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# Program to construct Newton's Divided Difference Interpolation Formula from the given distinct data points and estimate the value of the function

Posted By: Easy Tutor     Category: C++ Programming     Views: 16242

## Code for Program to construct Newton's Divided Difference Interpolation Formula from the given distinct data points and estimate the value of the function in C++ Programming

``` # include <iostream.h>
# include   <stdlib.h>
# include   <string.h>
# include    <stdio.h>
# include    <conio.h>
# include     <math.h>

constint max_size=13;

int n=0;
int top=-1;
int choice=0;

longdouble xn[max_size]={0};
longdouble px[max_size]={0};
longdouble dd_table[max_size][max_size]={0};

char Non_linear_equation[100]={NULL};
char Stack[30][30]={NULL};
char Postfix_expression[30][30]={NULL};

void push(constchar *);
void convert_infix_expression_to_postfix_expression(constchar *);

constchar* pop( );
constlongdouble evaluate_postfix_expression(constlongdouble);

void show_screen( );
void clear_screen( );
void get_input( );
void construct_dd_table( );
void show_dd_table( );
void compute_newtons_dd_interpolation_polynomial( );
void estimate_pnx( );

int main( )
{
clrscr( );
textmode(C4350);

show_screen( );
get_input( );
construct_dd_table( );
show_dd_table( );
compute_newtons_dd_interpolation_polynomial( );
estimate_pnx( );

return 0;
}

/*************************************************************************///--------------------------  show_screen( )  ---------------------------///*************************************************************************/void show_screen( )
{
cprintf("\n********************************************************************************");
cprintf("*************-                                                     -************");
cprintf("*------------- ");

textbackground(1);
cprintf(" Newton's Divided Difference Interpolation Formula ");
textbackground(8);

cprintf(" ------------*");
cprintf("*************-                                                     -************");
cprintf("********************************************************************************");

for(int count=0;count<42;count++)
cprintf("*                                                                              *");

gotoxy(1,46);
cprintf("********************************************************************************");
cprintf("*------------------------------------------------------------------------------*");
cprintf("********************************************************************************");

gotoxy(1,2);
}

/*************************************************************************///-------------------------  clear_screen( )  ---------------------------///*************************************************************************/void clear_screen( )
{
for(int count=0;count<37;count++)
{
gotoxy(3,8+count);
cout<<"                                                                            ";
}

gotoxy(1,2);
}

/*************************************************************************///--------------------------  push(const char*)  ------------------------///*************************************************************************/void push(constchar* Operand)
{
if(top==(max_size-1))
{
cout<<"Error : Stack is full."<<endl;
cout<<"\n        Press any key to exit.";

getch( );
exit(0);
}

else
{
top++;
strcpy(Stack[top],Operand);
}
}

/*************************************************************************///------------------------------  pop( )  -------------------------------///*************************************************************************/constchar* pop( )
{
char Operand[40]={NULL};

if(top==-1)
{
cout<<"Error : Stack is empty."<<endl;
cout<<"\n        Press any key to exit.";

getch( );
exit(0);
}

else
{
strcpy(Operand,Stack[top]);
strset(Stack[top],NULL);
top--;
}

return Operand;
}

/*************************************************************************///----  convert_infix_expression_to_postfix_expression(const char*)  ----///*************************************************************************/void convert_infix_expression_to_postfix_expression(constchar* Expression)
{
char Infix_expression[100]={NULL};
char Symbol_scanned[30]={NULL};

push("(");
strcpy(Infix_expression,Expression);
strcat(Infix_expression,"+0)");

int flag=0;
int count_1=0;
int count_2=0;
int equation_length=strlen(Infix_expression);

if(Infix_expression[0]=='(')
flag=1;

do
{
strset(Symbol_scanned,NULL);

if(flag==0)
{
int count_3=0;

do
{
Symbol_scanned[count_3]=Infix_expression[count_1];

count_1++;
count_3++;
}
while(count_1<=equation_length &&
Infix_expression[count_1]!='(' &&
Infix_expression[count_1]!='+' &&
Infix_expression[count_1]!='-' &&
Infix_expression[count_1]!='*' &&
Infix_expression[count_1]!='/' &&
Infix_expression[count_1]!='^' &&
Infix_expression[count_1]!=')');

flag=1;
}

elseif(flag==1)
{
Symbol_scanned[0]=Infix_expression[count_1];

count_1++;

if(Infix_expression[count_1]!='(' &&
Infix_expression[count_1]!='^' &&
Infix_expression[count_1]!='*' &&
Infix_expression[count_1]!='/' &&
Infix_expression[count_1]!='+' &&
Infix_expression[count_1]!='-' &&
Infix_expression[count_1]!=')')
flag=0;

if(Infix_expression[count_1-1]=='(' &&
(Infix_expression[count_1]=='-' ||
Infix_expression[count_1]=='+'))
flag=0;
}

if(strcmp(Symbol_scanned,"(")==0)
push("(");

elseif(strcmp(Symbol_scanned,")")==0)
{
while(strcmp(Stack[top],"(")!=0)
{
strcpy(Postfix_expression[count_2],pop( ));

count_2++;
}

pop( );
}

elseif(strcmp(Symbol_scanned,"^")==0 ||
strcmp(Symbol_scanned,"+")==0 ||
strcmp(Symbol_scanned,"-")==0 ||
strcmp(Symbol_scanned,"*")==0 ||
strcmp(Symbol_scanned,"/")==0)
{
if(strcmp(Symbol_scanned,"^")==0)
{  }

elseif(strcmp(Symbol_scanned,"*")==0 ||
strcmp(Symbol_scanned,"/")==0)
{
while(strcmp(Stack[top],"^")==0 ||
strcmp(Stack[top],"*")==0 ||
strcmp(Stack[top],"/")==0)
{
strcpy(Postfix_expression[count_2],pop( ));

count_2++;
}
}

elseif(strcmp(Symbol_scanned,"+")==0 ||
strcmp(Symbol_scanned,"-")==0)
{
while(strcmp(Stack[top],"(")!=0)
{
strcpy(Postfix_expression[count_2],pop( ));

count_2++;
}
}

push(Symbol_scanned);
}

else
{
strcat(Postfix_expression[count_2],Symbol_scanned);

count_2++;
}
}
while(strcmp(Stack[top],NULL)!=0);

strcat(Postfix_expression[count_2],"=");
count_2++;
}

/*************************************************************************///----------  evaluate_postfix_expression(const long double)  -----------///*************************************************************************/constlongdouble evaluate_postfix_expression(constlongdouble x)
{
longdouble function_value=0;

int count_1=-1;

char Symbol_scanned[30]={NULL};

do
{
count_1++;

strcpy(Symbol_scanned,Postfix_expression[count_1]);

if(strcmp(Symbol_scanned,"^")==0 ||
strcmp(Symbol_scanned,"*")==0 ||
strcmp(Symbol_scanned,"/")==0 ||
strcmp(Symbol_scanned,"+")==0 ||
strcmp(Symbol_scanned,"-")==0)

{
char Result[30]={NULL};
char Operand[2][30]={NULL};

strcpy(Operand[0],pop( ));
strcpy(Operand[1],pop( ));

longdouble operand[2]={0};
longdouble result=0;

char *endptr;

for(int count_2=0;count_2<2;count_2++)
{
int flag=0;

if(Operand[count_2][0]=='-')
{
int length=strlen(Operand[count_2]);

for(int count_3=0;count_3<(length-1);count_3++)
Operand[count_2][count_3]=Operand[count_2][(count_3+1)];

Operand[count_2][count_3]=NULL;

flag=1;
}

if(strcmp(Operand[count_2],"x")==0)
operand[count_2]=x;

elseif(strcmp(Operand[count_2],"e")==0)
operand[count_2]=2.718282;

elseif(strcmp(Operand[count_2],"sinx")==0)
operand[count_2]=sinl(x);

elseif(strcmp(Operand[count_2],"cosx")==0)
operand[count_2]=cosl(x);

elseif(strcmp(Operand[count_2],"tanx")==0)
operand[count_2]=tanl(x);

elseif(strcmp(Operand[count_2],"lnx")==0)
operand[count_2]=logl(x);

elseif(strcmp(Operand[count_2],"logx")==0)
operand[count_2]=log10l(x);

else
operand[count_2]=strtod(Operand[count_2],&endptr);

if(flag)
operand[count_2]*=-1;
}

switch(Symbol_scanned[0])
{
case'^' : result=powl(operand[1],operand[0]);
break;

case'*' : result=operand[1]*operand[0];
break;

case'/' : result=operand[1]/operand[0];
break;

case'+' : result=operand[1]+operand[0];
break;

case'-' : result=operand[1]-operand[0];
break;
}

gcvt(result,25,Result);

push(Result);
}

elseif(strcmp(Symbol_scanned,"=")!=0)
push(Symbol_scanned);
}
while(strcmp(Symbol_scanned,"=")!=0);

char Function_value[30]={NULL};
char *endptr;

strcpy(Function_value,pop( ));
function_value=strtod(Function_value,&endptr);

return function_value;
}

/*************************************************************************///-----------------------------  get_input( )  --------------------------///*************************************************************************/void get_input( )
{
do
{
clear_screen( );

gotoxy(6,9);
cout<<"Number of Distinct Data Points :";

gotoxy(6,10);
cout<<"ÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍ";

gotoxy(27,13);
cout<<"[ min. n = 2  |  max. n = 12 ]";

gotoxy(6,12);
cout<<"Enter the max. number of distinct data points = n = ";

cin>>n;

if(n<2 || n>12)
{
gotoxy(12,25);
cout<<"Error : Wrong Input. Press <Esc> to exit or any other key";

gotoxy(12,26);
cout<<"        to try again.";

n=int(getche( ));

if(n==27)
exit(0);
}
}
while(n<2 || n>12);

gotoxy(6,19);
cout<<"Input Mode :";

gotoxy(6,20);
cout<<"ÍÍÍÍÍÍÍÍÍÍÍÍ";

gotoxy(8,23);
cout<<"Press : ";

gotoxy(10,25);
cout<<"- 'Y' or <Enter> to enter function";

gotoxy(10,27);
cout<<"- 'N' or <Any other key> to enter values of the function";

gotoxy(8,30);
cout<<"Enter your choice : ";

char Choice=NULL;

Choice=getch( );

if(Choice=='y' || Choice=='Y' || int(Choice)==13)
{
choice=1;

gotoxy(28,30);
cout<<"Y";
}

else
{
gotoxy(28,30);
cout<<"N";
}

gotoxy(25,43);
cout<<"Press any key to continue...";

getch( );

if(choice)
{
clear_screen( );

gotoxy(6,11);
cout<<"Non-Linear Function :";

gotoxy(6,12);
cout<<"ÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍ";

gotoxy(6,37);
cout<<"Note : Write the function with proper Braces ( ) e.g; 2x+3 as (2*x)+3";

gotoxy(6,40);
cout<<"Available Operators  :  ^ (raised to power) , * , / , + , -";

gotoxy(6,42);
cout<<"Available Operands   :  x , e , sinx , cosx , tanx , lnx , logx ,";

gotoxy(6,44);
cout<<"                        n = any number";

gotoxy(6,14);
cout<<"Enter the Function : f(x) = ";

cin>>Non_linear_equation;

convert_infix_expression_to_postfix_expression(Non_linear_equation);
}

clear_screen( );

gotoxy(6,9);
cout<<"Data Points & Values of Function :";

gotoxy(6,10);
cout<<"ÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍ";

gotoxy(25,12);
cout<<"ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÂÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿";

gotoxy(25,13);
cout<<"³       x       ³     f(x)      ³";

gotoxy(25,14);
cout<<"ÃÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÅÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ´";

gotoxy(25,15);
cout<<"³               ³               ³";

for(int count_1=0;count_1<n;count_1++)
{
gotoxy(25,(wherey( )+1));
cout<<"³               ³               ³";

gotoxy(25,(wherey( )+1));
cout<<"³               ³               ³";
}

gotoxy(25,(wherey( )+1));
cout<<"ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÁÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ";

gotoxy(25,16);

for(int count_2=0;count_2<n;count_2++)
{
gotoxy(27,wherey( ));
cin>>xn[count_2];

if(choice)
{
dd_table[0][count_2]=evaluate_postfix_expression(xn[count_2]);

gotoxy(43,(wherey( )-1));
cout<<dd_table[0][count_2];
}

else
{
gotoxy(43,(wherey( )-1));
cin>>dd_table[0][count_2];
}

if(choice)
gotoxy(25,(wherey( )+2));

else
gotoxy(25,(wherey( )+1));
}

gotoxy(25,43);
cout<<"Press any key to continue...";

getch( );
}

/*************************************************************************///-----------------------  construct_dd_table( )  -----------------------///*************************************************************************/void construct_dd_table( )
{
for(int count_1=1;count_1<n;count_1++)
{
for(int count_2=count_1;count_2<n;count_2++)
{
longdouble fx1=dd_table[(count_1-1)][(count_2-1)];
longdouble fx2=dd_table[(count_1-1)][count_2];
longdouble x1=xn[(count_2-count_1)];
longdouble x2=xn[count_2];

dd_table[count_1][count_2]=((fx2-fx1)/(x2-x1));
}
}
}

/*************************************************************************///--------------------------  show_dd_table( )  -------------------------///*************************************************************************/void show_dd_table( )
{
clear_screen( );

gotoxy(4,10);
cout<<"Divided Difference Table :";

gotoxy(4,11);
cout<<"ÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍ";

gotoxy(4,13);
cout<<"ÚÄÄÄÄÄÄÄÄÂÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ";

gotoxy(4,14);
cout<<"³   x    ³     f(x)      ";

gotoxy(4,15);
cout<<"ÃÄÄÄÄÄÄÄÄÅÄÄÄÄÄÄÄÄÄÄÄÄÄÄ";

gotoxy(4,16);
cout<<"³        ³              ";

int x_cord=4;
int y_cord=17;

for(int count_1=0;count_1<n;count_1++)
{
gotoxy(x_cord,y_cord);
cout<<"³        ³              ";

gotoxy(x_cord,(y_cord+1));
cout<<"³        ³              ";

gotoxy((x_cord+2),y_cord);
cout<<xn[count_1];

gotoxy((x_cord+11),y_cord);
cout<<dd_table[0][count_1];

y_cord+=2;
}

gotoxy(x_cord,y_cord);
cout<<"ÀÄÄÄÄÄÄÄÄÁÄÄÄÄÄÄÄÄÄÄÄÄÄÄ";

x_cord=28;

int count_2=0;

for(int count_3=1;count_3<n;count_3++)
{
gotoxy(x_cord,13);
cout<<"ÂÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ";

gotoxy(x_cord,14);
cout<<"³";

gotoxy(x_cord,15);
cout<<"ÅÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ";

gotoxy(x_cord,16);
cout<<"³";

gotoxy(x_cord,y_cord);
cout<<"ÁÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ";

gotoxy((x_cord+6),14);
cout<<count_3;

if(count_3==1)
cout<<"st";

elseif(count_3==2)
cout<<"nd";

elseif(count_3==3)
cout<<"rd";

else
cout<<"th";

y_cord=17;

for(int count_4=0;count_4<n;count_4++)
{
gotoxy(x_cord,y_cord);
cout<<"³";

gotoxy(x_cord,(y_cord+1));
cout<<"³";

if(count_4>=count_3)
{
gotoxy((x_cord+2),(y_cord-count_3));
cout<<dd_table[count_3][count_4];
}

y_cord+=2;
}

x_cord+=16;
count_2++;

if((count_2%3)==0 && count_3<(n-1))
{
gotoxy(x_cord,13);
cout<<"ÂÄÄ";

gotoxy(x_cord,14);
cout<<"³";

gotoxy(x_cord,15);
cout<<"ÅÄÄ";

gotoxy(x_cord,16);
cout<<"³";

gotoxy(x_cord,y_cord);
cout<<"ÁÄÄ";

y_cord=17;

for(int count_5=0;count_5<n;count_5++)
{
gotoxy(x_cord,y_cord);
cout<<"³";

gotoxy(x_cord,(y_cord+1));
cout<<"³";

y_cord+=2;
}

gotoxy(30,44);
cout<<"Press any key to continue...";

getch( );

y_cord=13;
x_cord=28;

for(int count_6=0;count_6<=(n+2);count_6++)
{
gotoxy(x_cord,y_cord);
cout<<"                                                   ";

gotoxy(x_cord,(y_cord+1));
cout<<"                                                   ";

y_cord+=2;
}

y_cord-=2;
count_2=0;
count_3--;
}
}

gotoxy(x_cord,13);
cout<<"¿";

gotoxy(x_cord,14);
cout<<"³";

gotoxy(x_cord,16);
cout<<"³";

y_cord=17;

for(int count_6=0;count_6<n;count_6++)
{
gotoxy(x_cord,y_cord);
cout<<"³";

gotoxy(x_cord,(y_cord+1));
cout<<"³";

y_cord+=2;
}

gotoxy(x_cord,15);
cout<<"´";

gotoxy(x_cord,y_cord);
cout<<"Ù";

gotoxy(30,44);
cout<<"Press any key to continue...";

getch( );
}

/*************************************************************************///---------  compute_newtons_dd_interpolation_polynomial( )  ------------///*************************************************************************/void compute_newtons_dd_interpolation_polynomial( )
{
for(int count_1=0;count_1<n;count_1++)
xn[count_1]=(xn[count_1]*(-1));

px[0]=dd_table[0][0];

for(int count_2=1;count_2<n;count_2++)
{
longdouble fx[max_size]={0};

fx[0]=xn[0];
fx[1]=1;

for(int count_3=0;count_3<(count_2-1);count_3++)
{
fx[(count_3+2)]=1;

for(int count_4=(1+count_3);count_4>0;count_4--)
{
fx[count_4]*=xn[(count_3+1)];
fx[count_4]+=fx[(count_4-1)];
}

fx[0]*=xn[(count_3+1)];
}

for(int count_5=0;count_5<(count_2+1);count_5++)
fx[count_5]*=dd_table[count_2][count_2];

for(int count_6=0;count_6<(count_2+1);count_6++)
px[count_6]+=fx[count_6];
}
}

/*************************************************************************///----------------------------  estimate_pnx( )  ------------------------///*************************************************************************/void estimate_pnx( )
{
clear_screen( );

gotoxy(6,10);
cout<<"Newton's Divided Difference Interpolation Formula :";

gotoxy(6,11);
cout<<"ÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍ";

gotoxy(10,13);
cout<<"P"<<(n-1)<<"(x)  =  ";

for(int count_1=(n-1);count_1>=0;count_1--)
{
if(count_1==(n-1) && n>2)
cout<<px[count_1]<<" x"<<count_1;

elseif(count_1>1)
cout<<fabs(px[count_1])<<" x"<<count_1;

elseif(count_1==1)
cout<<fabs(px[count_1])<<" x";

elseif(count_1==0)
cout<<fabs(px[count_1]);

if(wherex( )>=70)
gotoxy(30,(wherey( )+2));

if(count_1>0)
{
if(px[(count_1-1)]>0)
cout<<"  +  ";

else
cout<<"  -  ";
}

if(wherex( )>=70)
gotoxy(30,(wherey( )+2));
}

gotoxy(6,19);
cout<<"Estimation of Pn(x) :";

gotoxy(6,20);
cout<<"ÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍ";

char Choice=NULL;

longdouble x=0;
longdouble pnx=0;

do
{
Choice=NULL;

x=0;
pnx=0;

gotoxy(10,22);
cout<<"Enter the value of x = ";

cin>>x;

pnx=px[0];
pnx+=(px[1]*x);

for(int count_2=2;count_2<n;count_2++)
{
longdouble temp=x;

for(int count_3=1;count_3<count_2;count_3++)
temp*=x;

pnx+=(px[count_2]*temp);
}

gotoxy(10,26);
cout<<"The estimated value of P"<<(n-1)<<"("<<x<<")  =  "<<pnx;

if(choice)
{
longdouble fx=0;

fx=evaluate_postfix_expression(x);

gotoxy(10,28);
cout<<"The Actual value of f"<<"("<<x<<")  =  "<<fx;

gotoxy(10,30);
cout<<"Absolute Error = E(abs) =  "<<fabs((fx-pnx));
}

gotoxy(15,40);
cout<<"Press <Esc> to exit, 'V' to view D.D. Table again";

gotoxy(25,42);
cout<<"or any other key to continue...";

Choice=getch( );

if(int(Choice)!=27 && Choice!='v' && Choice!='V')
{
gotoxy(10,22);
cout<<"                                                    ";

gotoxy(10,26);
cout<<"                                                    ";

gotoxy(10,28);
cout<<"                                                    ";

gotoxy(10,30);
cout<<"                                                    ";

gotoxy(15,40);
cout<<"                                                    ";

gotoxy(25,42);
cout<<"                                                    ";
}

elseif(Choice=='v' || Choice=='V')
{
show_dd_table( );
estimate_pnx( );
}

elseif(int(Choice)==27)
exit(0);
}
while(1);
}
```
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