Package org.opensourcephysics.numerics

Class Root

java.lang.Object
org.opensourcephysics.numerics.Root
public class Root extends Object
Class Root defines various root finding algorithms. This class cannot be subclassed or instantiated because all methods are static.
Author:
Wolfgang Christian

  • Method Summary

    Modifier and Type
    Method
    Description
    static double
    bisection(Function f, double x1, double x2, double tol)
    Implements the bisection method for finding the root of a function.
    static double[][]
    cubic(double a, double b, double c, double d)
    Solves for the roots of the cubic equation ax3+bx2+cx+d=0.
    static double[][]
    getJacobian(VectorFunction feqs, int n, double[] xx, double tol)
    Computes the Jacobian using a finite difference approximation.
    static double
    newton(Function f, double x, double tol)
    Implements Newton's method for finding the root of a function.
    static double
    newton(Function f, Function df, double x, double tol)
    Implements Newton's method for finding the root of a function.
    static double
    newtonBisection(Function f, double xleft, double xright, double tol, int icmax)
    Implements Newton's method for finding the root but switches to the bisection method if the the estimate is not between xleft and xright.
    static double
    newtonMultivar(VectorFunction feqs, double[] xx, int max, double tol)
     
    static double[][]
    polynomial(double[] c)
    Solves for the roots of the polynomial with the given coefficients c: c[0] + c[1] * x + c[2] * x^2 + ....
    static double[][]
    quadratic(double a, double b, double c)
    Solves for the complex roots of the quadratic equation ax2+bx+c=0.
    static double[]
    quadraticReal(double a, double b, double c)
    Solves for the real roots of the quadratic equation ax2+bx+c=0.

    Methods inherited from class java.lang.Object

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

  • Method Details

    • quadraticReal

      public static double[] quadraticReal(double a, double b, double c)
      Solves for the real roots of the quadratic equation ax2+bx+c=0.
      Parameters:
      a - double quadratic term coefficient
      b - double linear term coefficient
      c - double constant term
      Returns:
      double[] an array containing the two roots.

    • quadratic

      public static double[][] quadratic(double a, double b, double c)
      Solves for the complex roots of the quadratic equation ax2+bx+c=0.
      Parameters:
      a - double quadratic term coefficient
      b - double linear term coefficient
      c - double constant term
      Returns:
      double[] an array containing the two roots.

    • cubic

      public static double[][] cubic(double a, double b, double c, double d)
      Solves for the roots of the cubic equation ax3+bx2+cx+d=0.
      Parameters:
      a - double cubic term coefficient
      b - double quadratic term coefficient
      c - double linear term coefficient
      d - double constant term
      Returns:
      double[] an array containing the two roots.

    • polynomial

      public static double[][] polynomial(double[] c)
      Solves for the roots of the polynomial with the given coefficients c: c[0] + c[1] * x + c[2] * x^2 + .... The roots are the complex eigenvalues of the companion matrix.
      Parameters:
      double - [] c coefficients
      Returns:
      double[][] complex roots

    • newton

      public static double newton(Function f, double x, double tol)
      Implements Newton's method for finding the root of a function. The derivative is calculated numerically using the central difference approximation.
      Parameters:
      f - Function the function
      x - double guess the root
      tol - double computation tolerance
      Returns:
      double the root or NaN if root not found.

    • newton

      public static double newton(Function f, Function df, double x, double tol)
      Implements Newton's method for finding the root of a function.
      Parameters:
      f - Function the function
      df - Function the derivative of the function
      x - double guess the root
      tol - double computation tolerance
      Returns:
      double the root or NaN if root not found.

    • bisection

      public static double bisection(Function f, double x1, double x2, double tol)
      Implements the bisection method for finding the root of a function.
      Parameters:
      f - Function the function
      x1 - double lower
      x2 - double upper
      tol - double computation tolerance
      Returns:
      double the root or NaN if root not found

    • newtonBisection

      public static double newtonBisection(Function f, double xleft, double xright, double tol, int icmax)
      Implements Newton's method for finding the root but switches to the bisection method if the the estimate is not between xleft and xright. Method contributed by: J E Hasbun A Newton Raphson result is accepted if it is within the known bounds, else a bisection step is taken. Ref: Computational Physics by P. L. Devries (J. Wiley, 1993) input: [xleft,xright] is the interval wherein fx() has a root, icmax is the maximum iteration number, and tol is the tolerance level output: returns xbest as the value of the function Reasonable values of icmax and tol are 25, 5e-3. Returns the root or NaN if root not found.
      Parameters:
      xleft - double
      xright - double
      tol - double tolerance
      icmax - int number of trials
      Returns:
      double the root

    • newtonMultivar

      public static double newtonMultivar(VectorFunction feqs, double[] xx, int max, double tol)

    • getJacobian

      public static double[][] getJacobian(VectorFunction feqs, int n, double[] xx, double tol)
      Computes the Jacobian using a finite difference approximation. Contributed to OSP by J E Hasbun 2007.
      Parameters:
      feqs - VectorFunction - the function containing n equations
      n - int - number of equations
      xx - double[] - the variable array at which the Jacobian is calculated
      tol - double - the small change to find the derivatives
      Returns:
      double[][] J - the square matrix containing the Jacobian