How to resolve the algorithm Numerical integration step by step in the Common Lisp programming language

Published on 12 May 2024 09:40 PM
#Common lisp

How to resolve the algorithm Numerical integration step by step in the Common Lisp programming language

Table of Contents

Problem Statement

Write functions to calculate the definite integral of a function ƒ(x) using all five of the following methods: Your functions should take in the upper and lower bounds (a and b), and the number of approximations to make in that range (n). Assume that your example already has a function that gives values for ƒ(x) . Simpson's method is defined by the following pseudo-code:

Demonstrate your function by showing the results for:

Let's start with the solution:

Step by Step solution about How to resolve the algorithm Numerical integration step by step in the Common Lisp programming language

Source code in the common programming language

(defun left-rectangle (f a b n &aux (d (/ (- b a) n)))
  (* d (loop for x from a below b by d summing (funcall f x))))

(defun right-rectangle (f a b n &aux (d (/ (- b a) n)))
  (* d (loop for x from b above a by d summing (funcall f x))))

(defun midpoint-rectangle (f a b n &aux (d (/ (- b a) n)))
  (* d (loop for x from (+ a (/ d 2)) below b by d summing (funcall f x))))

(defun trapezium (f a b n &aux (d (/ (- b a) n)))
  (* (/ d 2)
     (+ (funcall f a)
        (* 2 (loop for x from (+ a d) below b by d summing (funcall f x)))
        (funcall f b))))

(defun simpson (f a b n)
  (loop with h = (/ (- b a) n)
        with sum1 = (funcall f (+ a (/ h 2)))
        with sum2 = 0
        for i from 1 below n
        do (incf sum1 (funcall f (+ a (* h i) (/ h 2))))
        do (incf sum2 (funcall f (+ a (* h i))))
        finally (return (* (/ h 6)
                           (+ (funcall f a)
                              (funcall f b)
                              (* 4 sum1)
                              (* 2 sum2))))))

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