拉格朗日，牛顿插值，高斯，龙贝格，牛顿迭代，牛顿-科特斯，雅克比，秦九昭，幂法，高斯塞德尔
（一）拉格朗日插值多项式
#include
#include
float lagrange(float *x,float *y,float xx,int n) /*拉格朗日插值算法*/
{ int i,j;
float *a,yy=0.0; /*a作为临时变量，记录拉格朗日插值多项式*/
a=(float *)malloc(n*sizeof(float));
for(i=0;i<=n-1;i++)
{ a[i]=y[i];
for(j=0;j<=n-1;j++)
if(j!=i) a[i]*=(xx-x[j])/(x[i]-x[j]);
yy+=a[i];
}
free(a);
return yy;
}
main()
{ int i,n;
float x[20],y[20],xx,yy;
printf("Input n:");
scanf("%d",&n);
if(n>=20) {printf("Error!The value of n must in (0,20)."); getch();return 1;}
if(n<=0) {printf("Error! The value of n must in (0,20)."); getch(); return 1;}
for(i=0;i<=n-1;i++)
{ printf("x[%d]:",i);
scanf("%f",&x[i]);
}
printf("\n");
for(i=0;i<=n-1;i++)
{ printf("y[%d]:",i);scanf("%f",&y[i]);}
printf("\n");
printf("Input xx:");
scanf("%f",&xx);
yy=lagrange(x,y,xx,n);
printf("x=%f,y=%f\n",xx,yy);
getch();
}
（二）牛顿插值多项式
#include
#include
void difference(float *x,float *y,int n)
{ float *f;
int k,i;
f=(float *)malloc(n*sizeof(float));
for(k=1;k<=n;k++)
{ f[0]=y[k];
for(i=0;i
f[i+1]=(f[i]-y[i])/(x[k]-x[i]);
y[k]=f[k];
}
return;
}
main()
{ int i,n;
float x[20],y[20],xx,yy;
printf("Input n:");
scanf("%d",&n);
if(n>=20) {printf("Error! The value of n must in (0,20)."); getch(); return 1;}
if(n<=0) {printf("Error! The value of n must in (0,20).");getch(); return 1;}
for(i=0;i<=n-1;i++)
{ printf("x[%d]:",i);
scanf("%f",&x[i]);
}
printf("\n");
for(i=0;i<=n-1;i++)
{ printf("y[%d]:",i);scanf("%f",&y[i]);}
printf("\n");
difference(x,(float *)y,n);
printf("Input xx:");
scanf("%f",&xx);
yy=y[20];
for(i=n-1;i>=0;i--) yy=yy*(xx-x[i])+y[i];
printf("NewtonInter(%f)=%f",xx,yy);
getch();
}
（三）高斯列主元消去法：
#include
#include
#define N 20
int main()
{ int n,i,j,k;
int mi,tmp,mx;
float a[N][N],b[N],x[N];
printf("\nInput n:");
scanf("%d",&n);
if(n>N)
{ printf("The input n should in(0,N)!\n");
getch();
return 1;
}
if(n<=0)
{ printf("The input n should in(0,N)!\n");
getch();
return 1;
}
printf("Now input a(i,j),i,j=0...%d:\n",n-1);
for(i=0;i
{ for(j=0;j
scanf("%f",&a[i][j]);}
printf("Now input b(i),i,j=0...%d:\n",n-1);
for(i=0;i
scanf("%f",&b[i]);
for(i=0;i
{ for(j=i+1,mi=i,mx=fabs(a[i][j]);j
if(fabs(a[j][i])>mx)
{ mi=j;
mx=fabs(a[j][i]);
}
if(i
{ tmp=b[i];b[i]=b[mi];b[mi]=tmp;
for(j=i;j
{ tmp=a[i][j];
a[i][j]=a[mi][j];
a[mi][j]=tmp;
}
}
for(j=i+1;j
{ tmp=-a[j][i]/a[i][i];
b[j]+=b[i]*tmp;
for(k=i;k
a[j][k]+=a[i][k]*tmp;
}
}
x[n-1]=b[n-1]/a[n-1][n-1];
for(i=n-2;i>=0;i--)
{ x[i]=b[i];
for(j=i+1;j
x[i]-=a[i][j]*x[j];
x[i]/=a[i][i];
}
for(i=0;i
printf("Answer:\n x[%d]=%f\n",i,x[i]);
getch();
return 0;
}
#include
#include
#define NUMBER 20
#define Esc 0x1b
#define Enter 0x0d
float A[NUMBER][NUMBER+1] ,ark;
int flag,n;
exchange(int r,int k);
float max(int k);
message();
main()
{
float x[NUMBER];
int r,k,i,j;
char celect;
clrscr();
printf("\n\nUse Gauss.");
printf("\n\n1.Jie please press Enter.");
printf("\n\n2.Exit press Esc.");
celect=getch();
if(celect==Esc)
exit(0);
printf("\n\n input n=");
scanf("%d",&n);
printf(" \n\nInput matrix A and B:");
for(i=1;i<=n;i++)
{
printf("\n\nInput a%d1--a%d%d and b%d:",i,i,n,i);
for(j=1;j<=n+1;j++) scanf("%f",&A[i][j]);
}
for(k=1;k<=n-1;k++)
{
ark=max(k);
if(ark==0)
{
printf("\n\nIt's wrong!");message();
}
else if(flag!=k)
exchange(flag,k);
for(i=k+1;i<=n;i++)
for(j=k+1;j<=n+1;j++)
A[i][j]=A[i][j]-A[k][j]*A[i][k]/A[k][k];
}
x[n]=A[n][n+1]/A[n][n];
for( k=n-1;k>=1;k--)
{
float me=0;
for(j=k+1;j<=n;j++)
{
me=me+A[k][j]*x[j];
}
x[k]=(A[k][n+1]-me)/A[k][k];
}
for(i=1;i<=n;i++)
{
printf(" \n\nx%d=%f",i,x[i]);
}
message();
}
exchange(int r,int k)
{
int i;
for(i=1;i<=n+1;i++)
A[0][i]=A[r][i];
for(i=1;i<=n+1;i++)
A[r][i]=A[k][i];
for(i=1;i<=n+1;i++)
A[k][i]=A[0][i];
}
float max(int k)
{
int i;
float temp=0;
for(i=k;i<=n;i++)
if(fabs(A[i][k])>temp)
{
temp=fabs(A[i][k]);
flag=i;
}
return temp;
}
message()
{
printf("\n\n Go on Enter ,Exit press Esc!");
switch(getch())
{
case Enter: main();
case Esc: exit(0);
default:{printf("\n\nInput error!");message();}
}
}
（四）龙贝格求积公式：
#include
#include
#define f(x) (sin(x)/x)
#define N 20
#define MAX 20
#define a 2
#define b 4
#define e 0.00001
float LBG(float p,float q,int n)
{ int i;
float sum=0,h=(q-p)/n;
for (i=1;i
sum+=f(p+i*h);
sum+=(f(p)+f(q))/2;
return(h*sum);
}
void main()
{ int i;
int n=N,m=0;
float T[MAX+1][2];
T[0][1]=LBG(a,b,n);
n*=2;
for(m=1;m
{ for(i=0;i
T[i][0]=T[i][1];
T[0][1]=LBG(a,b,n);
n*=2;
for(i=1;i<=m;i++)
T[i][1]=T[i-1][1]+(T[i-1][1]-T[i-1][0])/(pow(2,2*m)-1);
if((T[m-1][1]T[m][1]-e))
{ printf("Answer=%f\n",T[m][1]); getch();
return ;
}
}
}
（五）牛顿迭代公式：
#include
#include
#include
#define N 100
#define PS 1e-5
#define TA 1e-5
float Newton(float (*f)(float),float(*f1)(float),float x0 )
{ float x1,d=0;
int k=0;
do
{ x1= x0-f(x0)/f1(x0);
if((k++>N)||(fabs(f1(x1))
{ printf("\nFailed!");
getch();
exit();
}
d=(fabs(x1)<1?x1-x0:(x1-x0)/x1);
x0=x1;
printf("x(%d)=%f\n",k,x0);
}
while((fabs(d))>PS&&fabs(f(x1))>TA) ;
return x1;
}
float f(float x)
{ return x*x*x+x*x-3*x-3; }
float f1(float x)
{ return 3.0*x*x+2*x-3; }
void main()
{ float f(float);
float f1(float);
float x0,y0;
printf("Input x0: ");
scanf("%f",&x0);
printf("x(0)=%f\n",x0);
y0=Newton(f,f1,x0);
printf("\nThe root is x=%f\n",y0);
getch();
}
（六）牛顿-科特斯求积公式：
#include
#include
int NC(a,h,n,r,f)
float (*a)[];
float h;
int n,f;
float *r;
{ int nn,i;
float ds;
if(n>1000||n<2)
{ if (f)
printf("\n Faild! Check if 1\n",n);
return(-1);
}
if(n==2)
{ *r=0.5*((*a)[0]+(*a)[1])*(h);
return(0);
}
if (n-4==0)
{ *r=0;
*r=*r+0.375*(h)*((*a)[n-4]+3*(*a)[n-3]+3*(*a)[n-2]+(*a)[n-1]);
return(0);
}
if(n/2-(n-1)/2<=0)
nn=n;
else
nn=n-3;
ds=(*a)[0]-(*a)[nn-1];
for(i=2;i<=nn;i=i+2)
ds=ds+4*(*a)[i-1]+2*(*a)[i];
*r=ds*(h)/3;
if(n>nn)
*r=*r+0.375*(h)*((*a)[n-4]+3*(*a)[n-3]+3*(*a)[n-2]+(*a)[n-1]);
return(0);
}
main()
{
float h,r;
int n,ntf,f;
int i;
float a[16];
printf("Input the x[i](16):\n");
for(i=0;i<=15;i++)
scanf("%d",&a[i]);
h=0.2;
f=0;
ntf=NC(a,h,n,&r,f);
if(ntf==0)
printf("\nR=%f\n",r);
else
printf("\n Wrong!Return code=%d\n",ntf);
getch();
}
（七）雅克比迭代：
#include
#include
#define N 20
#define MAX 100
#define e 0.00001
int main()
{ int n;
int i,j,k;
float t;
float a[N][N],b[N][N],c[N],g[N],x[N],h[N];
printf("\nInput dim of n:"); scanf("%d",&n);
if(n>N)
{ printf("Faild! Check if 0
if(n<=0)
{printf("Faild! Check if 0
printf("Input a[i,j],i,j=0…%d:\n",n-1);
for(i=0;i
for(j=0;j
scanf("%f",&a[i][j]);
printf("Input c[i],i=0…%d:\n",n-1);
for(i=0;i
scanf("%f",&c[i]);
for(i=0;i
for(j=0;j
{ b[i][j]=-a[i][j]/a[i][i]; g[i]=c[i]/a[i][i]; }
for(i=0;i
{ for(j=0;j
h[j]=g[j];
{ for(k=0;k
{ if(j==k) continue; h[j]+=b[j][k]*x[k]; }
}
t=0;
for(j=0;j
if(t
for(j=0;j
x[j]=h[j];
if(t
{ printf("x_i=\n");
for(i=0;i
printf("x[%d]=%f\n",i,x[i]);
getch();
return 0;
}
printf("after %d repeat , return\n",MAX);
getch();
return 1;
}
getch();
}
(八)秦九昭算法：
#include
float qin(float a[],int n,float x)
{ float r=0;
int i;
for(i=n;i>=0;i--)
r=r*x+a[i];
return r;
}
main()
{ float a[50],x,r=0;
int n,i;
do
{ printf("Input frequency:");
scanf("%d",&n);
}
while(n<1);
printf("Input value:");
for(i=0;i<=n;i++)
scanf("%f",&a[i]);
printf("Input frequency:");
scanf("%f",&x);
r=qin(a,n,x);
printf("Answer:%f",r);
getch();
}
（九）幂法：
#include
#include
#define N 100
#define e 0.00001
#define n 3
float x[n]={0,0,1};
float a[n][n]={{2,3,2},{10,3,4},{3,6,1}};
float y[n];
main()
{ int i,j,k;
float xm,oxm;
oxm=0;
for(k=0;k
{ for(j=0;j
{ y[j]=0;
for(i=0;i
y[j]+=a[j][i]*x[i];
}
xm=0;
for(j=0;j
if(fabs(y[j])>xm) xm=fabs(y[j]);
for(j=0;j
y[j]/=xm;
for(j=0;j
x[j]=y[j];
if(fabs(xm-oxm)
{ printf("max:%f\n\n",xm);
printf("v[i]:\n");
for(k=0;k
break;
}
oxm=xm;
}
getch();
}
（十）高斯塞德尔：
#include
#include
#define N 20
#define M 99
float a[N][N];
float b[N];
int main()
{ int i,j,k,n;
float sum,no,d,s,x[N];
printf("\nInput dim of n:");
scanf("%d",&n);
if(n>N)
{ printf("Faild! Check if 0 "); getch();
return 1;
}
if(n<=0)
{ printf("Faild! Check if 0 ");getch();return 1;}
printf("Input a[i,j],i,j=0…%d:\n",n-1);
for(i=0;i
for(j=0;j
scanf("%f",&a[i][j]);
printf("Input b[i],i=0…%d:\n",n-1);
for(i=0;i
for(i=0;i
k=0;
printf("\nk=%dx=",k);
for(i=0;i
do
{ k++;
if(k>M){printf("\nError!\n”);getch();}
break;
}
no=0.0;
for(i=0;i
{ s=x[i];
sum=0.0;
for(j=0;j
if (j!=i) sum=sum+a[i][j]*x[j];
x[i]=(b[i]-sum)/a[i][i];
d=fabs(x[i]-s);
if (no
}
printf("\nk=%2dx=",k);
for(i=0;i
}
while (no>=0.1e-6);
if(no<0.1e-6)
{ printf("\n\n answer=\n");
printf("\nk=%d",k);
for (i=0;i
printf("\n x[%d]=%12.8f",i,x[i]);
}
getch();
} |
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