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Re: Re: Solve / NSolve
*To*: mathgroup at smc.vnet.net
*Subject*: [mg95314] Re: [mg95296] Re: Solve / NSolve
*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>
*Date*: Thu, 15 Jan 2009 06:11:14 -0500 (EST)
*References*: <gkgouo$3sg$1@smc.vnet.net> <200901141055.FAA18013@smc.vnet.net> <351FC0A3-BC60-4522-9D1F-5630A5073599@mimuw.edu.pl>
On 14 Jan 2009, at 17:31, Andrzej Kozlowski wrote:
>
> On 14 Jan 2009, at 11:55, Jean-Marc Gulliet wrote:
>
>>> Solve is a
>>> *symbolic* solver, i.e. it manipulates the equations in
>>> essentially an
>>> algebraic way (which does not mean that it does so in a similar
>>> fashion
>>> as a human being would do). OTOH, NSolve uses *numeric* algorithms.
>>> (Both sets of tools and algorithms have virtually nothing in
>>> common in
>>> terms of strategies; roughly speaking, symbolic manipulations for
>>> the
>>> former, iterative computations for the latter, for instance.)
>
>
> It is a common misconception that Solve is "algenriaic" while
> NSolve uses "iterative computations" (it generally does not,
> FindRoot does that), or that the algorithms used by Solve and NSolve
> have "nothing in common". Both Solve and NSolve are primarily
> intended for solving algebraic equations. They also have quite a lot
> in common. In fact, in when NSolve is given a non-algebraic system
> it simply passes it to Solve and does not attempt to solve it using
> iterative methods. For algebraic systems both Solve and NSolve rely
> on Groebner basis, but while Solve uses exact Groeber basis NSolve
> (at least with WorkingPrecision other than MachiePrecision) relies
> on Mathematica's implementation of numerical GroebnerBasis
> (GroebnerBasis[...,CoefficientDomain->InexactNumbers]), which in
> turns relies on Mathematica's "significance arithmetic".
> To sum up, while Solve and NSolve usually (but not always) use
> different algorithms, they are both essentially algebraic solvers.
> In fact NSolve is more "pure algebraic" solver since it won't even
> touch non-algebraic equations passing them to Solve to try its luck
> on them (which is usually lacking).
>
> Andrzej Kozlowski
>
>
I guess, the statement that NSolve "generally" does not use iterative
methods was a bit unclear. More accurately, for a single univariate
polynomial equations NSolve uses the Jenkins-Traub algorithm (as do
most other CAS), which is iterative. For general polynomial systems,
however, numerical Groebner basis is used, which ultimately leads to
solving a univariate polynomial equation, which of course, uses an
iterative method. So in this sense Jean-Marc was correct. However, the
Jenkins-Traub method, even though iterative, unlike the Newton-Raphson
method does not perform any differentiation, hence it can be
considered "algebraic". Compare this with FindRoot, which uses the
Newton-Raphson and works with any sufficiently smooth functions, not
necessarily algebraic ones.
Andrzej Kozlowski
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