/************************************************************************* A C++ Program to estimate the value of Second Derivative of the function at the given points from the given data using Central Difference Formula of Order Two. *************************************************************************/ # include <iostream.h> # include <stdlib.h> # include <string.h> # include <stdio.h> # include <conio.h> # include <math.h> //------------------------ Global Variables ---------------------------// const int max_size=13; int n=0; int top=-1; int choice=0; long double h=0; long double x0=0; long double xn[max_size]={0}; long double fx[max_size]={0}; char Fx[100]={NULL}; char D2fx[100]={NULL}; char Stack[30][30]={NULL}; char Postfix_expression[2][30][30]={NULL}; //------------------------ Funcion Prototypes -------------------------// void push(const char *); void convert_ie_to_pe(const char *,const int); const char* pop( ); const long double evaluate_postfix_expression(const long double,const int); void show_screen( ); void clear_screen( ); void get_input( ); void estimate_dfx( ); const int get_index(const long double); //------------------------------ main( ) ------------------------------// int main( ) { clrscr( ); textmode(C4350); show_screen( ); get_input( ); estimate_dfx( ); return 0; } //------------------------ Funcion Definitions ------------------------// //-------------------------- show_screen( ) ---------------------------// void show_screen( ) { cprintf(\"\\n********************************************************************************\"); cprintf(\"*************************- -************************\"); cprintf(\"*------------------------- \"); textbackground(1); cprintf(\" Numerical Differentiation \"); 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(5,8+count); cout<<\" \"; } gotoxy(1,2); } //-------------------------- push(const char*) ------------------------// void push(const char* 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( ) -------------------------------// const char* 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_ie_to_pe(const char*,const int) ------------// void convert_ie_to_pe(const char* Expression,const int index) { 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; } else if(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(\"(\"); else if(strcmp(Symbol_scanned,\")\")==0) { while(strcmp(Stack[top],\"(\")!=0) { strcpy(Postfix_expression[index][count_2],pop( )); count_2++; } pop( ); } else if(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) { } else if(strcmp(Symbol_scanned,\"*\")==0 || strcmp(Symbol_scanned,\"/\")==0) { while(strcmp(Stack[top],\"^\")==0 || strcmp(Stack[top],\"*\")==0 || strcmp(Stack[top],\"/\")==0) { strcpy(Postfix_expression[index][count_2],pop( )); count_2++; } } else if(strcmp(Symbol_scanned,\"+\")==0 || strcmp(Symbol_scanned,\"-\")==0) { while(strcmp(Stack[top],\"(\")!=0) { strcpy(Postfix_expression[index][count_2],pop( )); count_2++; } } push(Symbol_scanned); } else { strcat(Postfix_expression[index][count_2],Symbol_scanned); count_2++; } } while(strcmp(Stack[top],NULL)!=0); strcat(Postfix_expression[index][count_2],\"=\"); count_2++; } //----- evaluate_postfix_expression(const long double,const int) ------// const long double evaluate_postfix_expression( const long double x,const int index) { long double function_value=0; int count_1=-1; char Symbol_scanned[30]={NULL}; do { count_1++; strcpy(Symbol_scanned,Postfix_expression[index][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( )); long double operand[2]={0}; long double 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; else if(strcmp(Operand[count_2],\"e\")==0) operand[count_2]=2.718282; else if(strcmp(Operand[count_2],\"sinx\")==0) operand[count_2]=sinl(x); else if(strcmp(Operand[count_2],\"cosx\")==0) operand[count_2]=cosl(x); else if(strcmp(Operand[count_2],\"tanx\")==0) operand[count_2]=tanl(x); else if(strcmp(Operand[count_2],\"lnx\")==0) operand[count_2]=logl(x); else if(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); } else if(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 = 3 | max. n = 12 ]\"; gotoxy(6,12); cout<<\"Enter the max. number of distinct data points = n = \"; cin>>n; if(n<3 || 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<3 || 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>>Fx; gotoxy(6,17); cout<<\"Enter the Differential Function : f\\\"(x) = \"; cin>>D2fx; convert_ie_to_pe(Fx,0); convert_ie_to_pe(D2fx,1); } 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) { fx[count_3]=evaluate_postfix_expression(xn[count_3],0); gotoxy(43,wherey( )); cout<<fx[count_3]; } else { gotoxy(43,wherey( )); cin>>fx[count_3]; } if(choice) gotoxy(25,(wherey( )+2)); else gotoxy(25,(wherey( )+1)); } gotoxy(25,43); cout<<\"Press any key to continue...\"; getch( ); } //------------------- get_index(const long double) --------------------// const int get_index(const long double x) { for(int count=0;count<n;count++) { if(xn[count]==x) break; } return count; } //---------------------------- estimate_dfx( ) ------------------------// void estimate_dfx( ) { clear_screen( ); gotoxy(6,9); cout<<\"Centeral Difference Formula of Order 2 :\"; gotoxy(6,10); cout<<\"ÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍ\"; gotoxy(8,13); cout<<\"f\\\"(x) ÷ [f(x+h)-2f(x)+f(x-h)]/h^2\"; gotoxy(6,17); cout<<\"Estimation of f\\\"(x) :\"; gotoxy(6,18); cout<<\"ÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍ\"; char Choice=NULL; long double x=0; long double d2fx=0; long double actual_d2fx=0; int error_flag=0; do { Choice=NULL; x=0; d2fx=0; actual_d2fx=0; error_flag=0; gotoxy(10,20); cout<<\"Enter the value of x = \"; cin>>x; if(x<=xn[0] || x>=xn[(n-1)]) { error_flag=1; gotoxy(10,23); cout<<\"Error: Please enter x greater than x(0) and less than x(n).\"; } else { int index=0; index=get_index(x); long double fx=::fx[index]; long double fxph=::fx[(index+1)]; long double fxmh=::fx[(index-1)]; d2fx=((fxph-(2*fx)+fxmh)/(h*h)); gotoxy(10,23); cout<<\"The estimated value of f\\\"(\"<<x<<\") ÷ \"<<d2fx; } if(choice && !error_flag) { actual_d2fx=evaluate_postfix_expression(x,1); gotoxy(10,25); cout<<\"The Actual value of f\\\"(\"<<x<<\") = \"<<actual_d2fx; gotoxy(10,28); cout<<\"Absolute Error = E(abs) = \"<<fabs((actual_d2fx-d2fx)); } gotoxy(15,42); cout<<\"Press <Esc> to exit or any other key to continue...\"; Choice=getch( ); if(int(Choice)!=27) { gotoxy(10,20); cout<<\" \"; gotoxy(10,23); cout<<\" \"; gotoxy(10,25); cout<<\" \"; gotoxy(10,28); cout<<\" \"; gotoxy(15,42); cout<<\" \"; } else if(int(Choice)==27) exit(0); } while(1); }