com.numericalmethod.suanshu.analysis.uniroot
Class Newton

java.lang.Object
  com.numericalmethod.suanshu.analysis.uniroot.Uniroot
      com.numericalmethod.suanshu.analysis.uniroot.Newton

public class Newton
extends Uniroot

Newton–Raphson method is an iterative root finding method for univariate functions. It has the following properties.

• The function to be solved is assumed to be continuous and smooth (1st derivative exists).
• The solution converges quadratically, when the multiplicity of the root is 1; otherwise, it is linear.
• The solution may fail to converge when the derivative is or is close to 0.
• The solution may fail to converge if the initial guess is far away from the true value.

See Also:

Wikipedia: Newton's method

Nested Class Summary

Nested classes/interfaces inherited from class com.numericalmethod.suanshu.analysis.uniroot.Uniroot

Uniroot.NoRootFoundException

Field Summary

UnivariateRealFunction

df the 1st derivative of f, df/dx

Fields inherited from class com.numericalmethod.suanshu.analysis.uniroot.Uniroot

f, tol

Constructor Summary

Newton(UnivariateRealFunction f, double tol)
Construct an instance of Newton's root finding algorithm.

Newton(UnivariateRealFunction f, UnivariateRealFunction df, double tol)
Construct an instance of Newton's root finding algorithm.

Method Summary

double	solve(int maxIterations, double guess) Solve f(x) = 0 using Newton's algorithm with an initial guess for the maximum number of iterations.
double	solve(int maxIterations, double lower, double upper, double... guess) The implementation of a specific uniroot finding algorithm.

Methods inherited from class com.numericalmethod.suanshu.analysis.uniroot.Uniroot

solve

Methods inherited from class java.lang.Object

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

Field Detail

df

public final UnivariateRealFunction df

the 1st derivative of f, df/dx

Constructor Detail

Newton

public Newton(UnivariateRealFunction f,
              UnivariateRealFunction df,
              double tol)

Construct an instance of Newton's root finding algorithm.

Parameters:

f - the function to be solved

df - the derivative of the function

tol - the convergence tolerance

Newton

public Newton(UnivariateRealFunction f,
              double tol)

Construct an instance of Newton's root finding algorithm. The first derivative is automatically computed using the central finite difference.

Parameters:

f - the function to be solved

tol - the convergence tolerance

Method Detail

solve

public double solve(int maxIterations,
                    double guess)

Solve f(x) = 0 using Newton's algorithm with an initial guess for the maximum number of iterations.

Parameters:

maxIterations - the maximum number of iterations

guess - an initial guess

Returns:

an (approximate) root

solve

public double solve(int maxIterations,
                    double lower,
                    double upper,
                    double... guess)

Description copied from class: Uniroot

The implementation of a specific uniroot finding algorithm.

Specified by:

solve in class Uniroot

Parameters:

maxIterations - the maximum number of iterations

lower - the lower bound of the bracketing interval

upper - the upper bound of the bracketing interval

guess - an initial guess of the root. It should lie between [lower, upper]. Note that guess is a double[] array. This signature allows multiple initial guesses for certain types of uniroot algorithms, e.g., Brent's algorithm.

Returns:

an approximate root