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Re: [Axiom-developer] a problem, maybe with strict typing

From: William Sit
Subject: Re: [Axiom-developer] a problem, maybe with strict typing
Date: Mon, 18 Dec 2006 07:20:00 -0500


If I understand your question correctly, at least in the interpreter, there is no need to go to SUP F, especially for domains like F = EXPR INT. In the example you give,
Now let F be EXPR INT, or, in fact, any domain that has TRANFUN. Already series as simple as g +-> sin g - sin 1, g(0) = 1 fail:

it would be possible to "find g" with enough initial data using, with t standing for the next undetermined coefficient

t cos(1)x - O(x^2) Now you want this to be zero, say. so t = 0. You then iterate:

taylor(sin(1 + t*x^2) - sin(1), x=0)

and of course you get g = 1. In general, you may want to use Puiseaux series. For example, if you want sin(g(x))= sqrt(x), g(0) = 0, you would get

g(x) = sqrt(x)+(1/6)*x^(3/2)+(3/40)*x^(5/2)+t*x^(7/2) after a few iterations, which is the beginning terms of the series for asin(sqrt(x)) of course.

Now if you want to write a package, you do not want to fix the variable x (which a user of your package may use t, for example). However, assuming you are only using input functions (like z and initial data) that are univariate in EXPR INT, you can extract the main variable, and assign it to x (in your package). Let axiom generate a new symbol and assign it to t, which will be different from x (even if the user also uses t for his main variable, it will be a different t). Now you form your partial series for g involving t and x, and substitute g for x to form z g. Then compute the resulting series ending up in R:=UPXS(EXPR F,x,a) or UPS(EXPR F,x,a), where F can be simply INT. (If you require your user to give input functions directly from GR:=UPXS(F,x,a) or UPS(F,x,a), you will need to first "embed" gr to SGR:=UPXS(SUP F,x,a) or UPS(SUP F,x,a), and you can assign t to the main variable of SUP F; then you have to repack everything (just the normal part of the series up to the order you need) back into EXPR F before forming the series in R). In any case, you can now compute your undetermine coefficient one at a time using arithmetic in EXPR F. The series for g may be retracted to GR. The user of your package would not need to know anything about your auxilliary variable t.

Something similar to this can be found in IDEAL (PolynomialIdeals) package where we need an auxilliary variable (like your t) to turn inequations h \ne 0 into an equation of the form h*t-1 = 0 (here t must not be mixed up with user variables). A lot of type conversion is needed between different polynomial rings. So here, you need to write type conversion routines between EXPR INT and UPXS or UPS (one direction is given by series or taylor already).


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