# 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);
    }