There's also this - http://code.jsoftware.com/wiki/Scripts/nlls - an implementation of the Levenberg-Marquardt algo that I cribbed from someone who copied it from APL.

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On Thu, Aug 10, 2017 at 6:47 PM, Don Kelly <d...@shaw.ca> wrote: > I also suggest that you look at references to the use of NR for power > system load flow problems which are non-linear and generally expressed in > terms of complex numbers in the polar format. These do converge well. I > haven't written one in J but one written in APL has 11 readable lines-most > of which are setting up the matrices to solve (with no complex number > capability- J should be more compact). > > Don > > > > On 2017-08-10 1:10 PM, Ian Clark wrote: > >> Is it also possible to solve a system of equation like the following >>> >> one… ? >> >> Basically, yes. >> >> Because not only can x = (x1,x2) be a vector, but so can y = (y1,y2) in >> this adaptation of your equations: >> >> y1 = a*(1-x1) >> y2 = b*(x2-x1^2) >> >> TABULA is an app (distributed as a JAL "addon") which employs >> Newton-Raphson (occasionally) to "solve" systems of non-linear equations >> supplied by the user. "Solve" includes replacing y2, say, with (y2+∆y2) >> and >> relying on TABULA to adjust x2, x1 and y1 accordingly. >> >> …And to do so without forcing a debate with the clueless user as to what >> "accordingly" means here. >> >> As you'll be aware, N-R algorithms don't always converge, especially when >> you don't have control over what the user throws at you. Pure >> mathematicians do; engineers don't. Hyperbolic functions behave badly, >> e.g. >> y-->(y+∆y) in: y = k/x unless (∆y) is "small enough"… again there is no >> point asking the poor user what "small enough" is supposed to mean in this >> instance. >> >> For a problem that has no general solution, TABULA performs remarkably >> well >> with the systems of (sometimes non-linear) equations that physicists and >> engineers typically need solving. Occasionally TABULA throws up its hands >> in despair – as even the perfect app would have to, when faced with >> y-->(y+∆y) >> in: y = sin(x) whenever ∆y takes (y+∆y) outside the domain: [-1,1]. >> >> I suggest you read these articles: >> >> http://code.jsoftware.com/wiki/TABULA/LaunchElephant >> http://code.jsoftware.com/wiki/TABULA/ChurchClock-NEW >> >> as examples of the sort of empirical problem TABULA is designed to >> address. >> Then examine the code of the addon: CAL, the engine used by TABULA to do >> the real work. Particularly the verb: (inversion) and its ancillary (and >> alternative) verbs. >> >> … in a similar elegant manner? >>> >> Since TABULA is an empirical tool for non-mathematicians, such as K-12 >> pupils and college students training to be physicists, engineers and >> ecologists, it sacrifices a great deal of elegance in the interests of >> generality and practicality, not to mention giving the user a >> comprehensible answer when things go wrong. Thus in practice the "_" in " >> N^:_" needs replacing by a finite value simply to timeout the infinite >> iterations which can and do occur. >> >> Accordingly you'll see that the different variants of the verb (inversion) >> are nowhere near as pretty as: N=: 1 : '- u % u d. 1' >> >> You don't say whether you are a mathematician (in particular a functional >> analyst) or an engineer. Whichever it is, you'll either be baffled by, or >> scornful of, the code in CAL.ijs. But it should give you a bit of a start >> in whatever you want to do. >> >> >> On Thu, Aug 10, 2017 at 7:07 PM, Martin <d...@famic.de> wrote: >> >> Hi there, >>> >>> J looks very interesting. I have no previous experience with array >>> languages and, being curious, started to experiment. Now, I would >>> like to solve a system of non-linear equations. I could only examples >>> solving single equations like this one: >>> >>> N=: 1 : '- u % u d. 1' NB. Adverb implementing Newton-Raphson >>> iteration. >>> (_2 + *:) N^:_ ]1 NB. Find root of “0 = _2+x^2”, starting guess >>> of “1”. >>> >>> Is it also possible to solve a system of equation like the following >>> one in a similar elegant manner? >>> >>> f1(x) = a*(1-x1) >>> f2(x) = b*(x2-x1^2) >>> >>> Example from https://www.gnu.org/software/gsl/doc/html/multiroots.html# >>> examples >>> >>> Thanks for any ideas! >>> -Martin >>> >>> ---------------------------------------------------------------------- >>> For information about J forums see http://www.jsoftware.com/forums.htm >>> >> ---------------------------------------------------------------------- >> For information about J forums see http://www.jsoftware.com/forums.htm >> > > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm > -- Devon McCormick, CFA Quantitative Consultant ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm