HalleySolver Class

public sealed class HalleySolver : EquationSolver

Public NotInheritable Class HalleySolver
	Inherits EquationSolver

public ref class HalleySolver sealed : public EquationSolver

[<SealedAttribute>]
type HalleySolver = 
    class
        inherit EquationSolver
    end

Inheritance

Object  →  ManagedIterativeAlgorithm<Double, Double, SolutionReport<Double, Double>>  →  IterativeAlgorithm<Double>  →  EquationSolver  →  HalleySolver

Halley's algorithm starts from a single starting value. It then uses the derivative and second derivative of the target function to find successive approximations of the root.

Halley's method has cubic convergence. This means that once an approximation is close to a root, the number of correct digits will roughly triple in each iteration.

Halley's method therefore has superior convergence properties relative to Newton's method. However, this comes at a cost which is only justified if the second derivative is relatively cheap to compute. One example of this is functions that satisfy differential equations, like the Bessel functions.

HalleySolver inherits from EquationSolver, which in turn inherits from IterativeAlgorithm<T>. All properties of IterativeAlgorithm<T> are available. The AbsoluteTolerance and RelativeTolerance properties set the desired precision as specified by the ConvergenceCriterion property. The default value for both tolerances is SqrtEpsilon (roughly 10^-8). MaxIterations sets the maximum number of iterations.

The TargetFunction property is a function of one variable that specifies the function we want to find a root for. The LowerBound and UpperBound properties specify the bounds of the bracketing interval. The target function must have a different sign at each end of this interval.

The Solve() method performs the actual approximation of the root. This method returns the best approximation that was found. The Status property indicates whether the algorithm was successful. The EstimatedError property gives an upper bound for the difference between the approximated and the actual root.

Reference