Simpson’s Rule:
Simpson's rule is a method for numerical integration, the numerical approximation of definite integrals. Specifically, it is the following approximation:
/* Write C program to implement Simpson method. */#include<stdio.h>
#include<conio.h>
#include<math.h>
char postfix[80];float stack[80];char stack1[80];int top=-1,top1=-1;float eval(char postfix[], float x1);
void infix_postfix(char infix[]);
main()
{ float x0, xn, h, s,e1,e2, e3; char exp[80], arr[80]; int i,n,l=0;clrscr();
printf("\nEnter an expression: ");gets(exp);
puts("Enter x0, xn and number of sub-intervals: "); scanf("%f%f%d", &x0, &xn, &n);h=(xn-x0)/n;
if(exp[0]=='l'&& exp[1]=='o'&& exp[2]=='g')
{l=strlen(exp);
for(i=0;i<l-3; i++)arr[0]=exp[i+3];
arr[i]='\0';infix_postfix(arr);
e1=eval(postfix,x0);
e2=eval(postfix,xn);
e3=4*eval(postfix, x0+h);
s=log(e1)+log(e2)+log(e3);
for (i=3;i<=n-1;i+=2)s+=4*eval(postfix,x0+i*h)+2*eval(postfix, x0+(i-1)*h);
}
else{infix_postfix(exp);
s=eval(postfix,x0)+eval(postfix,xn)+4*eval(postfix, x0+h);
for (i=3;i<=n-1;i+=2)s+=4*eval(postfix,x0+i*h)+2*eval(postfix, x0+(i-1)*h);
}
printf("The value of integral is %6.3f\n",(h/3)*s);return(0);}
/*Inserting the operands in a stack. */void push(float item)
{if(top==99) { printf("\n\tThe stack is full");getch();
exit(0);
}
else{top++;
stack[top]=item;
}
return;}
/*Removing the operands from a stack. */float pop(){ float item; if(top==-1){ printf("\n\tThe stack is empty\n\t");getch();
}
item=stack[top];
top--;
return (item);}
void push1(char item)
{ if(top1==79){ printf("\n\tThe stack is full");getch();
exit(0);
}
else{top1++;
stack1[top1]=item;
}
return;}
/*Removing the operands from a stack. */char pop1(){ char item; if(top1==-1){ printf("\n\tThe stack1 is empty\n\t");getch();
}
item=stack1[top1];
top1--;
return (item);}
/*Converting an infix expression to a postfix expression. */void infix_postfix(char infix[])
{ int i=0,j=0,k; char ch; char token; for(i=0;i<79;i++) postfix[i]=' '; push1('?');i=0;
token=infix[i];
while(token!='\0')
{ if(isalnum(token)) {postfix[j]=token;
j++;
}
else if(token=='(')
{ push1('(');}
else if(token==')')
{while(stack1[top1]!='(')
{ch=pop1();
postfix[j]=ch;
j++;
}
ch=pop1();
}
else { while(ISPriority(stack1[top1])>=ICP(token)) {ch=pop1();
postfix[j]=ch;
j++;
}
push1(token);
}
i++;
token=infix[i];
}
while(top1!=0){ch=pop1();
postfix[j]=ch;
j++;
}
postfix[j]='\0';}
/*Determining the priority of elements that are placed inside the stack. */int ISPriority(char token)
{ switch(token) {case '(':return (0);
case ')':return (9);
case '+':return (7);
case '-':return (7);
case '*':return (8);
case '/':return (8);
case '?':return (0);
default: printf("Invalid expression");
}
return 0;}
/*Determining the priority of elements that are approaching towards the stack. */int ICP(char token)
{ switch(token) {case '(':return (10);
case ')':return (9);
case '+':return (7);
case '-':return (7);
case '*':return (8);
case '/':return (8);
case '\0':return (0);
default: printf("Invalid expression");
}
return 0;}
/*Calculating the result of expression, which is converted in postfix notation. */float eval(char p[], float x1)
{ float t1,t2,k,r; int i=0,l;l=strlen(p);
while(i<l) {if(p[i]=='x')
push(x1);
else if(isdigit(p[i])) { k=p[i]-'0';push(k);
}
else {t1=pop();
t2=pop();
switch(p[i]) {case '+':k=t2+t1;
break;case '-':k=t2-t1;
break;case '*':k=t2*t1;
break;case '/':k=t2/t1;
break;default: printf("\n\tInvalid expression");
}
push(k);
}
i++;
}
if(top>0) { printf("You have entered the operands more than the operators");exit(0);
}
else {r=pop();
return (r);}
return 0;}
Output would be like this:
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