三次样条插值算法c
㈠ Akima 插值和样条插值的C语言源代码，要有注释。
Akima插值
附带的图片为运行结果
#include"stdio.h"
#include"math.h"
#include"interpolation.h"
voidinterpolation_akima(AKINTEPap){
intnum,k,kk,m,l;
doublepio,*mtr,*x,*y,u[5],p,q;
num=ap->n;k=ap->k;
pio=ap->t;mtr=ap->s;
x=ap->x;y=ap->y;
if(num<1){
return;
}
elseif(num==1){
mtr[0]=mtr[4]=y[0];
return;
}
elseif(num==2){
mtr[0]=y[0];
mtr[1]=(y[1]-y[0])/(x[1]-x[0]);
if(k<0)
mtr[4]=(y[0]*(pio-x[1])-y[1]*(pio-x[0]))/(x[0]-x[1]);
return;
}
if(k<0){
if(pio<=x[1])kk=0;
elseif(pio>=x[num-1])kk=num-2;
else{
kk=1;m=num;
while(((kk-m)!=1)&&((kk-m)!=-1)){
l=(kk+m)/2;
if(pio<x[l-1])m=l;
elsekk=l;
}
kk--;
}
}
elsekk=k;
if(kk>=num-1)kk=num-2;
u[2]=(y[kk+1]-y[kk])/(x[kk+1]-x[kk]);
if(num==3){
if(kk==0){
u[3]=(y[2]-y[1])/(x[2]-x[1]);
u[4]=2.0*u[3]-u[2];
u[1]=2.0*u[2]-u[3];
u[0]=2.0*u[1]-u[2];
}
else{
u[1]=(y[1]-y[0])/(x[1]-x[0]);
u[0]=2.0*u[1]-u[2];
u[3]=2.0*u[2]-u[1];
u[4]=2.0*u[3]-u[2];
}
}
else{
if(kk<=1){
u[3]=(y[kk+2]-y[kk+1])/(x[kk+2]-x[kk+1]);
if(kk==1){
u[1]=(y[1]-y[0])/(x[1]-x[0]);
u[0]=2.0*u[1]-u[2];
if(num==4)u[4]=2.0*u[3]-u[2];
elseu[4]=(y[4]-y[3])/(x[4]-x[3]);
}
else{
u[1]=2.0*u[2]-u[3];
u[0]=2.0*u[1]-u[2];
u[4]=(y[3]-y[2])/(x[3]-x[2]);
}
}
elseif(kk>=(num-3)){
u[1]=(y[kk]-y[kk-1])/(x[kk]-x[kk-1]);
if(kk==(num-3)){
u[3]=(y[num-1]-y[num-2])/(x[num-1]-x[num-2]);
u[4]=2.0*u[3]-u[2];
if(num==4)u[0]=2.0*u[1]-u[2];
elseu[0]=(y[kk-1]-y[kk-2])/(x[kk-1]-x[kk-2]);
}
else{
u[3]=2.0*u[2]-u[1];
u[4]=2.0*u[3]-u[2];
u[0]=(y[kk-1]-y[kk-2])/(x[kk-1]-x[kk-2]);
}
}
else{
u[1]=(y[kk]-y[kk-1])/(x[kk]-x[kk-1]);
u[0]=(y[kk-1]-y[kk-2])/(x[kk-1]-x[kk-2]);
u[3]=(y[kk+2]-y[kk+1])/(x[kk+2]-x[kk+1]);
u[4]=(y[kk+3]-y[kk+2])/(x[kk+3]-x[kk+2]);
}
}
mtr[0]=fabs(u[3]-u[2]);
mtr[1]=fabs(u[0]-u[1]);
if((fabs(mtr[0])<0.0000001)&&(fabs(mtr[1])<0.0000001))
p=(u[1]+u[2])/2.0;
elsep=(mtr[0]*u[1]+mtr[1]*u[2])/(mtr[0]+mtr[1]);
mtr[0]=fabs(u[3]-u[4]);
mtr[1]=fabs(u[2]-u[1]);
if((fabs(mtr[0])<0.0000001)&&(fabs(mtr[1])<0.0000001))
q=(u[2]+u[3])/2.0;
elseq=(mtr[0]*u[2]+mtr[1]*u[3])/(mtr[0]+mtr[1]);
mtr[0]=y[kk];
mtr[1]=p;
mtr[3]=x[kk+1]-x[kk];
mtr[2]=(3.0*u[2]-2.0*p-q)/mtr[3];
mtr[3]=(q+p-2.0*u[2])/(mtr[3]*mtr[3]);
if(k<0){
p=pio-x[kk];
mtr[4]=mtr[0]+mtr[1]*p+mtr[2]*p*p+mtr[3]*p*p*p;
}
return;
}
main()
{
doublex[11]={3.0,5.0,8.0,13.0,17.0,25.0,27.0,29.0,31.0,35.0,39.0};
doubley[11]={7.0,10.0,11.0,17.0,23.0,18.0,13.0,6.0,3.0,1.0,0.0};
AKINTEaa={11,x,y,-1,14.0,{0}};
AKINTEab={11,x,y,-1,28.0,{0}};
printf(" ");
interpolation_akima(&aa);
printf("x=%6.3f,f(x)=%e ",aa.t,aa.s[4]);
printf("mtr0=%e,mtr1=%e,mtr2=%e,mtr3=%e ",aa.s[0],aa.s[1],aa.s[2],aa.s[3]);
printf(" ");
interpolation_akima(&ab);
printf("x=%6.3f,f(x)=%e ",ab.t,ab.s[4]);
printf("mtr0=%e,mtr1=%e,mtr2=%e,mtr3=%e ",ab.s[0],ab.s[1],ab.s[2],ab.s[3]);
printf(" ");
}
三次样条插值的实现
1、程序比较简单的：
#include<iostream>
#include<iomanip>
usingnamespacestd;
constintMAX=50;
floatx[MAX],y[MAX],h[MAX];
floatc[MAX],a[MAX],fxym[MAX];
floatf(intx1,intx2,intx3){
floata=(y[x3]-y[x2])/(x[x3]-x[x2]);
floatb=(y[x2]-y[x1])/(x[x2]-x[x1]);
return(a-b)/(x[x3]-x[x1]);
}//求差分
voidcal_m(intn){//用追赶法求解出弯矩向量M……
floatB[MAX];
B[0]=c[0]/2;
for(inti=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];
}
voidprintout(intn);
intmain(){
intn,i;charch;
do{
cout<<"Pleaseputinthenumberofthedots:";
cin>>n;
for(i=0;i<=n;i++){
cout<<"PleaseputinX"<<i<<':';
cin>>x[i];//cout<<endl;
cout<<"PleaseputinY"<<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 1:已知两端的一阶导数 2:两端的二阶导数已知 默认:自然边界条件 ";
intt;
floatf0,f1;
cin>>t;
switch(t){
case1:cout<<"PleaseputinY0'Y"<<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;
case2:cout<<"PleaseputinY0"Y"<<n<<"" ";
cin>>f0>>f1;
c[0]=a[n]=0;
fxym[0]=2*f0;fxym[n]=2*f1;
break;
default:cout<<"不可用 ";//待定
};//switch
for(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<<" 输出三次样条插值函数： ";
printout(n);
cout<<"Doyoutohaveanthertry?y/n:";
cin>>ch;
}while(ch=='y'||ch=='Y');
return0;
}
voidprintout(intn){
cout<<setprecision(6);
for(inti=0;i<n;i++){
cout<<i+1<<":["<<x[i]<<","<<x[i+1]<<"] "<<" ";
/*
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]<<") ";
cout<<endl;*/
floatt=fxym[i]/(6*h[i]);
if(t>0)cout<<t<<"*("<<x[i+1]<<"-x)^3";
elsecout<<-t<<"*(x-"<<x[i+1]<<")^3";
t=fxym[i+1]/(6*h[i]);
if(t>0)cout<<"+"<<t<<"*(x-"<<x[i]<<")^3";
elsecout<<"-"<<-t<<"*(x-"<<x[i]<<")^3";
cout<<" ";
t=(y[i]-fxym[i]*h[i]*h[i]/6)/h[i];
if(t>0)cout<<"+"<<t<<"*("<<x[i+1]<<"-x)";
elsecout<<"-"<<-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]<<")";
elsecout<<"-"<<-t<<"*(x-"<<x[i]<<")";
cout<<endl<<endl;
}
cout<<endl;
}
2、程序比较复杂的：
（程序前面的01.，02.，03.等等为语句编号，实际应用时请一一删除）01./*=======================================================================*/
02.#include<stdio.h>
03.////////////////////////////////////////////////////////////////////////////////
04.#defineMAXNUM50//定义样条数据区间个数最多为50个
05.typedefstructSPLINE//定义样条结构体，用于存储一条样条的所有信息
06.{//初始化数据输入
07.floatx[MAXNUM+1];//存储样条上的点的x坐标，最多51个点
08.floaty[MAXNUM+1];//存储样条上的点的y坐标，最多51个点
09.unsignedintpoint_num;//存储样条上的实际的点的个数
10.floatbegin_k1;//开始点的一阶导数信息
11.floatend_k1;//终止点的一阶导数信息
12.//floatbegin_k2;//开始点的二阶导数信息
13.//floatend_k2;//终止点的二阶导数信息
14.//计算所得的样条函数S(x)
15.floatk1[MAXNUM+1];//所有点的一阶导数信息
16.floatk2[MAXNUM+1];//所有点的二阶导数信息
17.//51个点之间有50个段，func[]存储每段的函数系数
18.floata3[MAXNUM],a1[MAXNUM];
19.floatb3[MAXNUM],b1[MAXNUM];
20.//分段函数的形式为Si(x)=a3[i]*{x(i+1)-x}^3+a1[i]*{x(i+1)-x}+
21.//b3[i]*{x-x(i)}^3+b1[i]*{x-x(i)}
22.//xi为x[i]的值，xi_1为x[i+1]的值
23.}SPLINE,*pSPLINE;
24.typedefintRESULT;//返回函数执行的结果状态，下面为具体的返回选项
25.#ifndefTRUE
26.#defineTRUE1
27.#endif
28.#ifndefFALSE
29.#defineFALSE-1
30.#endif
31.#ifndefNULL
32.#defineNULL0
33.#endif
34.#ifndefERR
35.#defineERR-2
36.#endif
37.//////////////////////////////////////////////////////////////////////////////////
38./*===============================================================================
39.***函数名称：Spline3()
40.***功能说明：完成三次样条差值，其中使用追赶法求解M矩阵
41.***入口参数：(pSPLINE)pLine样条结构体指针pLine中的x[],y[],num,begin_k1,end_k1
42.***出口参数：(pSPLINE)pLine样条结构体指针pLine中的函数参数
43.***返回参数：返回程序执行结果的状态TRUEorFALSE
44.================================================================================*/
45.RESULTSpline3(pSPLINEpLine)
46.{
47.floatH[MAXNUM]={0};//小区间的步长
48.floatFi[MAXNUM]={0};//中间量
49.floatU[MAXNUM+1]={0};//中间量
50.floatA[MAXNUM+1]={0};//中间量
51.floatD[MAXNUM+1]={0};//中间量
52.floatM[MAXNUM+1]={0};//M矩阵
53.floatB[MAXNUM+1]={0};//追赶法中间量
54.floatY[MAXNUM+1]={0};//追赶法中间变量
55.inti=0;
56.////////////////////////////////////////计算中间参数
57.if((pLine->point_num<3)||(pLine->point_num>MAXNUM+1))
58.{
59.returnERR;//输入数据点个数太少或太多
60.}
61.for(i=0;i<=pLine->point_num-2;i++)
62.{//求H[i]
63.H[i]=pLine->x[i+1]-pLine->x[i];
64.Fi[i]=(pLine->y[i+1]-pLine->y[i])/H[i];//求F[x(i),x(i+1)]
65.}
66.for(i=1;i<=pLine->point_num-2;i++)
67.{//求U[i]和A[i]和D[i]
68.U[i]=H[i-1]/(H[i-1]+H[i]);
69.A[i]=H[i]/(H[i-1]+H[i]);
70.D[i]=6*(Fi[i]-Fi[i-1])/(H[i-1]+H[i]);
71.}
72.//若边界条件为1号条件，则
73.U[i]=1;
74.A[0]=1;
75.D[0]=6*(Fi[0]-pLine->begin_k1)/H[0];
76.D[i]=6*(pLine->end_k1-Fi[i-1])/H[i-1];
77.//若边界条件为2号条件，则
78.//U[i]=0;
79.//A[0]=0;
80.//D[0]=2*begin_k2;
81.//D[i]=2*end_k2;
82./////////////////////////////////////////追赶法求解M矩阵
83.B[0]=A[0]/2;
84.for(i=1;i<=pLine->point_num-2;i++)
85.{
86.B[i]=A[i]/(2-U[i]*B[i-1]);
87.}
88.Y[0]=D[0]/2;
89.for(i=1;i<=pLine->point_num-1;i++)
90.{
91.Y[i]=(D[i]-U[i]*Y[i-1])/(2-U[i]*B[i-1]);
92.}
93.M[pLine->point_num-1]=Y[pLine->point_num-1];
94.for(i=pLine->point_num-1;i>0;i--)
95.{
96.M[i-1]=Y[i-1]-B[i-1]*M[i];
97.}
98.//////////////////////////////////////////计算方程组最终结果
99.for(i=0;i<=pLine->point_num-2;i++)
100.{
101.pLine->a3[i]=M[i]/(6*H[i]);
102.pLine->a1[i]=(pLine->y[i]-M[i]*H[i]*H[i]/6)/H[i];
103.pLine->b3[i]=M[i+1]/(6*H[i]);
104.pLine->b1[i]=(pLine->y[i+1]-M[i+1]*H[i]*H[i]/6)/H[i];
105.}
106.returnTRUE;
107.}
108.//////////////////////////////////////////////////////////////////////////////////
109.SPLINEline1;
110.pSPLINEpLine1=&line1;
111.//////////////////////////////////////////////////////////////////////////////////
112.main()
113.{
114.line1.x[0]=27.7;
115.line1.x[1]=28;
116.line1.x[2]=29;
117.line1.x[3]=30;
118.line1.y[0]=4.1;
119.line1.y[1]=4.3;
120.line1.y[2]=4.1;
121.line1.y[3]=3.0;
122.line1.point_num=4;
123.line1.begin_k1=3.0;
124.line1.end_k1=-4.0;
125.Spline3(pLine1);
126.return0;
127.}
128.//////////////////////////////////////////////////////////////////////////////////
㈡ 三次样条插值计算步骤
三次样条插值在实际中有着广泛的应用，在计算机上也容易实现。下面介绍用计算机求取三样条插值函数S（x）的算法步骤：
（1）输入初始节点离散数据xi，yi（i＝0，1，…，n）；
（2）依据式（6－46），计算hi＝xi－xi－1，λi和Ri（i＝1，…，n－1）；
（3）根据实际问题，从式（6－49）、式（6－51）和式（6－53）中选择一类对应的边界条件，求取v0，w0，u0，R0，un，vn，wn，Rn；
（4）根据形成的方程组（6－54）的特点，选用追赶法、高斯法等解方程组，求出Mi（i＝0，1，2，…，n）；
（5）依据式（6－41）、式（6－42），计算插值点的三样条插值函数值和该点的导数值。
㈢ C语言实现三次样条插值的子函数
void SPL(int n, double *x, double *y, int ni, double *xi, double *yi)； 是你所要。
已知 n 个点 x,y; x 必须已按顺序排好。要插值 ni 点，横坐标 xi[], 输出 yi[]。
程序里用double 型，保证计算精度。
SPL调用现成的程序。
现成的程序很多。端点处理方法不同，结果会有不同。想同matlab比较，你需 尝试 调用 spline（）函数 时，令 end1 为 1, 设 slope1 的值，令 end2 为 1 设 slope2 的值。
#include <stdio.h>
#include <math.h>
int spline (int n, int end1, int end2,
double slope1, double slope2,
double x[], double y[],
double b[], double c[], double d[],
int *iflag)
{
int nm1, ib, i, ascend;
double t;
nm1 = n - 1;
*iflag = 0;
if (n < 2)
{ /* no possible interpolation */
*iflag = 1;
goto LeaveSpline;
}
ascend = 1;
for (i = 1; i < n; ++i) if (x[i] <= x[i-1]) ascend = 0;
if (!ascend)
{
*iflag = 2;
goto LeaveSpline;
}
if (n >= 3)
{
d[0] = x[1] - x[0];
c[1] = (y[1] - y[0]) / d[0];
for (i = 1; i < nm1; ++i)
{
d[i] = x[i+1] - x[i];
b[i] = 2.0 * (d[i-1] + d[i]);
c[i+1] = (y[i+1] - y[i]) / d[i];
c[i] = c[i+1] - c[i];
}
/* ---- Default End conditions */
b[0] = -d[0];
b[nm1] = -d[n-2];
c[0] = 0.0;
c[nm1] = 0.0;
if (n != 3)
{
c[0] = c[2] / (x[3] - x[1]) - c[1] / (x[2] - x[0]);
c[nm1] = c[n-2] / (x[nm1] - x[n-3]) - c[n-3] / (x[n-2] - x[n-4]);
c[0] = c[0] * d[0] * d[0] / (x[3] - x[0]);
c[nm1] = -c[nm1] * d[n-2] * d[n-2] / (x[nm1] - x[n-4]);
}
/* Alternative end conditions -- known slopes */
if (end1 == 1)
{
b[0] = 2.0 * (x[1] - x[0]);
c[0] = (y[1] - y[0]) / (x[1] - x[0]) - slope1;
}
if (end2 == 1)
{
b[nm1] = 2.0 * (x[nm1] - x[n-2]);
c[nm1] = slope2 - (y[nm1] - y[n-2]) / (x[nm1] - x[n-2]);
}
/* Forward elimination */
for (i = 1; i < n; ++i)
{
t = d[i-1] / b[i-1];
b[i] = b[i] - t * d[i-1];
c[i] = c[i] - t * c[i-1];
}
/* Back substitution */
c[nm1] = c[nm1] / b[nm1];
for (ib = 0; ib < nm1; ++ib)
{
i = n - ib - 2;
c[i] = (c[i] - d[i] * c[i+1]) / b[i];
}
b[nm1] = (y[nm1] - y[n-2]) / d[n-2] + d[n-2] * (c[n-2] + 2.0 * c[nm1]);
for (i = 0; i < nm1; ++i)
{
b[i] = (y[i+1] - y[i]) / d[i] - d[i] * (c[i+1] + 2.0 * c[i]);
d[i] = (c[i+1] - c[i]) / d[i];
c[i] = 3.0 * c[i];
}
c[nm1] = 3.0 * c[nm1];
d[nm1] = d[n-2];
}
else
{
b[0] = (y[1] - y[0]) / (x[1] - x[0]);
c[0] = 0.0;
d[0] = 0.0;
b[1] = b[0];
c[1] = 0.0;
d[1] = 0.0;
}
LeaveSpline:
return 0;
}
double seval (int n, double u,
double x[], double y[],
double b[], double c[], double d[],
int *last)
{
int i, j, k;
double w;
i = *last;
if (i >= n-1) i = 0;
if (i < 0) i = 0;
if ((x[i] > u) || (x[i+1] < u))
{
i = 0;
j = n;
do
{
k = (i + j) / 2;
if (u < x[k]) j = k;
if (u >= x[k]) i = k;
}
while (j > i+1);
}
*last = i;
w = u - x[i];
w = y[i] + w * (b[i] + w * (c[i] + w * d[i]));
return (w);
}
void SPL(int n, double *x, double *y, int ni, double *xi, double *yi)
{
double *b, *c, *d;
int iflag,last,i;
b = (double *) malloc(sizeof(double) * n);
c = (double *)malloc(sizeof(double) * n);
d = (double *)malloc(sizeof(double) * n);
if (!d) { printf("no enough memory for b,c,d\n");}
else {
spline (n,0,0,0,0,x,y,b,c,d,&iflag);
if (iflag==0) printf("I got coef b,c,d now\n"); else printf("x not in order or other error\n");
for (i=0;i<ni;i++) yi[i] = seval(ni,xi[i],x,y,b,c,d,&last);
free(b);free(c);free(d);
};
}
main(){
double x[6]={0.,1.,2.,3.,4.,5};
double y[6]={0.,0.5,2.0,1.6,0.5,0.0};
double u[8]={0.5,1,1.5,2,2.5,3,3.5,4};
double s[8];
int i;
SPL(6, x,y, 8, u, s);
for (i=0;i<8;i++) printf("%lf %lf \n",u[i],s[i]);
return 0;
}
㈣ 涓夋℃牱𨱒℃彃鍊
璁维(x)婊¤冻镙锋湰镣硅佹眰锛屽垯鍙闇鍦ㄦ疮涓瀛愬尯闂碵 ]涓婄‘瀹1涓涓夋″氶”寮忥纴锅囱句负锛
锅囱炬湁n涓镣癸纴闇瑕乶-1𨱒＄嚎鎻忚堪锛屾疮𨱒＄嚎锲涗釜链鐭ユ暟锛 鍒欐湭鐭ユ暟涓鏁颁负4(n-1)銆傛樉铹朵腑闂达纸n-2锛変釜镣瑰叿链4涓绾︽潫𨱒′欢锛
涓ょ绔镣瑰瓨鍦ㄧ害𨱒烻( ) = f( )锛屽垯绾︽潫鏂圭▼链4(n-2)+2=4锛坣-1锛-2锛屾墍浠ワ纴镐荤殑链鐭ユ暟涓鏁版瘆鏂圭▼涓鏁板氢袱涓銆傛墍浠ラ渶瑕侀濆栫殑涓や釜绾︽潫锛屼簬鏄灏辨湁浜嗕笁绉嶈竟鐣屾浔浠剁殑鎻掑肩畻娉曘
S(x) 鍦 [ ]锛坖=1,2,⋯,n-1锛変笂鏄涓夋″氶”寮忥纴浜庢槸S"(x)鍦╗ ]涓婃槸涓娆″氶”寮忥纴锅囱维"(x) 鍦╗ ]锛坖=1,2,⋯,n-1锛変袱绔镣逛笂镄勫煎凡鐭ワ纴璁
鍏朵腑
瀵 杩涜屼袱娆＄Н鍒嗗彲寰楋细
浠ヤ笂鏄鍦 涓婃眰寰楃殑 钖岀悊鍙姹 锛屽皢 钖屾椂浠ｅ叆涓や釜鍑芥暟镵旂珛鏂圭▼锛屽彲浠ユ眰寰
灏 锛
姹傚煎悗寰楋细
钖岀悊鍒嗗埆鍐椤嚭 ,镵旂珛绛夊纺,绠鍖栧悗鍙寰楋细
鍦╩atlab瀹炵幇镞舵敞镒忥细n涓镣癸纴n-1𨱒＄嚎锛屼互涓婄烦阒垫槸鐢辩浉闾荤殑涓ゆ浔绾跨殑寰鍒嗘柟绋嬭仈绔嬭屾潵锛堜竴阒惰繛缁锛夛纴锲犳ゆ柟绋嬫讳釜鏁板噺灏戜简1锛岀烦阒典腑链塶-2涓鏂圭▼銆 鍙﹀栵纴鐢╩atlab瀹炵幇镞堕渶瑕佹敞镒忥纴matlab涓涓嬫爣浠1寮濮嬶纴鍏朵粬璇瑷涓嬫爣鍙鑳戒粠0寮濮嬨