Modified Euler's Method for Numerical Solution of first order Differential Equation

#include<stdio.h>
#include<malloc.h>
#include<stdlib.h>
#include<math.h>
double dfxy(double x,double y);
FILE *fp;
int main(int argc, char* argv[])
{
double x0,y0,h,x,*xi,*yi,temp=0.0;
char c;
int n,i;
if(argc==1)
    {
    printf("\n Enter the value of h (interval-width):");
    scanf("%lf",&h);
    printf("\n Enter the initial value of x:");
    scanf("%lf",&x0);
    printf("\n Enter the initial value of y:");
    scanf("%lf",&y0);
    printf("\n Enter the value of x at which y is calculated:");
    scanf("%lf",&x);
    xi=(double *)malloc((n+1)*sizeof(double));
    yi=(double *)malloc((n+1)*sizeof(double));
    }
else if(argc==2)
    {
    /*fp=fopen(argv[1],"r");
    fscanf(fp,"%d",&n);
    fscanf(fp,"%lf",&lower_limit);
    fscanf(fp,"%lf",&upper_limit);
    xi=(double *)malloc((n+1)*sizeof(double));
    yi=(double *)malloc((n+1)*sizeof(double));*/
    }
else
    {
    printf("\n Invalid arguments, program will terminate:");
    exit(1);
    }
n=(x-x0)/h;
xi[0]=x0;
yi[0]=y0;
printf("\n\tX\t\t\tY\n");
printf("\tf(%+lf)\t\t=%+lf",xi[0],yi[0]);
i=1;
while(x0!=x)
    {
    yi[i]=yi[i-1]+dfxy(xi[i-1],yi[i-1])*h;
    x0=x0+h;
    xi[i]=x0;
    //printf("\n >>>>>>>>%lf",yi[i]);
    do
    {
    temp=yi[i];
    yi[i]=yi[i-1]+(h/2)*(dfxy(xi[i-1],yi[i-1])+dfxy(xi[i],yi[i]));
    //printf("\n >>>>>>>>%lf",yi[i]);
    }while(fabs(temp-yi[i])>0.00001);
    i++;
    xi[i]=x0;
    yi[i]=temp;
    printf("\n\tf(%+lf)\t\t=%+lf",xi[i],yi[i]);
    }
printf("\n");
return 0;
}
double dfxy(double x,double y)
    {
    return (x*x+y);
    }

Euler's Method for Numerical Solution of first order Differential Equation

#include<stdio.h>
#include<malloc.h>
#include<stdlib.h>
#include<math.h>
double dfxy(double x,double y);
FILE *fp;
int main(int argc, char* argv[])
{
double x0,y0,h,x,*xi,*yi,temp=0.0;
int n,i;
if(argc==1)
    {
    printf("\n Enter the value of h (interval-width):");
    scanf("%lf",&h);
    printf("\n Enter the initial value of x:");
    scanf("%lf",&x0);
    printf("\n Enter the initial value of y:");
    scanf("%lf",&y0);
    printf("\n Enter the value of x at which y is calculated:");
    scanf("%lf",&x);
    xi=(double *)malloc((n+1)*sizeof(double));
    yi=(double *)malloc((n+1)*sizeof(double));
    }
else if(argc==2)
    {
    /*fp=fopen(argv[1],"r");
    fscanf(fp,"%d",&n);
    fscanf(fp,"%lf",&lower_limit);
    fscanf(fp,"%lf",&upper_limit);
    xi=(double *)malloc((n+1)*sizeof(double));
    yi=(double *)malloc((n+1)*sizeof(double));*/
    }
else
    {
    printf("\n Invalid arguments, program will terminate:");
    exit(1);
    }
n=(x-x0)/h;
xi[0]=x0;
yi[0]=y0;
printf("\n\tX\t\t\tY\n");
printf("\t%+lf\t\t%+lf",xi[0],yi[0]);
i=1;
while(x0!=x)
    {
    yi[i]=yi[i-1]+dfxy(xi[i-1],yi[i-1])*h;
    x0=x0+h;
    xi[i]=x0;
    printf("\n\t%+lf\t\t%+lf",xi[i],yi[i]);
    }
printf("\n");
//printf("\n Value of the integral using Trapezoidal Rule is %+.8lf\n",invert==0 ? (h/2.0)*temp:-(h/2.0)*temp);
return 0;
}
double dfxy(double x,double y)
    {
    return (x+y);
    }

Lagrangian Interpolation

#include<stdio.h>
#include<malloc.h>
#include<stdlib.h>
FILE *fp;
int main(int argc, char *argv[])
{
double **A,*b,*xn,*xnplus1,*p,temp=1.0,x,value=0.0,**U;
int ROW,i,j,k,t=1;
char ch;
if(argc==1)
    {
    printf("\n Enter the number of values:");
    scanf("%d",&ROW);
    printf("\n Enter the value at which value has to be aprroximated:");
    scanf("%lf",&x);
    A=(double **)malloc((ROW+1)*sizeof(double*));
    for(i=0;i<=ROW;i++)
        {
        A[i]=(double *)malloc((ROW+1)*sizeof(double));
        }
    b=(double *)malloc((ROW+1)*sizeof(double));
    p=(double *)malloc((ROW+1)*sizeof(double));
    xn=(double *)malloc((ROW+1)*sizeof(double));
    xnplus1=(double *)malloc((ROW+1)*sizeof(double));
    for(i=1;i<=ROW;i++)
        {
        printf("\n Enter the value of X[%d]:",i);
        scanf("%lf",&A[i][1]);
        }
    for(i=1;i<=ROW;i++)
        {
        printf("\n Enter the value of Y[%d]:",i);
        scanf("%lf",&A[i][2]);
        }
       
    }
else if (argc==2)
    {
    fp=fopen(argv[1],"r");
        if(fp==NULL)
            {
            printf("\n File not found, program will terminate:");
            exit(0);
            }
    fscanf(fp,"%d",&ROW);
    fscanf(fp,"%lf",&x);
    A=(double **)malloc((ROW+1)*sizeof(double*));
    for(i=0;i<=ROW;i++)
        {
        A[i]=(double *)malloc((ROW+1)*sizeof(double));
        }
    b=(double *)malloc((ROW+1)*sizeof(double));
    p=(double *)malloc((ROW+1)*sizeof(double));
    xn=(double *)malloc((ROW+1)*sizeof(double));
    while(!feof(fp))
    {
    for(i=1;i<=ROW;i++)
    {
        for(j=1;j<=2;j++)
        {
        fscanf(fp,"%lf ",&A[i][j]);
        }
       
    }
    }
    fclose(fp);
    }
    else
    {
    printf("\n Invalid Arguments,program will terminate:\n");
    exit(0);
    }
printf("\n         X          Y");
printf("\n------------------------------------------------------------------\n");
for(i=1;i<=ROW;i++)
    {
    for(j=1;j<=2;j++)
        {
       
        A[i][j];
        printf("   %+lf",A[i][j]);
        }
    printf("\n");
    }
printf("\n------------------------------------------------------------------\n");
for(i=1;i<=ROW;i++)
    {
     for(j=1;j<=ROW;j++)
         {
         if(i==j)
             continue;
         else
             temp = temp * ((x-A[j][1])/(A[i][1]-A[j][1]));
         }
     //xn[i]=temp;
     value += temp*A[i][2];
     //printf("\n ------------------------------->%lf",temp);   
     temp=1.0;
    }   
printf("\n Value at %+lf by Lagrangian Interpolation formula is %+lf\n",x,value);
return 0;
}

Divided Difference

#include<stdio.h>
#include<malloc.h>
#include<stdlib.h>
void differenceTable(double **A,double **L,int ROW);
double result(double **A,double **L,double x,int ROW);
double factorial(int n);
FILE *fp;
int main(int argc, char *argv[])
{
double **A,*b,*xn,*xnplus1,*p,temp,x,**L,**U,lambda,error,t1=0.0;
int ROW,i,j,k=0,t=1,m=1,bo=1;
char ch;
if(argc==1)
    {
    printf("\n Enter the number of values:");
    scanf("%d",&ROW);
    printf("\n Enter the value at which value has to be aprroximated:");
    scanf("%lf",&x);
    A=(double **)malloc((ROW+1)*sizeof(double*));
    L=(double **)malloc((ROW+1)*sizeof(double*));
    U=(double **)malloc((ROW+1)*sizeof(double*));
    for(i=0;i<=ROW;i++)
        {
        A[i]=(double *)malloc((ROW+1)*sizeof(double));
        L[i]=(double *)malloc((ROW+1)*sizeof(double));
        U[i]=(double *)malloc((ROW+1)*sizeof(double));
        }
    b=(double *)malloc((ROW+1)*sizeof(double));
    p=(double *)malloc((ROW+1)*sizeof(double));
    xn=(double *)malloc((ROW+1)*sizeof(double));
    xnplus1=(double *)malloc((ROW+1)*sizeof(double));
    for(i=1;i<=ROW;i++)
        {
        printf("\n Enter the value of X[%d]:",i);
        scanf("%lf",&A[i][1]);
        }
    for(i=1;i<=ROW;i++)
        {
        printf("\n Enter the value of Y[%d]:",i);
        scanf("%lf",&A[i][2]);
        }
       
    }
else if (argc==2)
    {
    fp=fopen(argv[1],"r");
        if(fp==NULL)
            {
            printf("\n File not found, program will terminate:");
            exit(0);
            }
    fscanf(fp,"%d",&ROW);
    fscanf(fp,"%lf",&x);
    A=(double **)malloc((ROW+1)*sizeof(double*));
    L=(double **)malloc((ROW+1)*sizeof(double*));
    U=(double **)malloc((ROW+1)*sizeof(double*));
    for(i=0;i<=ROW;i++)
        {
        A[i]=(double *)malloc((ROW+1)*sizeof(double));
        L[i]=(double *)malloc((ROW+1)*sizeof(double));
        U[i]=(double *)malloc((ROW+1)*sizeof(double));
        }
    b=(double *)malloc((ROW+1)*sizeof(double));
    p=(double *)malloc((ROW+1)*sizeof(double));
    xn=(double *)malloc((ROW+1)*sizeof(double));
    xnplus1=(double *)malloc((ROW+1)*sizeof(double));
   
    while(!feof(fp))
    {
    for(i=1;i<=ROW;i++)
    {
        for(j=1;j<=2;j++)
        {
        fscanf(fp,"%lf ",&A[i][j]);
        }
       
    }
    }
    fclose(fp);
    }
    else
    {
    printf("\n Invalid Arguments,program will terminate:\n");
    exit(0);
    }
printf("\n         X          Y");
printf("\n------------------------------------------------------------------\n");
for(i=1;i<=ROW;i++)
    {
    for(j=1;j<=2;j++)
        {
       
        U[i][j]=A[i][j];
        printf("   %+lf",U[i][j]);
        }
    printf("\n");
    }
for(i=1;i<=ROW-1;i++)
    {
    for(j=ROW-i,t=1;j>=1;j--,t++)
        {
        L[t][i]=(U[t+1][2]-U[t][2])/(A[t+1+k][1]-A[t][1]);
        //printf("\n (%d,%d)------>%lf",t-1,t+1-1+k,L[t][i]);
        }
    for(t=1;t<=ROW-i;t++)
        U[t][2]=L[t][i];
    //printf("\n>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>");
    k++;
    }   
differenceTable(A,L,ROW);
printf("\n Value at %+lf by Divied Difference Interpolation formula is %+lf\n",x,result(A,L,x,ROW));
return 0;
}
double result(double **A,double **L,double x,int ROW)
{
int i,j;
double value=0,tmp=1.0;
for(i=0;i<ROW;i++)
{
    if(i==0)
        value +=A[1][2];
    else
    {
    for(j=1;j<=i;j++)
        tmp=tmp*(x-A[j][1]);
    tmp = tmp/factorial(i);
    tmp = tmp*L[1][i];
    value +=tmp;
    }
    if(i==0)
    printf("%2.2lf ",A[1][2]);
    else
    printf("%2.2lf ",L[1][i]);
    tmp=1.0;
}
return value;
}
double factorial(int n)
{
 if(n<=1)
     return 1;
 else
     return n*factorial(n-1);
}
void differenceTable(double **A,double **L,int ROW)
{
int **position,*ap,j,m,i;
position=(int **)malloc((ROW+1)*sizeof(int*));
ap=(int *)malloc((ROW+1)*sizeof(int));
for(i=0;i<=ROW;i++)
    position[i]=(int *)malloc((ROW+1)*sizeof(int));
int an,tmp;
tmp=ROW;
for(i=1;i<=ROW;i++)
    {
    an=1+(i-1)*2;
    ap[i]=an;
    for(j=1;j<=tmp;j++)
        {
        position[i][j]=an+(j-1)*4;
        }
    tmp--;
    }
tmp=ROW;
tmp=ROW+(ROW-1)*3;
int *match,tmp1,l,*pos,flag,z,k;
match=(int *)malloc((ROW+1)*sizeof(int));
pos=(int *)malloc((ROW+1)*sizeof(int));
for(i=0;i<=ROW;i++)
    {
    match[i]=0;
    pos[i]=0;
    }

printf("\n-------------------------------------Divided Difference Table----------------------------------\n");
printf("\n X        Y");
for(i=1;i<=ROW-1;i++)
    printf("                D^%d",i);
printf("\n----------------------------------------------------------------------------------------------\n");
for(i=1;i<=tmp;i++)
{
tmp1=ROW;

    for(l=1;l<=ROW;l++)
    {
    flag=0;
    for(m=1;m<=tmp1;m++)
        {
        if(i==position[l][m])
            {
            flag=1;
            match[l]=1;
            pos[m]=position[l][m];
            break;
            }
        } //inner for
    if(flag==1)
    {
    for(k=1;k<=ROW;k++)
        {
    if(match[k]==0)
    {
        printf("");
    }
    else
        {
        if(k==1)
            //printf("(%d,%d)+(%d,%d)|%d",(position[l][m]/4)+1,k,(position[l][m]/4)+1,k+1,i);
            printf("%+2.4lf\t %+2.4lf",A[(position[l][m]/4)+1][k],A[(position[l][m]/4)+1][k+1]);
        else
            {
            z=(position[l][m]-ap[k])/4+1;
            //printf("\t\t(%d,%d)|%d",z,k-1,i);
            printf("\t\t\t%+lf",L[z][k-1]);
            }
       
        }
        } //end of k-loop
           
    } //end of flag==1
   
    else
        {printf(" ");}
    tmp1--;
    for(k=0;k<=ROW;k++)
    {
    pos[k]=0;
    match[k]=0;
    }
    }
   
printf("\n");
}
printf("\n----------------------------------------------------------------------------------------------\n");
}

Aitken's Interpolation

#include<stdio.h>
#include<malloc.h>
#include<stdlib.h>
void differenceTable(double **A,double **L,int ROW);
FILE *fp;
int main(int argc, char *argv[])
{
double **A,*b,*xn,*xnplus1,*p,temp=0.0,x,**L,**U,lambda,error,t1=0.0;
int ROW,i,j,k,t=1,m=1,bo=1;
char ch;
if(argc==1)
{
printf("\n Enter the number of values:");
scanf("%d",&ROW);
printf("\n Enter the value at which value has to be aprroximated:");
scanf("%lf",&x);
A=(double **)malloc((ROW+1)*sizeof(double*));
L=(double **)malloc((ROW+1)*sizeof(double*));
U=(double **)malloc((ROW+1)*sizeof(double*));
for(i=0;i<=ROW;i++)
{
A[i]=(double *)malloc((ROW+1)*sizeof(double));
L[i]=(double *)malloc((ROW+1)*sizeof(double));
U[i]=(double *)malloc((ROW+1)*sizeof(double));
}
b=(double *)malloc((ROW+1)*sizeof(double));
p=(double *)malloc((ROW+1)*sizeof(double));
xn=(double *)malloc((ROW+1)*sizeof(double));
xnplus1=(double *)malloc((ROW+1)*sizeof(double));
for(i=1;i<=ROW;i++)
{
printf("\n Enter the value of X[%d]:",i);
scanf("%lf",&A[i][1]);
}
for(i=1;i<=ROW;i++)
{
printf("\n Enter the value of Y[%d]:",i);
scanf("%lf",&A[i][2]);
}

}
else if (argc==2)
{
fp=fopen(argv[1],"r");
if(fp==NULL)
{
printf("\n File not found, program will terminate:");
exit(0);
}
fscanf(fp,"%d",&ROW);
fscanf(fp,"%lf",&x);
A=(double **)malloc((ROW+1)*sizeof(double*));
L=(double **)malloc((ROW+1)*sizeof(double*));
U=(double **)malloc((ROW+1)*sizeof(double*));
for(i=0;i<=ROW;i++)
{
A[i]=(double *)malloc((ROW+1)*sizeof(double));
L[i]=(double *)malloc((ROW+1)*sizeof(double));
U[i]=(double *)malloc((ROW+1)*sizeof(double));
}
b=(double *)malloc((ROW+1)*sizeof(double));
p=(double *)malloc((ROW+1)*sizeof(double));
xn=(double *)malloc((ROW+1)*sizeof(double));
xnplus1=(double *)malloc((ROW+1)*sizeof(double));

while(!feof(fp))
{
for(i=1;i<=ROW;i++)
{
for(j=1;j<=2;j++)
{
fscanf(fp,"%lf ",&A[i][j]);
}

}
}
fclose(fp);
}
else
{
printf("\n Invalid Arguments,program will terminate:\n");
exit(0);
}
printf("\n         X          Y");
printf("\n------------------------------------------------------------------\n");
for(i=1;i<=ROW;i++)
{
for(j=1;j<=2;j++)
{

U[i][j]=A[i][j];
printf("   %+lf",U[i][j]);
}
printf("\n");
}
k=0;
for(i=1;i<=ROW-1;i++)
{
for(j=ROW-i,t=1;j>=1;j--,t++)
{
//printf("\n %lf-----------%lf",U[t+1+k][1],U[i][1]);
temp=1/(U[t+1+k][1]-U[i][1]);
//printf("\n (%d,%d)----------(%d,%d)",t+k,1,i-1,1);
//printf("\n yi-1(%lf)*x1-x(%lf)-------yi(%lf)*x0-x(%lf)",U[1][2],U[t+1+k][1]-x,U[1+t][2],U[i][1]-x);
temp = temp * ((U[t+1+k][1]-x)*U[1][2]-(U[i][1]-x)*U[t+1][2]);
L[t][i]=temp;
//printf("\n-----------%lf",temp);
}
for(t=1;t<=ROW-i;t++)
{
U[t][2]=L[t][i];
//printf("\n>>>>>>>>>>>%lf",U[t][2]);
}
k++;
temp=0.0;
//printf("\n___________________________________________________");
}
differenceTable(A,L,ROW);
printf("\n Value at %+lf by Aitken's Interpolation formula is %+lf\n",x,U[1][2]);
return 0;
}
void differenceTable(double **A,double **L,int ROW)
{
int **position,*ap,j,m,i;
position=(int **)malloc((ROW+1)*sizeof(int*));
ap=(int *)malloc((ROW+1)*sizeof(int));
for(i=0;i<=ROW;i++)
position[i]=(int *)malloc((ROW+1)*sizeof(int));
int an,tmp;
tmp=ROW;
for(i=1;i<=ROW;i++)
{
an=1+(i-1)*2;
ap[i]=an;
for(j=1;j<=tmp;j++)
{
position[i][j]=an+(j-1)*4;
}
tmp--;
}
tmp=ROW;
tmp=ROW+(ROW-1)*3;
int *match,tmp1,l,*pos,flag,z,k;
match=(int *)malloc((ROW+1)*sizeof(int));
pos=(int *)malloc((ROW+1)*sizeof(int));
for(i=0;i<=ROW;i++)
{
match[i]=0;
pos[i]=0;
}

printf("\n-----------------------------Aitken's Table-----------------------------\n");
printf("\n X       Y");
for(i=1;i<=ROW-1;i++)
printf("           D_%d",i);
printf("\n----------------------------------------------------------------------------------\n");
for(i=1;i<=tmp;i++)
{
tmp1=ROW;

for(l=1;l<=ROW;l++)
{
flag=0;
for(m=1;m<=tmp1;m++)
{
if(i==position[l][m])
{
flag=1;
match[l]=1;
pos[m]=position[l][m];
break;
}
} //inner for
if(flag==1)
{
for(k=1;k<=ROW;k++)
{
if(match[k]==0)
{
printf("");
}
else
{
if(k==1)
//printf("(%d,%d)+(%d,%d)|%d",(position[l][m]/4)+1,k,(position[l][m]/4)+1,k+1,i);
printf("%+2.2lf\t%+2.2lf",A[(position[l][m]/4)+1][k],A[(position[l][m]/4)+1][k+1]);
else
{
z=(position[l][m]-ap[k])/4+1;
//printf("\t\t(%d,%d)|%d",z,k-1,i);
printf("\t\t%+lf",L[z][k-1]);
}

}
} //end of k-loop

} //end of flag==1

else
{printf(" ");}
tmp1--;
for(k=0;k<=ROW;k++)
{
pos[k]=0;
match[k]=0;
}
}

printf("\n");
}
printf("\n--------------------------------------------------------------------------\n");
}

Gauss Central Backward Interpolation

#include<stdio.h>
#include<malloc.h>
#include<stdlib.h>
void differenceTable(double **A,double **L,int ROW);
double result(double **A,double **L,double x,int ROW);
double factorial(int n);
FILE *fp;
int main(int argc, char *argv[])
{
double **A,*b,*xn,*xnplus1,*p,temp,x,**L,**U,lambda,error,t1=0.0;
int ROW,i,j,k,t=1,m=1,bo=1;
char ch;
if(argc==1)
{
printf("\n Enter the number of values:");
scanf("%d",&ROW);
printf("\n Enter the value at which value has to be aprroximated:");
scanf("%lf",&x);
A=(double **)malloc((ROW+1)*sizeof(double*));
L=(double **)malloc((ROW+1)*sizeof(double*));
U=(double **)malloc((ROW+1)*sizeof(double*));
for(i=0;i<=ROW;i++)
{
A[i]=(double *)malloc((ROW+1)*sizeof(double));
L[i]=(double *)malloc((ROW+1)*sizeof(double));
U[i]=(double *)malloc((ROW+1)*sizeof(double));
}
b=(double *)malloc((ROW+1)*sizeof(double));
p=(double *)malloc((ROW+1)*sizeof(double));
xn=(double *)malloc((ROW+1)*sizeof(double));
xnplus1=(double *)malloc((ROW+1)*sizeof(double));
for(i=1;i<=ROW;i++)
{
printf("\n Enter the value of X[%d]:",i);
scanf("%lf",&A[i][1]);
}
for(i=1;i<=ROW;i++)
{
printf("\n Enter the value of Y[%d]:",i);
scanf("%lf",&A[i][2]);
}

}
else if (argc==2)
{
fp=fopen(argv[1],"r");
if(fp==NULL)
{
printf("\n File not found, program will terminate:");
exit(0);
}
fscanf(fp,"%d",&ROW);
fscanf(fp,"%lf",&x);
A=(double **)malloc((ROW+1)*sizeof(double*));
L=(double **)malloc((ROW+1)*sizeof(double*));
U=(double **)malloc((ROW+1)*sizeof(double*));
for(i=0;i<=ROW;i++)
{
A[i]=(double *)malloc((ROW+1)*sizeof(double));
L[i]=(double *)malloc((ROW+1)*sizeof(double));
U[i]=(double *)malloc((ROW+1)*sizeof(double));
}
b=(double *)malloc((ROW+1)*sizeof(double));
p=(double *)malloc((ROW+1)*sizeof(double));
xn=(double *)malloc((ROW+1)*sizeof(double));
xnplus1=(double *)malloc((ROW+1)*sizeof(double));

while(!feof(fp))
{
for(i=1;i<=ROW;i++)
{
for(j=1;j<=2;j++)
{
fscanf(fp,"%lf ",&A[i][j]);
}

}
}
fclose(fp);
}
else
{
printf("\n Invalid Arguments,program will terminate:\n");
exit(0);
}
printf("\n         X          Y");
printf("\n------------------------------------------------------------------\n");
for(i=1;i<=ROW;i++)
{
for(j=1;j<=2;j++)
{

U[i][j]=A[i][j];
printf("   %+lf",U[i][j]);
}
printf("\n");
}
for(i=1;i<=ROW-1;i++)
{
for(j=ROW-i,t=1;j>=1;j--,t++)
{
L[t][i]=(U[t+1][2]-U[t][2]);
//printf("\n %lf-----%lf",U[t+1][2],U[t][2]);
}
for(t=1;t<=ROW-i;t++)
U[t][2]=L[t][i];
}
differenceTable(A,L,ROW);
printf("\n Value at %+lf by Gauss Backward Interpolation formula is %+lf\n",x,result(A,L,x,ROW));
return 0;
}
double result(double **A,double **L,double x,int ROW)
{
int i,j,flag=1,t,k,sta;
t=(ROW+1)/2;
double h,value=0,p,tmp=1.0;
h=A[2][1]-A[1][1];
//printf("\np is ***********%lf",h);
p=(x-A[t][1])/h;
//printf("\np is ***********%lf",p);
for(i=0;i<ROW;i++)
{
if(i==0)
{
value +=A[t][2];
//printf("\n*************%lf",value);
}
else
{
k=0;
sta=1;
for(j=1;j<=i;j++)
{
//printf("\nTTTTTTTTTTTTT %d",i);
if(j==1)
{
tmp=tmp*p;
//printf("\t>>>>>0");
}
else
{
tmp=tmp*(p+sta);
//printf("\t>>>>>%d",sta);
sta=sta*(-1);
k++;
}
if(k==2)
{
sta++;
k=0;
}
}
tmp = tmp/factorial(i);
//printf("\n Value of temp is %lf",tmp);
if(flag==1)
{
//printf("\n Value of t is %d",t);
t--;
tmp = tmp*L[t][i];
printf("\n******%d***************%lf",i,L[t][i]);
flag=0;
}
else
{
tmp = tmp*L[t][i];
flag=1;
printf("\n******%d***************%lf",i,L[t][i]);
}
//printf("\n Value of temp is %lf",tmp);
//printf("\n*****%d******%lf**********%lf",i,L[t][i],tmp);
//printf("\n***********%lf",tmp);
value +=tmp;
}
tmp=1.0;
}
return value;
}
double factorial(int n)
{
 if(n<=1)
  return 1;
 else
  return n*factorial(n-1);
}
void differenceTable(double **A,double **L,int ROW)
{
int **position,*ap,j,m,i;
position=(int **)malloc((ROW+1)*sizeof(int*));
ap=(int *)malloc((ROW+1)*sizeof(int));
for(i=0;i<=ROW;i++)
position[i]=(int *)malloc((ROW+1)*sizeof(int));
int an,tmp;
tmp=ROW;
for(i=1;i<=ROW;i++)
{
an=1+(i-1)*2;
ap[i]=an;
for(j=1;j<=tmp;j++)
{
position[i][j]=an+(j-1)*4;
}
tmp--;
}
tmp=ROW;
tmp=ROW+(ROW-1)*3;
int *match,tmp1,l,*pos,flag,z,k;
match=(int *)malloc((ROW+1)*sizeof(int));
pos=(int *)malloc((ROW+1)*sizeof(int));
for(i=0;i<=ROW;i++)
{
match[i]=0;
pos[i]=0;
}

printf("\n-----------------------------Central Difference Table-----------------------------\n");
printf("\n X       Y");
for(i=1;i<=ROW-1;i++)
printf("      D_%d",i);
printf("\n----------------------------------------------------------------------------------\n");
for(i=1;i<=tmp;i++)
{
tmp1=ROW;

for(l=1;l<=ROW;l++)
{
flag=0;
for(m=1;m<=tmp1;m++)
{
if(i==position[l][m])
{
flag=1;
match[l]=1;
pos[m]=position[l][m];
break;
}
} //inner for
if(flag==1)
{
for(k=1;k<=ROW;k++)
{
if(match[k]==0)
{
printf("");
}
else
{
if(k==1)
//printf("(%d,%d)+(%d,%d)|%d",(position[l][m]/4)+1,k,(position[l][m]/4)+1,k+1,i);
printf("%+2.2lf\t%+2.2lf",A[(position[l][m]/4)+1][k],A[(position[l][m]/4)+1][k+1]);
else
{
z=(position[l][m]-ap[k])/4+1;
//printf("\t\t(%d,%d)|%d",z,k-1,i);
printf("\t\t%+lf",L[z][k-1]);
}

}
} //end of k-loop

} //end of flag==1

else
{printf(" ");}
tmp1--;
for(k=0;k<=ROW;k++)
{
pos[k]=0;
match[k]=0;
}
}

printf("\n");
}
printf("\n--------------------------------------------------------------------------\n");
}

Gauss Central Forward Interpolation

#include<stdio.h>
#include<malloc.h>
#include<stdlib.h>
void differenceTable(double **A,double **L,int ROW);
double result(double **A,double **L,double x,int ROW);
double factorial(int n);
FILE *fp;
int main(int argc, char *argv[])
{
double **A,*b,*xn,*xnplus1,*p,temp,x,**L,**U,lambda,error,t1=0.0;
int ROW,i,j,k,t=1,m=1,bo=1;
char ch;
if(argc==1)
{
printf("\n Enter the number of values:");
scanf("%d",&ROW);
printf("\n Enter the value at which value has to be aprroximated:");
scanf("%lf",&x);
A=(double **)malloc((ROW+1)*sizeof(double*));
L=(double **)malloc((ROW+1)*sizeof(double*));
U=(double **)malloc((ROW+1)*sizeof(double*));
for(i=0;i<=ROW;i++)
{
A[i]=(double *)malloc((ROW+1)*sizeof(double));
L[i]=(double *)malloc((ROW+1)*sizeof(double));
U[i]=(double *)malloc((ROW+1)*sizeof(double));
}
b=(double *)malloc((ROW+1)*sizeof(double));
p=(double *)malloc((ROW+1)*sizeof(double));
xn=(double *)malloc((ROW+1)*sizeof(double));
xnplus1=(double *)malloc((ROW+1)*sizeof(double));
for(i=1;i<=ROW;i++)
{
printf("\n Enter the value of X[%d]:",i);
scanf("%lf",&A[i][1]);
}
for(i=1;i<=ROW;i++)
{
printf("\n Enter the value of Y[%d]:",i);
scanf("%lf",&A[i][2]);
}

}
else if (argc==2)
{
fp=fopen(argv[1],"r");
if(fp==NULL)
{
printf("\n File not found, program will terminate:");
exit(0);
}
fscanf(fp,"%d",&ROW);
fscanf(fp,"%lf",&x);
A=(double **)malloc((ROW+1)*sizeof(double*));
L=(double **)malloc((ROW+1)*sizeof(double*));
U=(double **)malloc((ROW+1)*sizeof(double*));
for(i=0;i<=ROW;i++)
{
A[i]=(double *)malloc((ROW+1)*sizeof(double));
L[i]=(double *)malloc((ROW+1)*sizeof(double));
U[i]=(double *)malloc((ROW+1)*sizeof(double));
}
b=(double *)malloc((ROW+1)*sizeof(double));
p=(double *)malloc((ROW+1)*sizeof(double));
xn=(double *)malloc((ROW+1)*sizeof(double));
xnplus1=(double *)malloc((ROW+1)*sizeof(double));

while(!feof(fp))
{
for(i=1;i<=ROW;i++)
{
for(j=1;j<=2;j++)
{
fscanf(fp,"%lf ",&A[i][j]);
}

}
}
fclose(fp);
}
else
{
printf("\n Invalid Arguments,program will terminate:\n");
exit(0);
}
printf("\n         X          Y");
printf("\n------------------------------------------------------------------\n");
for(i=1;i<=ROW;i++)
{
for(j=1;j<=2;j++)
{

U[i][j]=A[i][j];
printf("   %+lf",U[i][j]);
}
printf("\n");
}
for(i=1;i<=ROW-1;i++)
{
for(j=ROW-i,t=1;j>=1;j--,t++)
{
L[t][i]=(U[t+1][2]-U[t][2]);
//printf("\n %lf-----%lf",U[t+1][2],U[t][2]);
}
for(t=1;t<=ROW-i;t++)
U[t][2]=L[t][i];
}
differenceTable(A,L,ROW);
printf("\n Value at %+lf by Gauss Forward Interpolation formula is %+lf\n",x,result(A,L,x,ROW));
return 0;
}
double result(double **A,double **L,double x,int ROW)
{
int i,j,flag=1,t,k,sta;
t=(ROW+1)/2;
double h,value=0,p,tmp=1.0;
h=A[2][1]-A[1][1];
//printf("\np is ***********%lf",h);
p=(x-A[t][1])/h;
//printf("\np is ***********%lf",p);
for(i=0;i<ROW;i++)
{
if(i==0)
{
value +=A[t][2];
//printf("\n*************%lf",value);
}
else
{
k=0;
sta=1;
for(j=1;j<=i;j++)
{
//printf("\nTTTTTTTTTTTTT %d",i);
if(j==1)
{
tmp=tmp*p;
//printf("\t>>>>>0");
}
else
{
tmp=tmp*(p-sta);
//printf("\t>>>>>%d",sta);
sta=sta*(-1);
k++;
}
if(k==2)
{
sta++;
k=0;
}
}
tmp = tmp/factorial(i);
//printf("\n Value of temp is %lf",tmp);
if(flag==1)
{
tmp = tmp*L[t][i];
flag=0;
}
else
{
t=t-1;
tmp = tmp*L[t][i];
flag=1;
}
//printf("\n Value of temp is %lf",tmp);
//printf("\n*****%d******%lf**********%lf",i,L[t][i],tmp);
//printf("\n***********%lf",tmp);
value +=tmp;
}
tmp=1.0;
}
return value;
}
double factorial(int n)
{
 if(n<=1)
  return 1;
 else
  return n*factorial(n-1);
}
void differenceTable(double **A,double **L,int ROW)
{
int **position,*ap,j,m,i;
position=(int **)malloc((ROW+1)*sizeof(int*));
ap=(int *)malloc((ROW+1)*sizeof(int));
for(i=0;i<=ROW;i++)
position[i]=(int *)malloc((ROW+1)*sizeof(int));
int an,tmp;
tmp=ROW;
for(i=1;i<=ROW;i++)
{
an=1+(i-1)*2;
ap[i]=an;
for(j=1;j<=tmp;j++)
{
position[i][j]=an+(j-1)*4;
}
tmp--;
}
tmp=ROW;
tmp=ROW+(ROW-1)*3;
int *match,tmp1,l,*pos,flag,z,k;
match=(int *)malloc((ROW+1)*sizeof(int));
pos=(int *)malloc((ROW+1)*sizeof(int));
for(i=0;i<=ROW;i++)
{
match[i]=0;
pos[i]=0;
}

printf("\n-----------------------------Central Difference Table-----------------------------\n");
printf("\n X       Y");
for(i=1;i<=ROW-1;i++)
printf("      D_%d",i);
printf("\n----------------------------------------------------------------------------------\n");
for(i=1;i<=tmp;i++)
{
tmp1=ROW;

for(l=1;l<=ROW;l++)
{
flag=0;
for(m=1;m<=tmp1;m++)
{
if(i==position[l][m])
{
flag=1;
match[l]=1;
pos[m]=position[l][m];
break;
}
} //inner for
if(flag==1)
{
for(k=1;k<=ROW;k++)
{
if(match[k]==0)
{
printf("");
}
else
{
if(k==1)
//printf("(%d,%d)+(%d,%d)|%d",(position[l][m]/4)+1,k,(position[l][m]/4)+1,k+1,i);
printf("%+2.2lf\t%+2.2lf",A[(position[l][m]/4)+1][k],A[(position[l][m]/4)+1][k+1]);
else
{
z=(position[l][m]-ap[k])/4+1;
//printf("\t\t(%d,%d)|%d",z,k-1,i);
printf("\t\t%+lf",L[z][k-1]);
}

}
} //end of k-loop

} //end of flag==1

else
{printf(" ");}
tmp1--;
for(k=0;k<=ROW;k++)
{
pos[k]=0;
match[k]=0;
}
}

printf("\n");
}
printf("\n--------------------------------------------------------------------------\n");
}

Newton's Backward Interpolation

#include<stdio.h>
#include<malloc.h>
#include<stdlib.h>
void differenceTable(double **A,double **L,int ROW);
double result(double **A,double **L,double x,int ROW);
double factorial(int n);
FILE *fp;
int main(int argc, char *argv[])
{
double **A,*b,*xn,*xnplus1,*p,temp,x,**L,**U,lambda,error,t1=0.0;
int ROW,i,j,k,t=1,m=1,bo=1;
char ch;
if(argc==1)
{
printf("\n Enter the number of values:");
scanf("%d",&ROW);
printf("\n Enter the value at which value has to be aprroximated:");
scanf("%lf",&x);
A=(double **)malloc((ROW+1)*sizeof(double*));
L=(double **)malloc((ROW+1)*sizeof(double*));
U=(double **)malloc((ROW+1)*sizeof(double*));
for(i=0;i<=ROW;i++)
{
A[i]=(double *)malloc((ROW+1)*sizeof(double));
L[i]=(double *)malloc((ROW+1)*sizeof(double));
U[i]=(double *)malloc((ROW+1)*sizeof(double));
}
b=(double *)malloc((ROW+1)*sizeof(double));
p=(double *)malloc((ROW+1)*sizeof(double));
xn=(double *)malloc((ROW+1)*sizeof(double));
xnplus1=(double *)malloc((ROW+1)*sizeof(double));
for(i=1;i<=ROW;i++)
{
printf("\n Enter the value of X[%d]:",i);
scanf("%lf",&A[i][1]);
}
for(i=1;i<=ROW;i++)
{
printf("\n Enter the value of Y[%d]:",i);
scanf("%lf",&A[i][2]);
}

}
else if (argc==2)
{
fp=fopen(argv[1],"r");
if(fp==NULL)
{
printf("\n File not found, program will terminate:");
exit(0);
}
fscanf(fp,"%d",&ROW);
fscanf(fp,"%lf",&x);
A=(double **)malloc((ROW+1)*sizeof(double*));
L=(double **)malloc((ROW+1)*sizeof(double*));
U=(double **)malloc((ROW+1)*sizeof(double*));
for(i=0;i<=ROW;i++)
{
A[i]=(double *)malloc((ROW+1)*sizeof(double));
L[i]=(double *)malloc((ROW+1)*sizeof(double));
U[i]=(double *)malloc((ROW+1)*sizeof(double));
}
b=(double *)malloc((ROW+1)*sizeof(double));
p=(double *)malloc((ROW+1)*sizeof(double));
xn=(double *)malloc((ROW+1)*sizeof(double));
xnplus1=(double *)malloc((ROW+1)*sizeof(double));

while(!feof(fp))
{
for(i=1;i<=ROW;i++)
{
for(j=1;j<=2;j++)
{
fscanf(fp,"%lf ",&A[i][j]);
}

}
}
fclose(fp);
}
else
{
printf("\n Invalid Arguments,program will terminate:\n");
exit(0);
}
printf("\n         X          Y");
printf("\n------------------------------------------------------------------\n");
for(i=1;i<=ROW;i++)
{
for(j=1;j<=2;j++)
{

U[i][j]=A[i][j];
printf("   %+lf",U[i][j]);
}
printf("\n");
}
for(i=1;i<=ROW-1;i++)
{
for(j=ROW-i,t=1;j>=1;j--,t++)
{
L[t][i]=U[t+1][2]-U[t][2];
}
for(t=1;t<=ROW-i;t++)
U[t][2]=L[t][i];
}
differenceTable(A,L,ROW);
printf("\n Value at %+lf by Newton's Backward Interpolation formula is %+lf\n",x,result(A,L,x,ROW));
return 0;
}
double result(double **A,double **L,double x,int ROW)
{
int i,j,flag=1,t;

double h,value=0,p,tmp=1.0;
h=A[2][1]-A[1][1];
p=(A[ROW][1]-x)/h;
for(i=0,t=ROW+1;i<=ROW;i++,t--)
{
if(i==0)
value +=A[ROW][2];
else
{
for(j=1;j<=i;j++)
tmp=tmp*(p-j+1)*(-1);
tmp = tmp/factorial(i);
tmp = tmp*L[t-1][i];
value +=tmp;
}
tmp=1.0;
}
return value;
}
double factorial(int n)
{
 if(n<=1)
  return 1;
 else
  return n*factorial(n-1);
}
void differenceTable(double **A,double **L,int ROW)
{
int **position,*ap,j,m,i;
position=(int **)malloc((ROW+1)*sizeof(int*));
ap=(int *)malloc((ROW+1)*sizeof(int));
for(i=0;i<=ROW;i++)
position[i]=(int *)malloc((ROW+1)*sizeof(int));
int an,tmp;
tmp=ROW;
for(i=1;i<=ROW;i++)
{
an=1+(i-1)*2;
ap[i]=an;
for(j=1;j<=tmp;j++)
{
position[i][j]=an+(j-1)*4;
}
tmp--;
}
tmp=ROW;
tmp=ROW+(ROW-1)*3;
int *match,tmp1,l,*pos,flag,z,k;
match=(int *)malloc((ROW+1)*sizeof(int));
pos=(int *)malloc((ROW+1)*sizeof(int));
for(i=0;i<=ROW;i++)
{
match[i]=0;
pos[i]=0;
}

printf("\n-----------------------------Difference Table-----------------------------\n");
printf("\n X       Y");
for(i=1;i<=ROW-1;i++)
printf("      D_%d",i);
printf("\n--------------------------------------------------------------------------\n");
for(i=1;i<=tmp;i++)
{
tmp1=ROW;

for(l=1;l<=ROW;l++)
{
flag=0;
for(m=1;m<=tmp1;m++)
{
if(i==position[l][m])
{
flag=1;
match[l]=1;
pos[m]=position[l][m];
break;
}
} //inner for
if(flag==1)
{
for(k=1;k<=ROW;k++)
{
if(match[k]==0)
{
printf("");
}
else
{
if(k==1)
//printf("(%d,%d)+(%d,%d)|%d",(position[l][m]/4)+1,k,(position[l][m]/4)+1,k+1,i);
printf("%+2.2lf\t%+2.2lf",A[(position[l][m]/4)+1][k],A[(position[l][m]/4)+1][k+1]);
else
{
z=(position[l][m]-ap[k])/4+1;
//printf("\t\t(%d,%d)|%d",z,k-1,i);
printf("\t\t%+2.2lf",L[z][k-1]);
}

}
} //end of k-loop

} //end of flag==1

else
{printf(" ");}
tmp1--;
for(k=0;k<=ROW;k++)
{
pos[k]=0;
match[k]=0;
}
}

printf("\n");
}
printf("\n--------------------------------------------------------------------------\n");
}