org.opensourcephysics.numerics
Class Root

java.lang.Object
  org.opensourcephysics.numerics.Root

public class Root
extends java.lang.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

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 Detail

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