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https://issues.apache.org/jira/browse/MATH-797?page=com.atlassian.jira.plugin.system.issuetabpanels:comment-tabpanel&focusedCommentId=13288163#comment-13288163
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Sébastien Brisard edited comment on MATH-797 at 6/3/12 2:04 PM:
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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: Gauss-Legendre,
Gauss-Chebyshev (first and second kinds), Gauss-Hermite. 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 built-in 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 on-the-fly), and I
wrote a high accuracy computation of integration points and weights for the
Gauss-Legendre rule, using {{BigDecimal}} (at that time, I knew but did not use
Commons-Math, 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 naming 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 Gauss-Legendre...
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: Gauss-Legendre,
Gauss-Chebyshev (first and second kinds), Gauss-Hermite. 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 built-in 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 on-the-fly), and I
wrote a high accuracy computation of integration points and weights for the
Gauss-Legendre rule, using {{BigDecimal}} (at that time, I knew but did not use
Commons-Math, 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 Gauss-Legendre...
> Single step integrators
> -----------------------
>
> Key: MATH-797
> URL: https://issues.apache.org/jira/browse/MATH-797
> 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 use-cases 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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