * 400 + Rnd(1) * 100 + 12'X(i) = i * 20 + Rnd(1) * 10 + 12'Y(i) = i * 10 + Rnd(1) * 300 + 22 - Rnd(1) * 200Next iEnd SubSub tspLine(ByVal n As Integer, ByVal ch As Integer, ByVal tx1 As Single, ByVal tx2 As Single, ByVal ty1 As Single, ByVal ty2 As Single)Dim a(1000) As Single, b(1000) As Single, c(1000) As Single, dX(1000) As Single, dY(1000) As SingleDim qx(1000) As Single, qy(1000) As SingleDim tt As Single, bx3 As Single, bx4 As Single, by3 As Single, by4 As Single Dim cx As Single, cy As Single, t(1000) As Single, px(1000) As Single,py(1000) As SingleDim u(3000) As Single, v(3000) As Single, i As Integernum = 0For i = 1 To nt(i) = hypot(X(i) - X(i - 1), Y(i) - Y(i - 1))Next iSelect Case chCase 0 '抛物条件u(0) = (X(1) - X(0)) / t(1): u(1) = (X(2) - X(1)) / t(2)u(2) = (u(1) - u(0)) / (t(2) + t(1))tx1 = u(0) - u(2) * t(1)u(0) = (Y(1) - Y(0)) / t(1): u(1) = (Y(2) - Y(1)) / t(2)u(2) = (u(1) - u(0)) / (t(2) + t(1))ty1 = u(0) - u(2) * t(1)u(0) = (X(n) - X(n - 1)) / t(n): u(1) = (X(n - 1) - X(n - 2)) / t(n - 1)u(2) = (u(0) - u(1)) / (t(n) + t(n - 1))tx2 = u(0) + u(2) * t(n)u(0) = (Y(n) - Y(n - 1)) / t(n): u(1) = (Y(n - 1) - Y(n - 2)) / t(n - 1)u(2) = (u(0) - u(1)) / (t(n) + t(n - 1))ty2 = u(0) + u(2) * t(n)Case 1 '夹持条件a(0) = 1: c(0) = 0: dX(0) = tx1: dY(0) = ty1a(n) = 1: b(n) = 0: dX(n) = tx2: dY(n) = ty2Case 2 '自由条件a(0) = 2: c(0) = 1dX(0) = 3 * (X(1) - X(0)) / t(1): dY(0) = 3 * (Y(1) - Y(0)) / t(1)a(n) = 2: b(n) = 1dX(n) = 3 * (X(n) - X(n - 1)) / t(n): dY(n) = 3 * (Y(n) - Y(n - 1)) / t(n)Case 3 '循环条件a(0) = 2: c(0) = 1dX(0) = 3 * (X(1) - X(0)) / t(1) - (t(1) * (X(2) - X(1)) / t(2) - X(1) + X(0)) / (t(1) + t(2))dY(0) = 3 * (Y(1) - Y(0)) / t(1) - (t(1) * (Y(2) - Y(1)) / t(2) - Y(1) + Y(0)) / (t(1) + t(2))a(n) = 2: b(n) = 1dX(n) = 3 * (X(n) - X(n - 1)) / t(n)dX(n) = dX(n) + (X(n) - X(n - 1) - t(n) * (X(n - 1) - X(n - 2)) / t(n - 1)) / (t(n) + t(n - 1))dY(n) = 3 * (Y(n) - Y(n - 1)) / t(n)dY(n) = dY(n) + (Y(n) - Y(n - 1) - t(n) * (Y(n - 1) - Y(n - 2)) / t(n - 1)) / (t(n) + t(n - 1))End Select'计算方程组系数阵和常数阵For i = 1 To n - 1a(i) = 2 * (t(i) + t(i + 1)): b(i) = t(i + 1): c(i) = t(i)dX(i) = 3 * (t(i) * (X(i + 1) - X(i)) / t(i + 1) + t(i + 1) * (X(i) - X(i - 1)) / t(i))dY(i) = 3 * (t(i) * (Y(i + 1) - Y(i)) / t(i + 1) + t(i + 1) * (Y(i) - Y(i - 1)) / t(i))Next i'采用追赶法解方程组c(0) = c(0) / a(0)For i = 1 To n - 1a(i) = a(i) - b(i) * c(i - 1): c(i) = c(i) / a(i)Next ia(n) = a(n) - b(n) * c(i - 1)qx(0) = dX(0) / a(0): qy(0) = dY(0) / a(0)For i = 1 To nqx(i) = (dX(i) - b(i) * qx(i - 1)) / a(i)qy(i) = (dY(i) - b(i) * qy(i - 1)) / a(i)Next ipx(n) = qx(n): py(n) = qy(n)For i = n - 1 To 0 Step -1px(i) = qx(i) - c(i) * px(i + 1)py(i) = qy(i) - c(i) * py(i + 1)Next i'计算曲线上点的坐标For i = 0 To n - 1bx3 = (3 * (X(i + 1) - X(i)) / t(i + 1) - 2 * px(i) - px(i + 1)) / t(i + 1)bx4 = ((2 * (X(i) - X(i + 1)) / t(i + 1) + px(i) + px(i + 1)) / t(i + 1)) / t(i + 1) by3 = (3 * (Y(i + 1) - Y(i)) / t(i + 1) - 2 * py(i) - py(i + 1)) / t(i + 1)by4 = ((2 * (Y(i) - Y(i + 1)) / t(i + 1) + py(i) + py(i + 1)) / t(i + 1)) / t(i + 1) tt = 0While (tt <= t(i + 1))cx = X(i) + (px(i) + (bx3 + bx4 * tt) * tt) * ttcy = Y(i) + (py(i) + (by3 + by4 * tt) * tt) * ttu1(num) = cx: v1(num) = cy: num = num + 1: tt = tt + 0.5Wendu1(num) = X(i + 1): v1(num) = Y(i + 1): num = num + 1Next iEnd SubPrivate Sub Form_Load()Dim i As Integert = 30ToInit = False'Picture1.Scale (0, 0)-(640, 550)Randomize Timermand2.Caption = "开始初始化"For i = 0 To tX(i) = Rnd(1) * 500 + Rnd(1) * 50 + 12Y(i) = Rnd(1) * 400 + Rnd(1) * 100 + 12Next iFor i = 0 To tX(i) = i * 30 + 20Y(i) = i * 20 + 20Next i'Me.Picture1.Picture = LoadPicture("c:\my documents\MenuBack.bmp") Me.Picture1.BackColor = QBColor(0)End SubPrivate Sub Form_Resize()On Error Resume NextMe.Picture1.Height = Me.ScaleHeight - 40End SubPrivate Sub Form_Unload(Cancel As Integer)EndEnd SubPrivate Sub Picture2_MouseDown(Index As Integer, Button As Integer, Shift As Integer, X As Single, Y As Single)On Error Resume NextIf Button = 1 ThenDownX = XDownY = YPicture2(Index).ZOrder 0Picture2(Index - 1).BackColor = QBColor(12)Picture2(Index + 1).BackColor = QBColor(12)lblX.Caption = "X: " + CStr(CInt(Picture2(Index).Left + 4))lblY.Caption = "Y: " + CStr(CInt(Picture2(Index).Top + 4))lblX.RefreshlblY.RefreshMovePic IndexDrawposi IndexEnd IfEnd SubPrivate Sub Picture2_MouseMove(Index As Integer, Button As Integer, Shift As Integer, X As Single, Y As Single)If Button = 1 ThenPicture2(Index).Left = Picture2(Index).Left - DownX + XPicture2(Index).Top = Picture2(Index).Top - DownY + YMovePic IndexCommand1_ClickDrawposi IndexEnd IfEnd SubPrivate Sub Picture2_MouseUp(Index As Integer, Button As Integer, Shift As Integer, X As Single, Y As Single)On Error Resume NextIf Button = 1 ThenDownX = XDownY = YPicture2(Index - 1).BackColor = QBColor(15)Picture2(Index + 1).BackColor = QBColor(15)'MovePic IndexlblX.Caption = "X:"lblY.Caption = "Y:"lblX.RefreshlblY.RefreshCommand1_ClickEnd IfEnd Sub三次样条插值函数高次插值函数的计算量大,有剧烈振荡,且数值稳定性差；在分段插值中，分段线性插值在分段点上仅连续而不可导，分段三次埃尔米特插值有连续的一阶导数，如此光滑程度常不能满足物理问题的需要，样条函数可以同时解决这两个问题,使插值函数既是低阶分段函数,又是光滑的函数，并且只需在区间端点提供某些导数信息。