> 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 <[email protected]> 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
