Lisp for the C++ programmer: Numerical Integration

In the series of Lisp for the C++ programmer, we present some Common Lisp examples in a way similar to examples which are written in C++ or some other languages belong the same family tree of C++.

Here is the example of Riemann sum in Common Lisp. We first define a function f which takes a single parameter x and the function y = f(x) = x, for simplicity. We will change this function later. It is known that integration of this function from 0 to 1 is 0.50.

Common Lisp code for numerical integration (approximatly result) is:

 (defun f (x)  
     x  
 )  
 (defun integrate (f start stop)  
     (setq epsilon 0.0001)  
     (setq sum 0.0)  
     (loop for i from start to stop by epsilon do  
         (setq sum (+ sum (* (funcall f i) epsilon)))  
     )  
 sum  
 )  
 (print (integrate 'f 0 1))  

The result is 0.4999532. Now we can use a more complex function, for example a normal distribution function with zero mean and unit variance. This function can be defined in Common Lisp as

 (defun normal (x)  
     (setq a (* (/ 1 (sqrt (* 2 3.141592))) (exp (* -0.5 (* x x)))))  
 a  
 )  
 (defun integrate (f start stop)  
     (setq epsilon 0.0001)  
     (setq sum 0.0)  
     (loop for i from start to stop by epsilon do  
         (setq sum (+ sum (* (funcall f i) epsilon)))  
     )  
 sum  
 )  
 (print (integrate 'normal -1 1))  

In the code above, as it can clearly be seen, we integrate the standard normal distribution from -1 to 1 and the result is 0.6826645.