📄 newton.cpp
字号:
//xi 0.5 1.0 1.5 2.0 2.5 3.0
//f(xi) 0.541688 -0.042060 0.375089 0.624706 -0.887597 0.099809
//输入节点数及节点数据，计算并输出差商表；
//输入待使用的插值多项式次数，待求的自变量 x，输出内插需要选择的节点及 Newton 插值多项式；
//用 3 次 Newton 插值多项式计算并输出 f (1.25)；
//用 5 次 Newton 插值多项式计算并输出 f (1.987)。
//
//
#include <stdio.h>
#include <malloc.h>
#include <conio.h>
#include <stdlib.h>
typedef struct vex
{
double x;
double y;
}vex,*pVex;
void function(pVex dvex,double F[5][5],int style,double x,double vex[],char* Newton,double result);
void main()
{
int N=6;
pVex dvex=(pVex)malloc(N*sizeof(vex));
dvex[0].x=0.5;dvex[0].y=0.541688;
dvex[1].x=1.0;dvex[1].y=-0.042060;
dvex[2].x=1.5;dvex[2].y=0.375089;
dvex[3].x=2.0;dvex[3].y=0.624706;
dvex[4].x=2.5;dvex[4].y=-0.887597;
dvex[5].x=3.0;dvex[5].y=0.099809;
//////////////////////////
int i,j;
double F[5][5];
///////////////////////
//差商表的形成过程
for(i=0;i<N-1;i++)
F[i][0]=(dvex[i+1].y-dvex[i].y)/(dvex[i+1].x-dvex[i].x);
for(j=1;j<N-1;j++)
for(i=j;i<N-1;i++)
F[i][j]=(F[i][j-1]-F[i-1][j-1])/(dvex[i+1].x-dvex[i-j].x);
/////////////////////////
printf("下面输出Newton差商表:\n");
printf("x值\ty值\t\t一阶差商\t二阶差商\t三阶差商\t四阶差商\t五阶差商\n");
printf("%.1f\t%f\n",dvex[0].x,dvex[0].y);
printf("%.1f\t%f\t%f\n",dvex[1].x,dvex[1].y,F[0][0]);
printf("%.1f\t%f\t%f\t%f\n",dvex[2].x,dvex[2].y,F[1][0],F[1][1]);
printf("%.1f\t%f\t%f\t%f\t%f\n",dvex[3].x,dvex[3].y,F[2][0],F[2][1],F[2][2]);
printf("%.1f\t%f\t%f\t%f\t%f\t%f\n",dvex[4].x,dvex[4].y,F[3][0],F[3][1],F[3][2],F[3][3]);
printf("%.1f\t%f\t%f\t%f\t%f\t%f\t%f\n",dvex[5].x,dvex[5].y,F[4][0],F[4][1],F[4][2],F[4][3],F[4][4]);
/////////////////////////
double pVex[5]={0,0,0,0,0};
char pNewton[200]="";
double result=0;
////////////////////////////////
printf("------------------------------------------------------------\n");
printf("下面用3次Newton插值多项式计算并输出f(1.125)\n");
function(dvex,F,3,1.125,pVex,pNewton,result);
printf("------------------------------------------------------------\n");
printf("下面用5次Newton插值多项式计算并输出f(1.987)");
function(dvex,F,5,1.987,pVex,pNewton,result);
////////////////////////////////
printf("------------------------------------------------------------\n");
printf("下面，请输入待使用的插值多项式次数，待求的自变量 x\n");
int nN;float dX;
for(;;)
{
printf("\n待使用的插值多项式次数(输入0退出程序):");
scanf("%d",&nN);
if(nN==0)exit(0);
while(nN<1||nN>5)
{
printf("输入的次数建议在1~5之间\n");
scanf("%d",&nN);
}
printf("待求的自变量 x:");
scanf("%f",&dX);
while(dX<dvex[0].x||dX>dvex[5].x)
{
printf("输入的自变量不在%f和%f之间\n",dvex[0].x,dvex[5].x);
scanf("%f",&dX);
}
printf("下面开始显示结果\n");
function(dvex,F,nN,dX,pVex,pNewton,result);
}
getch();
}
void function(pVex dvex,double F[5][5],int style,double x,double vex[],char* Newton,double result)
{
int i=0;
int vex_i;
int j=0;
switch(style)
{
case 1:
{
while(x>dvex[i].x)i++;
printf("\n所选节点为:\n");
for(vex_i=i-1,j=0;vex_i<i+1;vex_i++,j++)printf("%1.1f\t",vex[j]=dvex[vex_i].x);
sprintf(
Newton,
"f(x)=\n\t\t%f+\n\t\t%f*(x-%1.1f)",
dvex[i-1].y,
F[i-1][0],dvex[i-1].x
);
printf("\n表达式为:%s",Newton);
result=dvex[i-1].y+F[i-1][0]*(x-dvex[i-1].x);
printf("\n其f(%.6f)结果为%2.6f\n",x,result);
break;
}
case 2:
{
while(x>dvex[i].x)i++;
if(i==5)
{
printf("\n所选节点为:\n%1.1f\t%1.1f\t%1.1f\n",dvex[3].x,dvex[4].x,dvex[5].x);
sprintf(
Newton,
"f(x)=\n\t\t%f+\n\t\t%f*(x-%1.1f)+\n\t\t%f*(x-%1.1f)*(x-%1.1f)",
dvex[3].y,
F[3][0],dvex[3].x,
F[4][1],dvex[3].x,dvex[4].x
);
printf("\n表达式为:%s",Newton);
result=dvex[3].y+F[3][0]*(x-dvex[3].x)+F[4][1]*(x-dvex[4].x);
}
else
{
printf("\n所选节点为:\n");
for(vex_i=i-1,j=0;vex_i<i+2;vex_i++,j++)printf("%1.1f\t",vex[j]=dvex[vex_i].x);
sprintf(
Newton,
"f(x)=\n\t\t%f+\n\t\t%f*(x-%1.1f)+\n\t\t%f*(x-%1.1f)*(x-%1.1f)",
dvex[i-1].y,
F[i-1][0],dvex[i-1].x,
F[i][1],dvex[i-1].x,dvex[i].x
);
printf("\n表达式为:%s",Newton);
result=dvex[i-1].y+F[i-1][0]*(x-dvex[i-1].x)+F[i][1]*(x-dvex[i-1].x)*(x-dvex[i].x);
}
printf("\n其f(%.6f)结果为%2.6f\n",x,result);
break;
}
case 3:
{
while(x>dvex[i].x)i++;
//x[0] y[0] F[i][j]
//x[1] y[1] F[0][0]
//x[2] y[2] F[1][0] F[1][1]
//x[3] y[3] F[2][0] F[2][1] F[2][2]
//x[4] y[4] F[3][0] F[3][1] F[3][2] F[3][3]
//x[5] y[5] F[4][0] F[4][1] F[4][2] F[4][3] F[4][4]
if(i==5||x==4)
{
printf("\n所选节点为:\n%1.1f\t%1.1f\t%1.1f\t%1.1f\n",dvex[2].x,dvex[3].x,dvex[4].x,dvex[5].x);
sprintf(
Newton,
"f(x)=\n\t\t%f+\n\t\t%f*(x-%1.1f)+\n\t\t%f*(x-%1.1f)*(x-%1.1f)+\n\t\t%f*(x-%1.1f)*(x-%1.1f)*(x-%1.1f)",
dvex[2].y,
F[2][0],dvex[2].x,
F[3][1],dvex[2].x,dvex[3].x,
F[4][2],dvex[2].x,dvex[3].x,dvex[4].x
);
printf("\n表达式为:%s",Newton);
result=dvex[2].y+F[2][0]*(x-dvex[2].x)+F[3][1]*(x-dvex[2].x)*(x-dvex[3].x)+F[4][2]*(x-dvex[2].x)*(x-dvex[3].x)*(x-dvex[4].x);
}
else
{
printf("\n所选节点为:\n");
for(vex_i=i-1,j=0;vex_i<i+3;vex_i++,j++)printf("%1.1f\t",vex[j]=dvex[vex_i].x);
sprintf(
Newton,
"f(x)=\n\t\t%f+\n\t\t%f*(x-%1.1f)+\n\t\t%f*(x-%1.1f)*(x-%1.1f)+\n\t\t%f*(x-%1.1f)*(x-%1.1f)*(x-%1.1f)",
dvex[i-1].y,
F[i-1][0],dvex[i-1].x,
F[i][1],dvex[i-1].x,dvex[i].x,
F[i+1][2],dvex[i-1].x,dvex[i].x,dvex[i+1].x
);
printf("\n表达式为:%s",Newton);
result=dvex[i-1].y+F[i-1][0]*(x-dvex[i-1].x)+F[i][1]*(x-dvex[i-1].x)*(x-dvex[i].x)+F[i+1][2]*(x-dvex[i-1].x)*(x-dvex[i].x)*(x-dvex[i+1].x);
}
printf("\n其f(%.6f)结果为%2.6f\n",x,result);
break;
}
case 4:
{
while(x>dvex[i].x)i++;
if(i==5||i==4||x==3)
{
printf("\n所选节点为:\n%1.1f\t%1.1f\t%1.1f\t%1.1f\t%1.1f\n",dvex[1].x,dvex[2].x,dvex[3].x,dvex[4].x,dvex[5].x);
sprintf(
Newton,
"f(x)=\n\t\t%f+\n\t\t%f*(x-%1.1f)+\n\t\t%f*(x-%1.1f)*(x-%1.1f)+\n\t\t%f*(x-%1.1f)*(x-%1.1f)*(x-%1.1f)+\n\t\t%f*(x-%1.1f)*(x-%1.1f)*(x-%1.1f)*(x-%1.1f)",
dvex[1].y,
F[1][0],dvex[1].x,
F[2][1],dvex[1].x,dvex[2].x,
F[3][2],dvex[1].x,dvex[2].x,dvex[3].x,
F[4][3],dvex[1].x,dvex[3].x,dvex[3].x,dvex[4].x
);
printf("\n表达式为:%s",Newton);
result=dvex[2].y+F[2][0]*(x-dvex[2].x)+F[3][1]*(x-dvex[2].x)*(x-dvex[3].x)+F[4][1]*(x-dvex[2].x)*(x-dvex[3].x)*(x-dvex[4].x);
}
else
{
printf("\n所选节点为:\n");
for(vex_i=i-1;vex_i<i+4;vex_i++,j++)printf("%1.1f\t",vex[j]=dvex[vex_i].x);
sprintf(
Newton,
"f(x)=\n\t\t%f+\n\t\t%f*(x-%1.1f)+\n\t\t%f*(x-%1.1f)*(x-%1.1f)+\n\t\t%f*(x-%1.1f)*(x-%1.1f)*(x-%1.1f)+\n\t\t%f*(x-%1.1f)*(x-%1.1f)*(x-%1.1f)*(x-%1.1f)",
dvex[i-1].y,
F[i-1][0], dvex[i-1].x,
F[i][1], dvex[i-1].x, dvex[i].x,
F[i+1][2], dvex[i-1].x, dvex[i].x, dvex[i+1].x,
F[i+2][3], dvex[i-1].x, dvex[i].x, dvex[i+1].x, dvex[i+2].x
);
printf("\n表达式为:%s",Newton);
result=dvex[i-1].y+F[i-1][0]*(x-dvex[i-1].x)+F[i][1]*(x-dvex[i-1].x)*(x-dvex[i].x)+
F[i+1][2]*(x-dvex[i-1].x)*(x-dvex[i].x)*(x-dvex[i+1].x)+F[i+2][3]*(x-dvex[i-1].x)*(x-dvex[i].x)*(x-dvex[i+1].x)*(x-dvex[i+2].x);
}
printf("\n其f(%.6f)结果为%2.6f\n",x,result);
break;
}
case 5:
{
printf("\n所选节点为:\n");
for(j=0;j<6;j++)printf("%1.1f\t",dvex[j].x);
for(vex_i=0;vex_i<6;vex_i++)vex[i]=dvex[i].x;
sprintf(
Newton,
"f(x)=\n\t\t%f+\n\t\t%f*(x-%1.1f)+\n\t\t%f*(x-%1.1f)*(x-%1.1f)+\n\t\t%f*(x-%1.1f)*(x-%1.1f)*(x-%1.1f)+\n\t\t%f*(x-%1.1f)*(x-%1.1f)*(x-%1.1f)*(x-%1.1f)+\n\t\t%f*(x-%1.1f)*(x-%1.1f)*(x-%1.1f)*(x-%1.1f)*(x-%1.1f)",
dvex[0].y,
F[0][0],dvex[0].x,
F[1][1],dvex[0].x,dvex[1].x,
F[2][2],dvex[0].x,dvex[1].x,dvex[2].x,
F[3][3],dvex[0].x,dvex[1].x,dvex[2].x,dvex[3].x,
F[4][4],dvex[0].x,dvex[1].x,dvex[2].x,dvex[3].x,dvex[4].x
);
printf("\n表达式为:%s",Newton);
result=dvex[0].y+
F[0][0]*(x-dvex[0].x)+
F[1][1]*(x-dvex[0].x)*(x-dvex[1].x)+
F[2][2]*(x-dvex[0].x)*(x-dvex[1].x)*(x-dvex[2].x)+
F[3][3]*(x-dvex[0].x)*(x-dvex[1].x)*(x-dvex[2].x)*(x-dvex[3].x),
F[4][4]*(x-dvex[0].x)*(x-dvex[1].x)*(x-dvex[2].x)*(x-dvex[3].x)*(x-dvex[4].x);
printf("\n其f(%.6f)结果为%2.6f\n",x,result);
break;
}
}
}
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