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From "Fran Lattanzio (JIRA)" <j...@apache.org>
Subject [jira] [Commented] (MATH-1325) Improve finite differencing infrastructure
Date Thu, 18 Feb 2016 02:33:18 GMT

    [ https://issues.apache.org/jira/browse/MATH-1325?page=com.atlassian.jira.plugin.system.issuetabpanels:comment-tabpanel&focusedCommentId=15151611#comment-15151611
] 

Fran Lattanzio commented on MATH-1325:
--------------------------------------

I will generate a git patch for this - that would probably make it easier to load up than
a pull request. I'll also add some more unit tests, since that will show how to bring everything
together from a coding perspective.

In terms of multivariate derivatives: I haven't submitted anything yet. It's basically trivial
to generate the multidimensional stencils themselves (either by solving the mutlivariate equations
or, more simply, computing them as a suitable tensor product of univariate ones). The trickier
part(s) is in the bandwidth selection. I will commit something to at least generate the multivariate
coefficients and add a fixed multivariate bandwidth selection strategy, so at least there
will be something. Smarter/more complex multivariate bandwidth strategies can come later.

> Improve finite differencing infrastructure
> ------------------------------------------
>
>                 Key: MATH-1325
>                 URL: https://issues.apache.org/jira/browse/MATH-1325
>             Project: Commons Math
>          Issue Type: New Feature
>            Reporter: Fran Lattanzio
>            Priority: Minor
>
> The existing finite difference framework in commons math is a limiting because it accepts
only fixed bandwidth parameters. Furthermore, the finite difference coefficients/descriptions
are not exposed to the user in any reasonable fashion (e.g. a user doing a numerical ODE solve
probably wants to just grab suitable coefficients from somewhere). 
> Conceptually, I think the work of finite difference can be broadly divided into three
tasks:
> 1. Generation of finite difference coefficients. Again, one should be able to do this
and get the results outside of the context of taking an actual derivative. Ideally, we could
generate coefficients for any flavor (forward, central, backward) and order.
> 2. Selection of the bandwidth. This is, to be honest, the trickiest part of computing
a numerical derivative. There is some "art" to picking a proper bandwidth that will generate
an accurate numerical derivative - there are two competing sources of error (roundoff, due
to the finite representation of floating points; and truncation, due to the inherent nature
of finite differences). Ideally, we want to pick a bandwidth that will minimize the *total*
error.
> 3. Actually computing the finite difference derivative estimate. This is really easy
once you have 1. and 2.
> 4. Extend 1-3 to include support for multivariate finite differences.



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