NMath User's Guide

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13.3 Differentiation (.NET, C#, CSharp, VB, Visual Basic, F#)

The Differentiate() method on OneVariableFunction (Section 13.1) computes the derivative of a function at a given x-value. For example, if f is OneVariableFunction, this code estimates the derivative at 0:

Code Example – C# calculus

double d = f.Differentiate( 0 );

Code Example – VB calculus

Dim D As Double = F.Differentiate(0)

NOTE—Class Polynomial provides a method for constructing the exact derivative of a polynomial. See Section 13.4 for more information.

To perform differentiation, every OneVariableFunction has an IDifferentiator object associated with it. NMath provides class RiddersDifferentiator, which computes the derivative of a given function at a given x-value by Ridders' method of polynomial extrapolation, and implements the IDifferentiator interface.

Extrapolations of higher and higher order are produced. Iteration stops when either the estimated error is less than a specified error tolerance, the error estimate is significantly worse than the previous order, or the maximum order is reached.

The default IDifferentiator for a OneVariableFunction is an instance of RiddersDifferentiator. To achieve more control over how differentiation is performed, you can instantiate your own RiddersDifferentiator. For instance, this code uses the Tolerance property to set the error tolerance to a non-default value, and the MaximumOrder property to set the maximum order, then calls the Differentiate() method to differentiate function f at π:

Code Example – C# calculus

var ridders = new RiddersDifferentiator();

ridders.Tolerance = 1e-6;

ridders.MaximumOrder = 20;

double d = ridders.Differentiate( f, Math.PI );

Code Example – VB calculus

Dim Ridders As New RiddersDifferentiator()

Ridders.Tolerance = "1e-6"

Ridders.MaximumOrder = 20

Dim D As Double = Ridders.Differentiate(F, Math.PI)

Setting the error tolerance to a value less than zero ensures that the Ridders differentiation is of the maximum order:

Code Example – C# calculus

var ridders = new RiddersDifferentiator();

ridders.Tolerance = -1;

double d = ridders.Differentiate( f, 1 );

Code Example – VB calculus

Dim Ridders As New RiddersDifferentiator()

Ridders.Tolerance = -1

Dim D As Double = Ridders.Differentiate(F, 1)

Read-only properties are provided for accessing information about a derivative approximation, once it has been computed:

● ErrorEstimate gets an estimate of the error of the derivative just computed.

● Order gets the order of the final polynomial extrapolation.

● ToleranceMet gets a boolean value indicating whether or not the error estimate for the derivative approximation is less than the tolerance.

● Tableau gets a matrix of successive approximations produced while computing the derivative. Successive columns in the matrix contain higher orders of extrapolation; successive rows decreasing step size.

For instance, this code checks whether the error tolerance was met before proceeding:

Code Example – C# calculus

var ridders = new RiddersDifferentiator();

double d = ridders.Differentiate( f, Math.PI );





if ( ridders.ToleranceMet ) {




  // Do something here...




}


Code Example – VB calculus


Dim Ridders As New RiddersDifferentiator()




Dim D As Double = ridders.Differentiate(F, Math.PI)








If Ridders.ToleranceMet Then




  ' Do something here...




End If







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