Package dev.nm.analysis.function.rn2r1.univariate

Class ContinuedFraction

java.lang.Object

dev.nm.analysis.function.rn2r1.AbstractRealScalarFunction

dev.nm.analysis.function.rn2r1.univariate.AbstractUnivariateRealFunction

dev.nm.analysis.function.rn2r1.univariate.ContinuedFraction

All Implemented Interfaces:

Function<Vector,Double>, RealScalarFunction, UnivariateRealFunction

public class ContinuedFraction
extends AbstractUnivariateRealFunction

A continued fraction representation of a number has this form: \[ z = b_0 + \cfrac{a_1}{b_1 + \cfrac{a_2}{b_2 + \cfrac{a_3}{b_3 + \cfrac{a_4}{b_4 + \ddots\,}}}} \] a_i and b_i can be functions of x, which in turn makes z a function of x.

The sequence z_n may or may not converge. In theory, z_n can be written as a fraction: \(z_n = \frac{A_n}{B_n}\). A_n and B_n can be computed by the fundamental recurrence formulas. In practice, we compute z_n using the modified Lentz's method from Thompson and Barnett. This method may suffer from the "false convergence" problem. That is, differences between successive convergents become small, seeming to indicate convergence, but then increase again by many orders of magnitude before finally converging.

See Also:

• • "I. J. Thompson, A. R. Barnett, "Coulomb and Bessel functions of complex arguments and order," J. Comput. Phys, 1986."
  • "W. J. Lentz, "Generating Bessel Functions In Mie Scattering Calculations Using Continued Fractions," Applied Optics, Vol. 15, Issue 3, p. 668-671."
  • "W. Gautschi, Math. Comput, 31, 994(1997)."
  • Wikipedia: Generalized continued fraction
  • Wikipedia: Fundamental recurrence formulas

Nested Class Summary

Nested Classes

Modifier and Type	Class	Description
static class	ContinuedFraction.MaxIterationsExceededException	RuntimeException thrown when the continued fraction fails to converge for a given epsilon before a certain number of iterations.
static interface	ContinuedFraction.Partials	This interface defines a continued fraction in terms of the partial numerators an, and the partial denominators bn.

Nested classes/interfaces inherited from interface dev.nm.analysis.function.Function

Function.EvaluationException

Constructor Summary

Constructors

Constructor	Description
ContinuedFraction(ContinuedFraction.Partials partials)	Construct a continued fraction.
ContinuedFraction(ContinuedFraction.Partials partials, double epsilon, int maxIterations)	Construct a continued fraction.
ContinuedFraction(ContinuedFraction.Partials partials, int scale, int maxIterations)	Construct a continued fraction.

Method Summary

Modifier and Type	Method	Description
double	evaluate(double x)	Evaluate y = f(x).
BigDecimal	evaluate(BigDecimal x)	Evaluate z.

Methods inherited from class dev.nm.analysis.function.rn2r1.univariate.AbstractUnivariateRealFunction

evaluate

Methods inherited from class dev.nm.analysis.function.rn2r1.AbstractRealScalarFunction

dimensionOfDomain, dimensionOfRange

Methods inherited from class java.lang.Object

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

Methods inherited from interface dev.nm.analysis.function.Function

dimensionOfDomain, dimensionOfRange

Constructor Details

ContinuedFraction

public ContinuedFraction(ContinuedFraction.Partials partials,
 double epsilon,
 int maxIterations)

Construct a continued fraction.

Parameters:

partials - the definition in terms of partial numerators and partial denominators

epsilon - a precision parameter: when a number |x| ≤ ε, it is considered 0

maxIterations - the maximum number of iterations

ContinuedFraction

public ContinuedFraction(ContinuedFraction.Partials partials,
 int scale,
 int maxIterations)

Construct a continued fraction.

Parameters:

partials - the definition in terms of partial numerators and partial denominators

scale - the accuracy

maxIterations - the maximum number of iterations

ContinuedFraction

public ContinuedFraction(ContinuedFraction.Partials partials)

Construct a continued fraction.

Parameters:

partials - the definition in terms of partial numerators and partial denominators

Method Details

evaluate

public double evaluate(double x)

Evaluate y = f(x). This implementation adopts the modified Lentz's method, using double arithmetics. It is quick. However, the precision is limited by the double precision of the intermediate results. This (and probably other implementations using double precision math) may give poor results for some continued fraction.

Parameters:

x - x

Returns:

an approximation of z

Throws:

ContinuedFraction.MaxIterationsExceededException - if it does not converge before the maximum number of iterations; repeat with a bigger epsilon, or use the BigDecimal version of the algorithm

evaluate

public BigDecimal evaluate(BigDecimal x)

Evaluate z. This implementation adopts the modified Lentz's method using arbitrary precision arithmetics BigDecimal.

Parameters:

x - x

Returns:

an approximation of z

Throws:

ContinuedFraction.MaxIterationsExceededException - if it does not converge before the maximum number of iterations; repeat with a bigger epsilon