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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