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

java.lang.Object
  extended by 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 1422195 2012-12-15 06:45:18Z psteitz $

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[])


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