Re: why VB integration challenges are silly.

The words and vision of a computer scientist...

;)

rjf wrote:
VB claims definite integration to be a challenging issue of high
importance and divine beauty.

The vast majority of people interested in the computation of definite
integrals are probably engineers or physical scientists who find that
numeric (quadrature) programs produce an answer faster and perhaps more
useful that the symbolic expression, (should that even exist in terms
of familiar functions). The symbolic expression, unless it is
miraculously small, is usually evaluated to a number for human
consumption. Evaluating complicated expressions may be subject to
computational error comparable to the error in the quadrature
procedure, and may even take more time to evaluate.

That is why many people have devised clever algorithms for quadrature
and published them in scientific software libraries. The exact
symbolic solution of definite improper integrals is a search for some
pretty result, but to claim that it is of high practical importance is
not at all obvious. If it were important, why do so few people keep
working on this "challenging issue" and why is research funding for it
essentially non-existent?

Historically (that is, before computers) cleverness in this area and
other areas of "applied mathematics" was much more valuable, and some
of this cleverness could be considered beautiful. It is even true that
the ability to be clever in this way should be incorporated into CAS,
to the extent possible, at an appropriate level of priority compared to
other efforts. If VB wants to write programs to do integrals properly,
that's his choice. If some CAS vendor spends time/money on something
that paying customers have requested, that's his (or her) choice.

But refusal to compute a symbolic definite integral does not seem to me
to be much of a disqualification, especially since the 'challenges'
offered by VB are often just singularity-laced quirky simplification
problems.

Semi-symbolic solutions such as expansion in taylor series, asymptotic
approximations, and (as indicated above) plain old numerical quadrature
(perhaps enhanced by very high precision), may be far more interesting
to CAS 'customers'.

.

Relevant Pages

  • why VB integration challenges are silly.
    ... numeric (quadrature) programs produce an answer faster and perhaps more ... useful that the symbolic expression, (should that even exist in terms ... That is why many people have devised clever algorithms for quadrature ... symbolic solution of definite improper integrals is a search for some ...
    (sci.math.symbolic)
  • Computing the Integral.
    ... Tool:- Quadrature methods (Trapezoidal Rule) ... equation based on the replacement of integrals by finite ... Fredholm second kind. ...
    (sci.math.num-analysis)
  • Re: A trap: TWO HOWLERS in Mathematica 6 making a dangerous match
    ... dealing with integrals convergent only in the Hadamard ... divergent integral not realizing that it diverges... ... crushing quadrature schemes have the effect, ...
    (sci.math.symbolic)
  • Re: Integrable singularity
    ... one can separately compute the integrals on ... Some quadpack codes have been translated to F90-style codes by Alan ... Many Fortran codes for numerical quadrature by Alan Genz, ...
    (sci.math.num-analysis)
  • Re: Integrable singularity
    ... one can separately compute the integrals on ... Some quadpack codes have been translated to F90-style codes by Alan ... Many Fortran codes for numerical quadrature by Alan Genz, ...
    (comp.lang.fortran)