org.apache.commons.math3.analysis.polynomials
Class PolynomialFunctionNewtonForm

java.lang.Object
  org.apache.commons.math3.analysis.polynomials.PolynomialFunctionNewtonForm

All Implemented Interfaces:

UnivariateDifferentiableFunction, UnivariateFunction

public class PolynomialFunctionNewtonForm
extends Object
implements UnivariateDifferentiableFunction

Implements the representation of a real polynomial function in Newton Form. For reference, see Elementary Numerical Analysis, ISBN 0070124477, chapter 2.

The formula of polynomial in Newton form is p(x) = a[0] + a[1](x-c[0]) + a[2](x-c[0])(x-c[1]) + ... + a[n](x-c[0])(x-c[1])...(x-c[n-1]) Note that the length of a[] is one more than the length of c[]

Since:

1.2

Version:

$Id: PolynomialFunctionNewtonForm.java 7721 2013-02-14 14:07:13Z CardosoP $

Constructor Summary

PolynomialFunctionNewtonForm(double[] a, double[] c)
Construct a Newton polynomial with the given a[] and c[].

Method Summary

protected void	computeCoefficients() Calculate the normal polynomial coefficients given the Newton form.
int	degree() Returns the degree of the polynomial.
static double	evaluate(double[] a, double[] c, double z) Evaluate the Newton polynomial using nested multiplication.
double[]	getCenters() Returns a copy of the centers array.
double[]	getCoefficients() Returns a copy of the coefficients array.
double[]	getNewtonCoefficients() Returns a copy of coefficients in Newton form formula.
DerivativeStructure	value(DerivativeStructure t) Simple mathematical function.
double	value(double z) Calculate the function value at the given point.
protected static void	verifyInputArray(double[] a, double[] c) Verifies that the input arrays are valid.

Methods inherited from class java.lang.Object

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

Constructor Detail

PolynomialFunctionNewtonForm

public PolynomialFunctionNewtonForm(double[] a,
                                    double[] c)

Construct a Newton polynomial with the given a[] and c[]. The order of centers are important in that if c[] shuffle, then values of a[] would completely change, not just a permutation of old a[].

The constructor makes copy of the input arrays and assigns them.

Parameters:

a - Coefficients in Newton form formula.

c - Centers.

Throws:

NullArgumentException - if any argument is null.

NoDataException - if any array has zero length.

DimensionMismatchException - if the size difference between a and c is not equal to 1.

Method Detail

value

public double value(double z)

Calculate the function value at the given point.

Specified by:

value in interface UnivariateFunction

Parameters:

z - Point at which the function value is to be computed.

Returns:

the function value.

value

public DerivativeStructure value(DerivativeStructure t)

Simple mathematical function.

UnivariateDifferentiableFunction classes compute both the value and the first derivative of the function.

Specified by:

value in interface UnivariateDifferentiableFunction

Parameters:

t - function input value

Returns:

function result

Since:

3.1

degree

public int degree()

Returns the degree of the polynomial.

Returns:

the degree of the polynomial

getNewtonCoefficients

public double[] getNewtonCoefficients()

Returns a copy of coefficients in Newton form formula.

Changes made to the returned copy will not affect the polynomial.

Returns:

a fresh copy of coefficients in Newton form formula

getCenters

public double[] getCenters()

Returns a copy of the centers array.

Changes made to the returned copy will not affect the polynomial.

Returns:

a fresh copy of the centers array.

getCoefficients

public double[] getCoefficients()

Returns a copy of the coefficients array.

Changes made to the returned copy will not affect the polynomial.

Returns:

a fresh copy of the coefficients array.

evaluate

public static double evaluate(double[] a,
                              double[] c,
                              double z)

Evaluate the Newton polynomial using nested multiplication. It is also called Horner's Rule and takes O(N) time.

Parameters:

a - Coefficients in Newton form formula.

c - Centers.

z - Point at which the function value is to be computed.

Returns:

the function value.

Throws:

NullArgumentException - if any argument is null.

NoDataException - if any array has zero length.

DimensionMismatchException - if the size difference between a and c is not equal to 1.

computeCoefficients

protected void computeCoefficients()

Calculate the normal polynomial coefficients given the Newton form. It also uses nested multiplication but takes O(N^2) time.

verifyInputArray

protected static void verifyInputArray(double[] a,
                                       double[] c)

Verifies that the input arrays are valid.

The centers must be distinct for interpolation purposes, but not for general use. Thus it is not verified here.

Parameters:

a - the coefficients in Newton form formula

c - the centers

Throws:

NullArgumentException - if any argument is null.

NoDataException - if any array has zero length.

DimensionMismatchException - if the size difference between a and c is not equal to 1.

See Also:

DividedDifferenceInterpolator.computeDividedDifference(double[], double[])