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Sébastien Brisard edited comment on MATH797 at 6/3/12 1:58 PM:

I have a jQuadrature project sitting in my workspace for almost two years now, which I'm very
happy to refactor and contribute if we settle on a proper interface for Gauss integrators.
I have currently implemented: GaussLegendre, GaussChebyshev (first and second kinds), GaussHermite.
Integration points and weights are computed and cached if needed, so the order is not limited.
In NR for example, the integration points were computed with builtin Newton iterations. I
chose bracketing instead. It may be a bit slower, but it's not called very often (only when
new Gaussian rules are needed), and accuracy of the computed roots is then garanteed (to within
1ulp). It's more difficult to insure the accuracy of the weights (when you compute them onthefly),
and I wrote a high accuracy computation of integration points and weights for the GaussLegendre
rule, using {{BigDecimal}} (at that time, I knew but did not use CommonsMath, shame on me!).
If any of this seems interesting to you, it's right here !
One point worth noting: with Gauss integration schemes, I always wondered whether calling
the integration method {{integrate(f, a, b)}} was a good idea. Indeed, generic integration
schemes do not integrate the provided univariate function {{f( x )}}, but the weighted function
{{f( x ) * w( x )}}, which might be confusing. Of course, there is no such ambiguity with
GaussLegendre...
was (Author: celestin):
I have a jQuadrature project sitting in my workspace for almost two years now, which I'm
very happy to refactor and contribute if we settle on a proper interface for Gauss integrators.
I have currently implemented: GaussLegendre, GaussChebyshev (first and second kinds), GaussHermite.
Integration points and weights are computed and cached if needed, so the order is not limited.
In NR for example, the integration points were computed with builtin Newton iterations. I
chose bracketing instead. It may be a bit slower, but it's not called very often (only when
new Gaussian rules are needed), and accuracy of the computed roots is then garanteed (to within
1ulp). It's more difficult to insure the accuracy of the weights (when you compute them onthefly),
and I wrote a high accuracy computation of integration points and weights for the GaussLegendre
rule, using {{BigDecimal}} (at that time, I knew but did not use CommonsMath, shame on me!).
If any of this seems interesting to you, it's right here !
One point worth noting: with Gauss integration schemes, I always wondered whether calling
the integration method {{integrate(f, a, b)}} was a good idea. Indeed, generic integration
schemes do not integrate the provided univariate function {{f(x)}}, but the weighted function
{{f(x) * w(x)}}, which might be confusing. Of course, there is no such ambiguity with GaussLegendre...
> Single step integrators
> 
>
> Key: MATH797
> URL: https://issues.apache.org/jira/browse/MATH797
> Project: Commons Math
> Issue Type: Wish
> Affects Versions: 3.0
> Reporter: Gilles
> Assignee: Gilles
> Priority: Trivial
> Fix For: 3.1
>
>
> CM assumes that the user wants to integrate a complex function on a large interval, so
the large interval has to be subdivided into many subintervals. CM does the partition, and
performs convergence checks, using an iterative approach.
> However, if the function is smooth enough, no subdivision of the integration interval
is required. Those usecases could benefit from the efficiency gain of not performing a convergence
check.
> The proposal is to provide a new interface "UnivariateSingleStepIntegrator":
> {code}
> interface SingleIntervalIntegrator {
> /**
> * Method for implementing a single interval integration.
> * There is no convergence checks because it is not iterative.
> *
> * @param f Function to integrate.
> * @param lower Lower bound of the interval over which to integrate.
> * @param upper Upper bound of the interval over which to integrate.
> * @return the integrated value.
> */
> double integrate(UnivariateFunction f,
> double lower,
> double upper);
> }
> {code}
> In effect, the implementation of the above "integrate" method of a new "LegendreGaussIntegratorSingleStepIntegrator"
would the equivalent of "stage(1)" in the current "LegendreGaussIntegrator".

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