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 1455194 2013-03-11 15:45:54Z luc $

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)
                             throws NullArgumentException,
                                    NoDataException,
                                    DimensionMismatchException
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)
                       throws NullArgumentException,
                              DimensionMismatchException,
                              NoDataException
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)
                                throws NullArgumentException,
                                       NoDataException,
                                       DimensionMismatchException
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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