PHP Class MathPHP\NumericalAnalysis\NumericalDifferentiation\SecondDerivativeMidpointFormula

In numerical analysis, the Second Derivative Midpoint formula is used for approximating the second derivative of a function at a point in its domain. 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 3 points between a start and end point.
Inheritance: extends NumericalDifferentiation
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Public Methods

Method Description
differentiate ( numbers $target, $source, $args ) : number Use the Second Derivative Midpoint Formula to aproximate the second derivative of a function at our $target. 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 derivative we are approximating.

Method Details

differentiate() public static method

The Second Derivative Midpoint Formula requires we supply 3 points that are evenly spaced apart, and that our target equals the x-components of the midpoint. Example: differentiate(2, function($x) {return $x**2;}, 0, 4 ,3) will produce a set of arrays by evaluating the callback at 3 evenly spaced points between 0 and 4. Then, this array will be used in our approximation. Second Derivative Midpoint Formula: 1 h² f″(x₀) = - [f(x₀-h) - 2f(x₀) + f(x₀+h)] - - f⁽⁴⁾(ζ) h² 12 where ζ lies between x₀ - h and x₀ + h
public static differentiate ( numbers $target, $source, $args ) : number
$target numbers The value at which we are approximating the derivative
$source The source of our approximation. Should be either a callback function or a set of arrays. Each array (point) contains precisely two numbers, an x and y. Example array: [[1,2], [2,3], [3,4]]. Example callback: function($x) {return $x**2;}
return number The approximation of f'($target), i.e. the derivative of our input at our target point