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From Eric Barnhill <ericbarnh...@gmail.com>
Subject Re: [math] New finite differencing framework for math4
Date Wed, 17 Feb 2016 20:13:19 GMT
Dear Fran, this is very interesting especially the bandwidth control 
which I would find useful. I will be happy to look this code over and 
test it on some problems in my field. Right now that will be in a month 
or two due to some personal issues.

I added a couple of questions to the JIRA.

Best,
Eric

On 17/02/16 03:41, Fran Lattanzio wrote:
> Hello all,
>
> I have created a pull request with a new univariate finite difference
> framework for [math]. The main features of this framework are:
> 1. *Exact* calculation of the coefficients for any stencil type
> (forward, backward, central), derivative order, and accuracy order.
> This is done by solving the system of linear equations generated by
> repeated Taylor expansion, using the field LU decomp over
> BigFractions. This is actually an important feature because it ensures
> there is no roundoff error in the coefficients; and we can tell if a
> coefficient is *exactly* zero.
> 2. Separation of derivative calculation and bandwidth selection -
> specifically, there is a strategy interface that decides the
> bandwidth, based on point at which the derivative is desired (and the
> nature of the stencil, etc).
> 3. Several canned bandwidth strategies:
>   a. A fixed bandwidth.
>   b. A "rule-of-thumb" strategy (which covers far more general cases
> than the rule-of-thumb suggested in a certain numerical cookbook).
>   c. An adaptive strategy that estimates the error scale of the finite
> difference function in order to determine the optimal bandwidth. This
> is done using a technique akin to Richardson extrapolation.
>   d. A decorator strategy to round bandwidths to a power-of-two. (Using
> powers-of-two bandwidths is good because they have exact IEEE floating
> point representations, so x +/- h will be correct to full precision.)
>
> Strategies b. and c. can also accept a function condition error
> parameter. This is quite important if you know that the underlying
> function is not computed to full machine precision, e.g. the function
> is an approximation accurate to 1e-10; or done over the IEEE 32s
> rather than 64s.
>
> Bandwidth strategy c. is derived from a thesis published by Ravi
> Mathur. Mathur provides formulae that shows how to find the optimal
> bandwidth - optimal in the sense that it will minimize the total error
> (which is a delicate balancing act!). His thesis can be found here:
> https://repositories.lib.utexas.edu/bitstream/handle/2152/ETD-UT-2012-05-5275/MATHUR-DISSERTATION.pdf?sequence=1
>
> (This is the first implementation of Mathur's ideas that I have seen.
> Most of his work is thorough, so although this implementation is "new"
>   and thus not considered an industry standard, I think it is
> definitely worth including. Some initial numerical experiments confirm
> as much.)
>
> These strategies should cover the vast majority of use cases, and of
> course the strategy interface allows users to implement anything
> bespoke if the need arises.
>
> I am working on generating more unit tests now. For now, there are
> only relatively simple tests for the coefficient generation and
> "integration tests" that compare analytical vs. numerical derivatives.
> If this is something that the community wants to include in math4, I
> will be happy to fully flesh them out.
>
> I am also working on a multivariate version. Generating the
> coefficients is easy... the tricky part for the multivariate case is
> the bandwidth strategies. The fixed bandwidth strategy is obviously
> trivial. I am trying to expand Mathur's work to the multivariate case
> to derive analogous rule-of-thumb and adaptive strategies but the
> equations are bit ticklish in the multivariate world.
>
> In any case, the JIRA ticket and pull request are here:
> https://issues.apache.org/jira/browse/MATH-1325
> https://github.com/apache/commons-math/pull/24
>
> Any suggestions are welcome,
> Fran.
>
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