PHP Class MathPHP\NumericalAnalysis\NumericalIntegration\SimpsonsThreeEighthsRule

In numerical analysis, Simpson's 3/8 rule is a technique for approximating the definite integral of a function. Simpson's 3/8 rule belongs to the closed Newton-Cotes formulas, a group of methods for numerical integration which approximate the integral of a function. We can either directly supply a set of inputs and their corresponding outputs for said function, or if we explicitly know the function, we can define it as a callback function and then generate a set of points by evaluating that function at n points between a start and end point. We then use these values to interpolate a Lagrange polynomial. Finally, we integrate the Lagrange polynomial to approximate the integral of our original function. Simpson's 3/8 rule is produced by integrating the third Lagrange polynomial. https://en.wikipedia.org/wiki/Simpson%27s_rule#Simpson.27s_3.2F8_rule http://mathworld.wolfram.com/Simpsons38Rule.html http://www.efunda.com/math/num_integration/num_int_newton.cfm

Inheritance: extends NumericalIntegration

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Public Methods

Method	Description
approximate ( $source, $args ) : number	Use Simpson's 3/8 Rule to aproximate the definite integral of a function f(x). Our input can support either a set of arrays, or a callback function with arguments (to produce a set of arrays). Each array in our input contains two numbers which correspond to coordinates (x, y) or equivalently, (x, f(x)), of the function f(x) whose definite integral we are approximating.

Method Details

approximate() public static method

Note: Simpson's 3/8 rule requires that our number of subintervals is a factor of three (we must supply an n points such that n-1 is a multiple of three) and also that the size of each subinterval is equal (spacing between each point is equal). The bounds of the definite integral to which we are approximating is determined by the our inputs. Example: approximate([0, 10], [2, 5], [4, 7], [6,3]) will approximate the definite integral of the function that produces these coordinates with a lower bound of 0, and an upper bound of 6. Example: approximate(function($x) {return $x**2;}, [0, 3 ,4]) will produce a set of arrays by evaluating the callback at 4 evenly spaced points between 0 and 3. Then, this array will be used in our approximation. Simpson's 3/8 Rule: xn ⁿ⁻¹ xᵢ₊₁ ∫ f(x)dx = ∑ ∫ f(x)dx x₁ ⁱ⁼¹ xᵢ ⁽ⁿ⁻¹⁾/³ 3h = ∑ - [f⟮x₂ᵢ₋₁⟯ + 3f⟮x₂ᵢ⟯ + 3f⟮x₂ᵢ₊₁⟯ + f⟮x₂ᵢ₊₂⟯] + O(h⁵f⁗(x)) ⁱ⁼¹ 8 where h = (xn - x₁) / (n - 1)

public static approximate ( $source, $args ) : number
$source		The source of our approximation. Should be either a callback function or a set of arrays. Each array (point) contains precisely two numbers, an x and y. Example array: [[1,2], [2,3], [3,4], [4,5]]. Example callback: function($x) {return $x**2;}
return	number	The approximation to the integral of f(x)