com.numericalmethod.suanshu.analysis.uniroot
Class Halley

java.lang.Object
  com.numericalmethod.suanshu.analysis.uniroot.Uniroot
      com.numericalmethod.suanshu.analysis.uniroot.Halley

public class Halley
extends Uniroot

Halley's method is an iterative root finding method for a univariate function with a continuous second derivative, i.e., a C² function.

It has the following properties.

• The function to be solved is assumed to be continuous and smooth (1st derivative exists).
• The 1st derivative is assumed to be continuous and smooth (2nd derivative exists).
• The rate of convergence for solution is cubic.

See Also:

Wikipedia: Halley's method

Nested Class Summary

Nested classes/interfaces inherited from class com.numericalmethod.suanshu.analysis.uniroot.Uniroot

Uniroot.NoRootFoundException

Field Summary

UnivariateRealFunction	d2f the 2nd derivative of f, d2f/dx2
UnivariateRealFunction	df the 1st derivative of f, df/dx

Fields inherited from class com.numericalmethod.suanshu.analysis.uniroot.Uniroot

f, tol

Constructor Summary

Halley(UnivariateRealFunction f, double tol)
Construct an instance of Halley's root finding algorithm.

Halley(UnivariateRealFunction f, UnivariateRealFunction df, UnivariateRealFunction d2f, double tol)
Construct an instance of Halley's root finding algorithm.

Method Summary

double	solve(int maxIterations, double guess) Solve f(x) = 0 using Halley's algorithm with an initial guess for the maximum number of iterations.
double	solve(int iterations, double lower, double upper, double... guess) The implementation of a specific uniroot finding algorithm.

Methods inherited from class com.numericalmethod.suanshu.analysis.uniroot.Uniroot

solve

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Field Detail

df

public final UnivariateRealFunction df

the 1st derivative of f, df/dx

d2f

public final UnivariateRealFunction d2f

the 2nd derivative of f, d2f/dx2

Constructor Detail

Halley

public Halley(UnivariateRealFunction f,
              UnivariateRealFunction df,
              UnivariateRealFunction d2f,
              double tol)

Construct an instance of Halley's root finding algorithm.

Parameters:

f - the function to be solved

df - the first derivative of the function

d2f - the second derivative of the function

tol - the convergence tolerance

Halley

public Halley(UnivariateRealFunction f,
              double tol)

Construct an instance of Halley's root finding algorithm. The first and second derivatives are automatically computed using the central finite difference.

Parameters:

f - the function to be solved

tol - the convergence tolerance

Method Detail

solve

public double solve(int maxIterations,
                    double guess)

Solve f(x) = 0 using Halley's algorithm with an initial guess for the maximum number of iterations.

Parameters:

maxIterations - the maximum number of iterations

guess - an initial guess

Returns:

an (approximate) root

solve

public double solve(int iterations,
                    double lower,
                    double upper,
                    double... guess)

Description copied from class: Uniroot

The implementation of a specific uniroot finding algorithm.

Specified by:

solve in class Uniroot

Parameters:

iterations - the maximum number of iterations

lower - the lower bound of the bracketing interval

upper - the upper bound of the bracketing interval

guess - an initial guess of the root. It should lie between [lower, upper]. Note that guess is a double[] array. This signature allows multiple initial guesses for certain types of uniroot algorithms, e.g., Brent's algorithm.

Returns:

an approximate root