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

#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

#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;jscanf("%f",&a[i][j]);}

printf("Now input b(i),i,j=0...%d:\n",n-1);

for(i=0;iscanf("%f",&b[i]);

for(i=0;i{ for(j=i+1,mi=i,mx=fabs(a[i][j]);jif(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;iprintf("Answer:\n x[%d]=%f\n",i,x[i]);

getch();

return 0;

}

#include

#include

#define NUMBER 20

#define Esc0x1b

#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;isum+=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("\nFaild! 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 0if(n<=0)

{printf("Faild! Check if 0printf("Input a[i,j],i,j=0…%d:\n",n-1);

for(i=0;ifor(j=0;j

scanf("%f",&a[i][j]);

printf("Input c[i],i=0…%d:\n",n-1);

for(i=0;iscanf("%f",&c[i]);

for(i=0;ifor(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();

}