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https://issues.apache.org/jira/browse/MATH-1201?page=com.atlassian.jira.plugin.system.issuetabpanels:comment-tabpanel&focusedCommentId=14304003#comment-14304003
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Jason Sachs commented on MATH-1201:
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It's also not exactly clear from the documentation whether the tolerance is in
the function's output or in the function's input. After reading the
documentation and source several times I figured out that relative accuracy and
absolute accuracy are intended to refer to the function's input value (that the
difference in the function's input value between the computed root and the
exact value of a nearby root will be less than the tolerance), whereas function
value accuracy is intended to refer to the function's output value (the
absolute value of the function at the computed root will be less than the
function value accuracy).
> Please clarify tolerance semantics of
> org.apache.commons.math3.analysis.solvers
> --------------------------------------------------------------------------------
>
> Key: MATH-1201
> URL: https://issues.apache.org/jira/browse/MATH-1201
> Project: Commons Math
> Issue Type: Improvement
> Reporter: Jason Sachs
> Priority: Minor
>
> The documentation for
> [BrentSolver|http://commons.apache.org/proper/commons-math/apidocs/org/apache/commons/math3/analysis/solvers/BrentSolver.html]
> is somewhat vague and doesn't seem to agree with the source code:
> >The {{solve}} method returns a zero {{x}} of the function f in the given
> >interval {{[a, b]}} to within a tolerance {{6 eps abs x + t}} where {{eps}}
> >is the relative accuracy and {{t}} is the absolute accuracy. The given
> >interval must bracket the root.
> A couple of issues:
> - the default tolerance values are not clearly specified. The documentation
> says "default accuracy (1e-6)" but does not state whether it's absolute,
> relative, or function value accuracy. If I dig into the [source
> code|https://git-wip-us.apache.org/repos/asf?p=commons-math.git;a=blob;f=src/main/java/org/apache/commons/math3/analysis/solvers/BrentSolver.java],
> it is the default absolute accuracy. It is unclear what the default values
> for relative and function value accuracy are. I have to dig into the class
> tree and find
> [BaseAbstractUnivariateSolver|https://git-wip-us.apache.org/repos/asf?p=commons-math.git;a=blob;f=src/main/java/org/apache/commons/math3/analysis/solvers/BaseAbstractUnivariateSolver.java]
> to find out that the default relative accuracy is 10^-14 and the default
> function value accuracy is 10^-15. These constants in the code are never
> mentioned in the documentation for
> [BaseAbstractUnivariateSolver|https://commons.apache.org/proper/commons-math/javadocs/api-3.1/org/apache/commons/math3/analysis/solvers/BaseAbstractUnivariateSolver.html]
> but should be there.
> - the code appears not to use function value accuracy at all.
> - the [code for
> BrentSolver|https://git-wip-us.apache.org/repos/asf?p=commons-math.git;a=blob;f=src/main/java/org/apache/commons/math3/analysis/solvers/BrentSolver.java#l165]
> has the expression {{tol = 2*eps * abs(b) + t}}, not {{tol = 6*eps * abs(b)
> + t}} as would be implied by the documentation. Is this an error, or is there
> a magic feature of Brent's algorithm that effectively turns the 2 into a 6?
> ----
> Suggest you:
> - include the default relative and function value tolerances in the
> documentation for BaseAbstractUnivariateSolver
> - amend the documentation for
> [BaseUnivariateSolver|https://commons.apache.org/proper/commons-math/javadocs/api-3.1/org/apache/commons/math3/analysis/solvers/BaseUnivariateSolver.html]
> to expand upon the three tolerances: Are they always used by each of the
> solver implementations? (no they aren't) Do they add together, or is the
> minimum error of the three used? (it seems dependent on each of the solver
> algorithms; in BrentSolver the relative and absolute tolerances add)
> - amend the documentation for BrentSolver to state clearly that the default
> absolute accuracy is 10^-6 and the other default tolerances are defined in
> the documentation for BaseAbstractUnivariateSolver (with a link)
> - amend the documentation for BrentSolver to state that it does not use
> function value accuracy
> - address the discrepancy in the total tolerance formula between the
> documentation and the code: is the relevant constant 2 or 6?)
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