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Program to construct Newton's Backward Difference Interpolation Formula from the given distinct equally spaced data points

Posted By: Easy Tutor Category: C++ Programming Views: 4363

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

Download Sourcecode for Program to construct Newton's Backward Difference Interpolation Formula from the given distinct equally spaced data points (Size: 4.19 KB)

Code for Program to construct Newton's Backward Difference Interpolation Formula from the given distinct equally spaced data points 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 h=0;
 longdouble x0=0;

 longdouble xn[max_size]={0};
 longdouble difference_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_difference_table( );
 void show_difference_table( );
 void estimate_pnx( );

 constlongdouble compute_newtons_bdif(constlongdouble,constint);

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

       show_screen( );
       get_input( );
       construct_difference_table( );
       show_difference_table( );
       estimate_pnx( );

       return 0;
     }

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

       textbackground(1);
       cprintf(" Newton's Backward 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(6,16);
       cout<<"Enter the value of x0 = ";

       cin>>x0;

       gotoxy(6,18);
       cout<<"Enter the value of h = ";

       cin>>h;

       gotoxy(6,24);
       cout<<"Input Mode :";

       gotoxy(6,25);
       cout<<"ÍÍÍÍÍÍÍÍÍÍÍÍ";

       gotoxy(8,28);
       cout<<"Press : ";

       gotoxy(10,30);
       cout<<"- 'Y' or <Enter> to enter function";

       gotoxy(10,32);
       cout<<"- 'N' or <Any other key> to enter values of the function";

       gotoxy(8,35);
       cout<<"Enter your choice : ";

       char Choice=NULL;

       Choice=getch( );

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

         gotoxy(28,35);
         cout<<"Y";
      }

       else
      {
         gotoxy(28,35);
         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";

       xn[0]=x0;

       for(int count_2=0;count_2<(n-1);count_2++)
      xn[(count_2+1)]=(xn[count_2]+h);

       gotoxy(25,16);

       for(int count_3=0;count_3<n;count_3++)
      {
         gotoxy(27,wherey( ));
         cout<<xn[count_3];

         if(choice)
        {
           difference_table[0][count_3]=evaluate_postfix_expression(xn[count_3]);

           gotoxy(43,wherey( ));
           cout<<difference_table[0][count_3];
        }

         else
        {
           gotoxy(43,wherey( ));
           cin>>difference_table[0][count_3];
        }

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

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

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

       getch( );
    }

 /*************************************************************************///-----------------------  construct_difference_table( )  -----------------------///*************************************************************************/void construct_difference_table( )
    {
       for(int count_1=1;count_1<n;count_1++)
      {
         for(int count_2=count_1;count_2<n;count_2++)
        {
           longdouble fx1=difference_table[(count_1-1)][(count_2-1)];
           longdouble fx2=difference_table[(count_1-1)][count_2];

           difference_table[count_1][count_2]=fx2-fx1;
        }
      }
    }

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

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

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

       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<<difference_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<<difference_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_bdif(const long double,const int)  --------///*************************************************************************/constlongdouble compute_newtons_bdif(constlongdouble x,constint n)
    {
       longdouble s=0;
       longdouble pnx=0;
       longdouble temp=0;
       longdouble n_factorial=1;

       int temp_n=::n;

       s=((x-xn[(temp_n-1)])/h);

       pnx=difference_table[0][(temp_n-1)];

       for(int count_1=0;count_1<n;count_1++)
      {
         temp=s;
         temp*=difference_table[(count_1+1)][(temp_n-1)];
         temp/=n_factorial;

         for(int count_2=0;count_2<count_1;count_2++)
        temp*=(s+(count_2+1));

         pnx+=temp;
         n_factorial*=(count_1+2);
      }

       return pnx;
    }

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

       gotoxy(6,10);
       cout<<"Estimation of Pn(x) :";

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

       char Choice=NULL;

       int n=0;

       longdouble x=0;
       longdouble fx=0;
       longdouble pnx=0;

       do
      {
         Choice=NULL;

         n=0;
         x=0;
         fx=0;
         pnx=0;

         gotoxy(10,13);
         cout<<"Enter the value of n = ";

         cin>>n;

         gotoxy(10,15);
         cout<<"Enter the value of x = ";

         cin>>x;

         pnx=compute_newtons_bdif(x,n);

         gotoxy(10,20);
         cout<<"The estimated value of P"<<n<<"("<<x<<")  =  "<<pnx;

         if(choice)
        {
           fx=evaluate_postfix_expression(x);

           gotoxy(10,23);
           cout<<"The Actual value of f"<<"("<<x<<")  =  "<<fx;

           gotoxy(10,25);
           cout<<"Absolute Error = E(abs) =  "<<fabs((fx-pnx));
        }

         gotoxy(12,40);
         cout<<"Press <Esc> to exit, 'V' to view Difference Table again";

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

         Choice=getch( );

         if(int(Choice)!=27 && Choice!='v' && Choice!='V')
        {
           gotoxy(10,13);
           cout<<"                                                    ";

           gotoxy(10,15);
           cout<<"                                                    ";

           gotoxy(10,20);
           cout<<"                                                    ";

           gotoxy(10,23);
           cout<<"                                                    ";

           gotoxy(10,25);
           cout<<"                                                    ";

           gotoxy(12,40);
           cout<<"                                                         ";

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

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

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

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C++ Programming Articles

Program to construct Newton's Backward Difference Interpolation Formula from the given distinct equally spaced data points

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

Code for Program to construct Newton's Backward Difference Interpolation Formula from the given distinct equally spaced data points in C++ Programming

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Other Interesting Articles in C++ Programming:

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C++ Programming Articles

Program to construct Newton's Backward Difference Interpolation Formula from the given distinct equally spaced data points

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

Code for Program to construct Newton's Backward Difference Interpolation Formula from the given distinct equally spaced data points in C++ Programming

Related Articles and Code:

Other Interesting Articles in C++ Programming:

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