Package org.apache.commons.math.analysis.solvers

Class MullerSolver

  • All Implemented Interfaces:
    UnivariateRealSolver, ConvergingAlgorithm
    public class MullerSolver
extends UnivariateRealSolverImpl
    Implements the Muller's Method for root finding of real univariate functions. For reference, see Elementary Numerical Analysis, ISBN 0070124477, chapter 3.

    Muller's method applies to both real and complex functions, but here we restrict ourselves to real functions. Methods solve() and solve2() find real zeros, using different ways to bypass complex arithmetics.

    Since:
    1.2
    Version:
    $Revision: 1070725 $ $Date: 2011-02-15 02:31:12 +0100 (mar. 15 févr. 2011) $

    • Method Detail

      • solve

        @Deprecated
public double solve​(double min,
                    double max)
             throws ConvergenceException,
                    FunctionEvaluationException
        Deprecated.
        Solve for a zero root in the given interval.

        A solver may require that the interval brackets a single zero root. Solvers that do require bracketing should be able to handle the case where one of the endpoints is itself a root.

        Parameters:
        min - the lower bound for the interval.
        max - the upper bound for the interval.
        Returns:
        a value where the function is zero
        Throws:
        ConvergenceException - if the maximum iteration count is exceeded or the solver detects convergence problems otherwise.
        FunctionEvaluationException - if an error occurs evaluating the function

      • solve

        @Deprecated
public double solve​(double min,
                    double max,
                    double initial)
             throws ConvergenceException,
                    FunctionEvaluationException
        Deprecated.
        Solve for a zero in the given interval, start at startValue.

        A solver may require that the interval brackets a single zero root. Solvers that do require bracketing should be able to handle the case where one of the endpoints is itself a root.

        Parameters:
        min - the lower bound for the interval.
        max - the upper bound for the interval.
        initial - the start value to use
        Returns:
        a value where the function is zero
        Throws:
        ConvergenceException - if the maximum iteration count is exceeded or the solver detects convergence problems otherwise.
        FunctionEvaluationException - if an error occurs evaluating the function

      • solve

        public double solve​(int maxEval,
                    UnivariateRealFunction f,
                    double min,
                    double max,
                    double initial)
             throws MaxIterationsExceededException,
                    FunctionEvaluationException
        Find a real root in the given interval with initial value.

        Requires bracketing condition.

        Overrides:
        solve in class UnivariateRealSolverImpl
        Parameters:
        f - the function to solve
        min - the lower bound for the interval
        max - the upper bound for the interval
        initial - the start value to use
        maxEval - Maximum number of evaluations.
        Returns:
        the point at which the function value is zero
        Throws:
        MaxIterationsExceededException - if the maximum iteration count is exceeded or the solver detects convergence problems otherwise
        FunctionEvaluationException - if an error occurs evaluating the function
        java.lang.IllegalArgumentException - if any parameters are invalid

      • solve

        @Deprecated
public double solve​(UnivariateRealFunction f,
                    double min,
                    double max,
                    double initial)
             throws MaxIterationsExceededException,
                    FunctionEvaluationException
        Deprecated.
        in 2.2 (to be removed in 3.0).
        Find a real root in the given interval with initial value.

        Requires bracketing condition.

        Parameters:
        f - the function to solve
        min - the lower bound for the interval
        max - the upper bound for the interval
        initial - the start value to use
        Returns:
        the point at which the function value is zero
        Throws:
        MaxIterationsExceededException - if the maximum iteration count is exceeded or the solver detects convergence problems otherwise
        FunctionEvaluationException - if an error occurs evaluating the function
        java.lang.IllegalArgumentException - if any parameters are invalid

      • solve

        public double solve​(int maxEval,
                    UnivariateRealFunction f,
                    double min,
                    double max)
             throws MaxIterationsExceededException,
                    FunctionEvaluationException
        Find a real root in the given interval.

        Original Muller's method would have function evaluation at complex point. Since our f(x) is real, we have to find ways to avoid that. Bracketing condition is one way to go: by requiring bracketing in every iteration, the newly computed approximation is guaranteed to be real.

        Normally Muller's method converges quadratically in the vicinity of a zero, however it may be very slow in regions far away from zeros. For example, f(x) = exp(x) - 1, min = -50, max = 100. In such case we use bisection as a safety backup if it performs very poorly.

        The formulas here use divided differences directly.

        Overrides:
        solve in class UnivariateRealSolverImpl
        Parameters:
        f - the function to solve
        min - the lower bound for the interval
        max - the upper bound for the interval
        maxEval - Maximum number of evaluations.
        Returns:
        the point at which the function value is zero
        Throws:
        MaxIterationsExceededException - if the maximum iteration count is exceeded or the solver detects convergence problems otherwise
        FunctionEvaluationException - if an error occurs evaluating the function
        java.lang.IllegalArgumentException - if any parameters are invalid

      • solve

        @Deprecated
public double solve​(UnivariateRealFunction f,
                    double min,
                    double max)
             throws MaxIterationsExceededException,
                    FunctionEvaluationException
        Deprecated.
        in 2.2 (to be removed in 3.0).
        Find a real root in the given interval.

        Original Muller's method would have function evaluation at complex point. Since our f(x) is real, we have to find ways to avoid that. Bracketing condition is one way to go: by requiring bracketing in every iteration, the newly computed approximation is guaranteed to be real.

        Normally Muller's method converges quadratically in the vicinity of a zero, however it may be very slow in regions far away from zeros. For example, f(x) = exp(x) - 1, min = -50, max = 100. In such case we use bisection as a safety backup if it performs very poorly.

        The formulas here use divided differences directly.

        Parameters:
        f - the function to solve
        min - the lower bound for the interval
        max - the upper bound for the interval
        Returns:
        the point at which the function value is zero
        Throws:
        MaxIterationsExceededException - if the maximum iteration count is exceeded or the solver detects convergence problems otherwise
        FunctionEvaluationException - if an error occurs evaluating the function
        java.lang.IllegalArgumentException - if any parameters are invalid

      • solve2

        @Deprecated
public double solve2​(double min,
                     double max)
              throws MaxIterationsExceededException,
                     FunctionEvaluationException
        Deprecated.
        replaced by solve2(UnivariateRealFunction, double, double) since 2.0
        Find a real root in the given interval.

        solve2() differs from solve() in the way it avoids complex operations. Except for the initial [min, max], solve2() does not require bracketing condition, e.g. f(x0), f(x1), f(x2) can have the same sign. If complex number arises in the computation, we simply use its modulus as real approximation.

        Because the interval may not be bracketing, bisection alternative is not applicable here. However in practice our treatment usually works well, especially near real zeros where the imaginary part of complex approximation is often negligible.

        The formulas here do not use divided differences directly.

        Parameters:
        min - the lower bound for the interval
        max - the upper bound for the interval
        Returns:
        the point at which the function value is zero
        Throws:
        MaxIterationsExceededException - if the maximum iteration count is exceeded or the solver detects convergence problems otherwise
        FunctionEvaluationException - if an error occurs evaluating the function
        java.lang.IllegalArgumentException - if any parameters are invalid

      • solve2

        @Deprecated
public double solve2​(UnivariateRealFunction f,
                     double min,
                     double max)
              throws MaxIterationsExceededException,
                     FunctionEvaluationException
        Deprecated.
        in 2.2 (to be removed in 3.0).
        Find a real root in the given interval.

        solve2() differs from solve() in the way it avoids complex operations. Except for the initial [min, max], solve2() does not require bracketing condition, e.g. f(x0), f(x1), f(x2) can have the same sign. If complex number arises in the computation, we simply use its modulus as real approximation.

        Because the interval may not be bracketing, bisection alternative is not applicable here. However in practice our treatment usually works well, especially near real zeros where the imaginary part of complex approximation is often negligible.

        The formulas here do not use divided differences directly.

        Parameters:
        f - the function to solve
        min - the lower bound for the interval
        max - the upper bound for the interval
        Returns:
        the point at which the function value is zero
        Throws:
        MaxIterationsExceededException - if the maximum iteration count is exceeded or the solver detects convergence problems otherwise
        FunctionEvaluationException - if an error occurs evaluating the function
        java.lang.IllegalArgumentException - if any parameters are invalid