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java.lang.Objectorg.apache.commons.math.analysis.UnivariateRealSolverImpl
org.apache.commons.math.analysis.MullerSolver
public class MullerSolver
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.
Field Summary | |
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private static long |
serialVersionUID
serializable version identifier |
Fields inherited from class org.apache.commons.math.analysis.UnivariateRealSolverImpl |
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absoluteAccuracy, defaultAbsoluteAccuracy, defaultFunctionValueAccuracy, defaultMaximalIterationCount, defaultRelativeAccuracy, f, functionValueAccuracy, iterationCount, maximalIterationCount, relativeAccuracy, result, resultComputed |
Constructor Summary | |
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MullerSolver(UnivariateRealFunction f)
Construct a solver for the given function. |
Method Summary | |
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double |
solve(double min,
double max)
Find a real root in the given interval. |
double |
solve(double min,
double max,
double initial)
Find a real root in the given interval with initial value. |
double |
solve2(double min,
double max)
Find a real root in the given interval. |
Methods inherited from class org.apache.commons.math.analysis.UnivariateRealSolverImpl |
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clearResult, getAbsoluteAccuracy, getFunctionValueAccuracy, getIterationCount, getMaximalIterationCount, getRelativeAccuracy, getResult, isBracketing, isSequence, resetAbsoluteAccuracy, resetFunctionValueAccuracy, resetMaximalIterationCount, resetRelativeAccuracy, setAbsoluteAccuracy, setFunctionValueAccuracy, setMaximalIterationCount, setRelativeAccuracy, setResult, verifyBracketing, verifyInterval, verifySequence |
Methods inherited from class java.lang.Object |
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clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait |
Field Detail |
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private static final long serialVersionUID
Constructor Detail |
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public MullerSolver(UnivariateRealFunction f)
f
- function to solveMethod Detail |
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public double solve(double min, double max, double initial) throws MaxIterationsExceededException, FunctionEvaluationException
Requires bracketing condition.
min
- the lower bound for the intervalmax
- the upper bound for the intervalinitial
- the start value to use
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 invalidpublic double solve(double min, double max) throws MaxIterationsExceededException, FunctionEvaluationException
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.
min
- the lower bound for the intervalmax
- the upper bound for the interval
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 invalidpublic double solve2(double min, double max) throws MaxIterationsExceededException, FunctionEvaluationException
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.
min
- the lower bound for the intervalmax
- the upper bound for the interval
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
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