Package 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

  • Constructor Details

    • 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 Details

    • 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: