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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: 4467

A C++ Program to construct Newton's Divided Difference
Interpolation Formula from the given distinct data points and estimate the value of the function at the given points.

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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Easy Tutor author of Program to construct Newton's Divided Difference Interpolation Formula from the given distinct data points and estimate the value of the function is from United States. Easy Tutor says

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