Re: Re: Solve / NSolve

*To*: mathgroup at smc.vnet.net*Subject*: [mg95378] Re: [mg95348] Re: Solve / NSolve*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>*Date*: Sat, 17 Jan 2009 05:27:59 -0500 (EST)*References*: <gkgouo$3sg$1@smc.vnet.net> <200901161109.GAA14111@smc.vnet.net>

Actually, the problem in your original example has virtually nothing to do with the difference between NSolve and Solve. In fact, NSolve is not used in your example at all. The reason is that when you call NSolve with equations involving parameters all it does is to refer your equations to Solve and then apply N. Just compare: Solve[{f[1, 2, 1/2, 4/5, t], df[1, 2, 1/2, 4/5, t]}, {x, y}] // N with NSolve[{f[1, 2, 1/2, 4/5, t], df[1, 2, 1/2, 4/5, t]}, {x, y}] where f is the function in your original post. You will see that you get essentially the same answer arranged in a somewhat different way. So the difference between your final answers derives from the order in which N and and Power Expand Simplify are applied. In the first case you perform algebraic operations first and apply N last, in the second case you apply N before algebraic operations. The second approach is definitely not the thing to do, since once you expression is no longer exact algebraic transformations cannot be performed exactly. In your second case there are no parameters and Solve and NSolve use different algorithms. Solve rationalizes the equations and uses exact GroebnerBasis, NSolve uses numerical Groebner basis. The first approach is faster in this case but the second approach shoudl be more reliable in cases involving numerical instability. Andrzej Kozlowski On 16 Jan 2009, at 12:09, SigmundV wrote: > Thanks to Andrzej and Jean-Marc for the discussion on the differences > between Solve and NSolve. Have any of you tried to run the code i > provided (including the missing function definition which I provided > later)? I'm aware that Solve and NSolve do not use the same algorithm > to solve an equation, but it does not seem "logical" to me that they > don't give the same solution to the same system of algebraic > equations. What could be the reason for this? I'm also concerned witht > he speed. Solve is much faster than NSolve for my problem. How come? > In addition to the code I provided earlier, I just tried with another > example of a system of two algebraic equations: > > {1.1 x^2 + y^2 == 1.0, x + 1.1 y == 0.0} > Solve[%, {x, y}] // Timing > NSolve[%%, {x, y}] // Timing > > When evaluating this cell you'll see that Solve is substantially > faster than NSolve, but they yield the same solution. However, in my > first example they do not yield the same solution in all cases. > > Hopefully some of you can shed some light on all this. It could also > be me comparing apples and oranges. Let me hear your thoughts on all > this. > > Kind regards, > Sigmund >

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**Re: Solve / NSolve**

**Re: Re: Solve / NSolve**