Re: Re: Solve Feature?

*To*: mathgroup at smc.vnet.net*Subject*: [mg52877] Re: [mg52846] Re: Solve Feature?*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>*Date*: Wed, 15 Dec 2004 04:27:26 -0500 (EST)*References*: <200412100123.UAA18967@smc.vnet.net> <opsir6lsn2iz9bcq@monster.ma.dl.cox.net> <002401c4dfb5$809b6a00$1802a8c0@Pentium4> <opsiu9j4nmiz9bcq@monster.ma.dl.cox.net> <opsiu90xstiz9bcq@monster.ma.dl.cox.net> <41BE2889.8030108@wolfram.com> <200412141100.GAA24711@smc.vnet.net> <28F087EF-4E26-11D9-A0F0-000A95B4967A@mimuw.edu.pl> <opsi0yvxd6iz9bcq@monster.ma.dl.cox.net>*Sender*: owner-wri-mathgroup at wolfram.com

Certianly, if you raionalize and try to use Solve any really complex system will take longer than you can wait. Even a single polynomial of high enough degree. NSolve it is another matter; Daniel can explain this better. It seems to me that rationalizing first and then using NSolve may sometiems be a good idea, but of ocurse you will still be using MachiePrecision or whatever precision you specify in NSolve. Andrzej On 15 Dec 2004, at 10:10, DrBob wrote: > > Is there ever any reason not to use Rationalize before passing > equations to Solve or NSolve? > > Bobby > > On Wed, 15 Dec 2004 08:16:18 +0900, Andrzej Kozlowski > <akoz at mimuw.edu.pl> wrote: > >> *This message was transferred with a trial version of CommuniGate(tm) >> Pro* >> FindRoot will not give you a "closed formula", which certainly has >> its >> uses, at least in algebraic cases. However, the point seems to me a >> fair one, numerical algebra based on machine arithmetic seems >> inherently unreliable and I am also not sure it is worth the trouble. >> Exact methods are much too slow except for relatively "small >> problems". >> But significance arithmetic works well here and seems reasonably fast. >> This, in fact, looks like the best justification I know of for having >> it at all. In fact the following works perfectly: >> >> >> {c1, c5} = {(-50 + x)^2 + (-50 + y)^2 == >> 156.25`16, >> (-50.00000000000002`16 + x)^2 + >> (-37.49999999999999`16 + y)^2 == >> 156.25`16}; >> >> >> NSolve[%, {x, y}] >> >> >> { >> {x -> >> 39.1746824526924698688078348607`14.855117414427017, >> y -> >> 43.749999999999977679491924308049646499`29.6989700043\ >> 3602}, >> {x -> 60.8253175473075501311921649336`15.04619631453911\ >> 1, y -> >> 43.750000000000012320508075692150353501`29.6989700043\ >> 3602}} >> >> >> {c1, c5} /. % >> >> >> {{True, True}, {True, True}} >> >> >> What's more, going down to much lower precision still gives perfect >> answers: >> >> In[14]:= >> {c1, c5} = {(-50 + x)^2 + (-50 + y)^2 == >> 156.25`8, >> (-50.00000000000002`8 + x)^2 + >> (-37.49999999999999`8 + y)^2 == >> 156.25`8}; >> >> In[15]:= >> NSolve[%, {x, y}] >> >> Out[15]= >> {{x -> 39.1746824526945169247`6.698970004336016, >> y -> 43.75`6.999999999999997}, >> {x -> 60.8253175473054829832`6.698970004336016, >> y -> 43.75`6.999999999999997}} >> >> In[13]:= >> {c1, c5} /. % >> >> Out[13]= >> {{True, True}, {True, True}} >> >> >> This is making me almost enthusiastic about significance arithmetic. >> Wouldn't it be a good idea perhaps to dispense with machine arithmetic >> for this type of problems entirely? >> >> Andrzej Kozlowski >> >> >> On 14 Dec 2004, at 20:00, DrBob wrote: >> >>> *This message was transferred with a trial version of CommuniGate(tm) >>> Pro* >>> OK, but it's a very simple polynomial system; if Solve and NSolve >>> give >>> wildly wrong answers without notifying us, it's difficult to justify >>> ever using them at all (with machine-precision coefficients). >>> >>> I think I'm forced to use FindRoot virtually every time, already; I >>> just hadn't thought about it. >>> >>> Bobby >>> >>> On Mon, 13 Dec 2004 17:40:57 -0600, Daniel Lichtblau >>> <danl at wolfram.com> wrote: >>> >>>> DrBob wrote: >>>>> By the way, it gets even worse if I use NSolve: >>>>> >>>>> solution = NSolve[{c1, c5}] >>>>> {c1, c5} /. solution >>>>> Apply[Subtract, {c1, c5}, {1}] /. solution >>>>> >>>>> {{x -> -4.27759132*^8, >>>>> y -> 43.74999927054117}, >>>>> {x -> 4.2775924*^8, >>>>> y -> 43.75000072945884}} >>>>> {{False, False}, {False, False}} >>>>> {{1.82977917785309*^17, >>>>> 1.82977917785309*^17}, >>>>> {1.8297792462945597*^17, >>>>> 1.8297792462945597*^17}} >>>>> >>>>> These are simple polynomial equations, so this is really puzzling. >>>>> >>>>> Bobby >>>>> >>>>> On Sat, 11 Dec 2004 17:15:30 -0600, DrBob <drbob at bigfoot.com> >>>>> wrote: >>>>> >>>>>> HELP!! Here are two very simple quadratic equations (circles): >>>>>> >>>>>> {c1, c5} = {(-50 + x)^2 + (-50 + y)^2 == 156.25, >>>>>> (-50.00000000000002 + x)^2 + >>>>>> (-37.49999999999999 + y)^2 == 156.25}; >>>>>> >>>>>> If we rationalize before solving, we get accurate solutions: >>>>>> >>>>>> solution = Solve@Rationalize@{c1, c5} >>>>>> solution // N >>>>>> {c1, c5} /. solution >>>>>> Subtract @@@ {c1, c5} /. solution >>>>>> >>>>>> {{x -> (25/4)*(8 - Sqrt[3]), y -> 175/4}, >>>>>> {x -> (25/4)*(8 + Sqrt[3]), y -> 175/4}} >>>>>> {{x -> 39.17468245269452, y -> 43.75}, >>>>>> {x -> 60.82531754730548, y -> 43.75}} >>>>>> {{True, True}, {True, True}} >>>>>> {{-4.263256414560601*^-14, 4.973799150320701*^-13}, >>>>>> {-4.263256414560601*^-14, -4.263256414560601*^-13}} >>>>>> >>>>>> You can check this visually with ImplicitPlot: >>>>>> >>>>>> ImplicitPlot[{c1, c5}, {x, -40, 70}] >>>>>> >>>>>> But if we solve without Rationalize, we get wildly inaccurate >>>>>> results: >>>>>> >>>>>> solution = Solve@{c1, c5} >>>>>> {c1, c5} /. solution >>>>>> Subtract @@@ {c1, c5} /. solution >>>>>> >>>>>> {{x -> 16., y -> 43.74999999999993}, >>>>>> {x -> 84., y -> 43.75000000000004}} >>>>>> {{False, False}, {False, False}} >>>>>> {{1038.812500000001, 1038.8125000000005}, >>>>>> {1038.8124999999995, 1038.8124999999993}} >>>>>> >>>>>> I'm using version 5.1. >>>>>> >>>>>> Bobby >>>> >>>> >>>> The behavior of Solve with these approximate inputs is subject to >>>> the >>>> vagaries of how it creates an infinite precision problem. As this >>>> one >>>> is >>>> poorly conditioned it is not too surprising that it does not give a >>>> good >>>> result. I'll point out that with VerifySolutions->True Solve will >>>> use >>>> significance arithmetic throughout, and this is sufficient (in this >>>> case) to get a viable result. >>>> >>>> The NSolve result is more troublesome. I tracked it to some >>>> difficulty >>>> in assessing either of two important facts. >>>> >>>> (1) The precision used in an eigen decomposition is not adequate for >>>> the >>>> problem at hand. >>>> >>>> (2) The result gives a large residual for the inputs when normalized >>>> in >>>> what I believe to be a reasonable way (obviously we can make the >>>> residual as large or small as we please by suitably rescaling the >>>> input). >>>> >>>> It is not too hard to fix the code for either of these but there are >>>> drawbacks. >>>> >>>> The second issue came about from a poor choice on my part of what to >>>> check: I used a numerical Groebner basis instead of the original >>>> inputs. >>>> So this can (and will) be addressed. In those cases where it >>>> catches >>>> trouble (as in this example) the fix is to redo certain computations >>>> at >>>> higher precision. As the code for this is already present the only >>>> issue >>>> is in the detection itself. The down side is that I am not certain >>>> it >>>> will be adequate to assess poor solutions in general, though it >>>> works >>>> reasonably well for this example. >>>> >>>> For the first issue I am right now at a loss. It seems that the >>>> numeric >>>> EigenXXX code gives a very poor result for a particular 2x2 matrix >>>> that >>>> appears in the course of the NSolve code. It is not too hard to >>>> figure >>>> out for a given matrix that the numeric eigen decomposition might be >>>> troublesome. Unfortunately that involves finding numeric singular >>>> values, and I believe that tends to be typically more expensive than >>>> the >>>> EigenXXX computation itself. So the problem becomes one of >>>> efficiency: >>>> detection of the problem cases could make all NSolve calls slower >>>> just >>>> to find the rare bad apples. >>>> >>>> I'll give this some more thought. >>>> >>>> Another possible issue is in how the numeric Groebner basis gets >>>> computed. This is something I also need to look at more closely. It >>>> appears that for one variable order the result is quite tame, but >>>> not >>>> for the other (NSolve[polys, {y,x}] works fine). I'm not yet certain >>>> but >>>> I think the nuemric instability at this step is essentially >>>> unavoidable. >>>> I believe this to be the case because I can find "nearby" polynomial >>>> systems for which a problematic Groebner basis will emerge (and in >>>> fact >>>> that happens in this example). It is not actually incorrect, but >>>> causes >>>> numeric problems at low precision for the eigen decomposition on a >>>> matrix formed from that Groebner basis. >>>> >>>> Whatever the outcome of investigating that numeric Groebner basis >>>> part, >>>> EigenXXX computation needs to sort out how to handle the resulting >>>> matrix. So this part at least is a bug to be fixed. >>>> >>>> >>>> Daniel Lichtblau >>>> Wolfram Research >>>> >>>> >>>> >>>> >>> >>> >>> >>> -- >>> DrBob at bigfoot.com >>> www.eclecticdreams.net >>> >> >> >> >> > > > > -- > DrBob at bigfoot.com > www.eclecticdreams.net >

**References**:**Solve bug?***From:*paul@selfreferral.com

**Re: Solve Feature?***From:*DrBob <drbob@bigfoot.com>

**Re: Re: multiple outputs from a function**

**Re: Remove the higher order terms in series expansion**

**Re: Re: Solve Feature?**

**Re: Solve Feature?**