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).


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

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:


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
   (_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#

Thanks for any ideas!

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