I have been perusing information on Groebner basis in Axiom and some
links are broken in:
https://lists.gnu.org/archive/html/axiom-math/2006-08/msg00001.html Bill Page Subject: RE: [Axiom-math] Groebernbases Date: Mon, 7 Aug 2006 11:37:42 -0400 In particular: http://wiki.axiom-developer.org/axiom--test--1/src/algebra/PgrobnerSpad http://wiki.axiom-developer.org/axiom--test--1/src/algebra/FrontPage/searchwiki?expr=Groebner Could somebody provide updates for the links? I know RTFM; but after 11 volumes of >1000 pages each that advice seems a little ??? Ray _______________________________________________ Axiom-math mailing list [hidden email] https://lists.nongnu.org/mailman/listinfo/axiom-math |
On 12/01/2014 05:24 PM, Raymond Rogers wrote:
> I have been perusing information on Groebner basis in Axiom and some > links are broken in: > > https://lists.gnu.org/archive/html/axiom-math/2006-08/msg00001.html > > Bill Page > Subject: RE: [Axiom-math] Groebernbases > Date: Mon, 7 Aug 2006 11:37:42 -0400 > In particular: > > http://wiki.axiom-developer.org/axiom--test--1/src/algebra/PgrobnerSpad > http://wiki.axiom-developer.org/axiom--test--1/src/algebra/FrontPage/searchwiki?expr=Groebner > Could somebody provide updates for the links? I cannot fix these links in the mailing list, but as far as I am informed, the current place of the axiom-wiki is http://axiom-wiki.newsynthesis.org (maintained by Waldek Hebisch) I don't know whether Tim maintains a copy of the old Wiki. Anyway, your links probably translate into http://axiom-wiki.newsynthesis.org/axiom--test--1/src/algebra/PgrobnerSpad http://axiom-wiki.newsynthesis.org/axiom--test--1/src/algebra/FrontPage/searchwiki?expr=Groebner You might also find this one useful (maintained by me). http://fricas.github.io/api/PolyGroebner.html Ralf _______________________________________________ Axiom-math mailing list [hidden email] https://lists.nongnu.org/mailman/listinfo/axiom-math |
On 12/01/2014 01:32 PM, Ralf Hemmecke wrote: > On 12/01/2014 05:24 PM, Raymond Rogers wrote: >> I have been perusing information on Groebner basis in Axiom and some >> links are broken in: >> >> https://lists.gnu.org/archive/html/axiom-math/2006-08/msg00001.html >> >> Bill Page >> Subject: RE: [Axiom-math] Groebernbases >> Date: Mon, 7 Aug 2006 11:37:42 -0400 >> In particular: >> >> http://wiki.axiom-developer.org/axiom--test--1/src/algebra/PgrobnerSpad >> http://wiki.axiom-developer.org/axiom--test--1/src/algebra/FrontPage/searchwiki?expr=Groebner >> Could somebody provide updates for the links? > I cannot fix these links in the mailing list, but as far as I am > informed, the current place of the axiom-wiki is > > http://axiom-wiki.newsynthesis.org > (maintained by Waldek Hebisch) > > I don't know whether Tim maintains a copy of the old Wiki. > > Anyway, your links probably translate into > http://axiom-wiki.newsynthesis.org/axiom--test--1/src/algebra/PgrobnerSpad > http://axiom-wiki.newsynthesis.org/axiom--test--1/src/algebra/FrontPage/searchwiki?expr=Groebner > > You might also find this one useful (maintained by me). > http://fricas.github.io/api/PolyGroebner.html > > Ralf > _______________________________________________ Axiom-math mailing list [hidden email] https://lists.nongnu.org/mailman/listinfo/axiom-math |
In reply to this post by Raymond Rogers-3
There are several examples in the src/input directory you might find useful.
In particular, see src/input/groebner.input.pamphlet src/input/groeb.input.pamphlet src/input/noonburg.input.pamphlet src/input/dop.input.pamphlet As an aside, you surprised me with the links you posted. > http://axiom-wiki.newsynthesis.org/axiom--test--1/src/algebra/PgrobnerSpad implies that you found Axiom code from when it was maintained under the GNU Arch source code control system (circa 2002). Tim _______________________________________________ Axiom-math mailing list [hidden email] https://lists.nongnu.org/mailman/listinfo/axiom-math |
In reply to this post by Ralf Hemmecke-4
On 12/01/2014 01:32 PM, Ralf Hemmecke wrote: > On 12/01/2014 05:24 PM, Raymond Rogers wrote: >> I have been perusing information on Groebner basis in Axiom and some >> links are broken in: >> >> https://lists.gnu.org/archive/html/axiom-math/2006-08/msg00001.html >> >> Bill Page >> Subject: RE: [Axiom-math] Groebernbases >> Date: Mon, 7 Aug 2006 11:37:42 -0400 >> In particular: >> >> http://wiki.axiom-developer.org/axiom--test--1/src/algebra/PgrobnerSpad >> http://wiki.axiom-developer.org/axiom--test--1/src/algebra/FrontPage/searchwiki?expr=Groebner >> Could somebody provide updates for the links? > I cannot fix these links in the mailing list, but as far as I am > informed, the current place of the axiom-wiki is > > http://axiom-wiki.newsynthesis.org > (maintained by Waldek Hebisch) > > I don't know whether Tim maintains a copy of the old Wiki. > > Anyway, your links probably translate into > http://axiom-wiki.newsynthesis.org/axiom--test--1/src/algebra/PgrobnerSpad > http://axiom-wiki.newsynthesis.org/axiom--test--1/src/algebra/FrontPage/searchwiki?expr=Groebner > > You might also find this one useful (maintained by me). > http://fricas.github.io/api/PolyGroebner.html > > Ralf > I wanted to convert quadratics p(x)->Q(x') by finding n,m: x'=m*x+n; so I put. p(x)-q(x')=0 x'-m*x-n=0 and got an answer in terms f(n,m)x^2+g(n,m)*x+h(n,m)=0 so then I had to set f(n,m)=0 g(n,m)=0 h(n,m)=0 as a separate step. Which worked fine to give me triangular output. I am questioning the second step; not that it's wrong, but IMO it shouldn't be necessary. Is there a technique to avoid that? I haven't come up with anything. Ray _______________________________________________ Axiom-math mailing list [hidden email] https://lists.nongnu.org/mailman/listinfo/axiom-math |
> Since you are maintaining Groebner information,
Oh, do I? I'm just maintaining the translation of the ++ documentation (available in the .spad files) to a web representation. > I wanted to convert quadratics p(x)->Q(x') by finding n,m: x'=m*x+n; so > I put. > p(x)-q(x')=0 > x'-m*x-n=0 > and got an answer in terms f(n,m)x^2+g(n,m)*x+h(n,m)=0 > so then I had to set > f(n,m)=0 > g(n,m)=0 > h(n,m)=0 > as a separate step. Which worked fine to give me triangular output. > I am questioning the second step; not that it's wrong, but IMO it > shouldn't be necessary. > Is there a technique to avoid that? Hmmm... in the first step you get a condition for m and n that still involves x. You get rid of the x by requiring that each coefficient is zero. Then you get 3 equations for m and n in the coefficients of the original quadratic polynomials that must simultaneously equate to zero. No you solve these equations in whatever way and express m and n in terms of the original coefficients. If you don't do the trick with removing the x and getting 3 equations, it would still be there. So why do you think there is another method for solving that problem? Ralf _______________________________________________ Axiom-math mailing list [hidden email] https://lists.nongnu.org/mailman/listinfo/axiom-math |
On 12/02/2014 03:08 AM, Ralf Hemmecke
wrote:
Since you are maintaining Groebner information,Oh, do I? I'm just maintaining the translation of the ++ documentation (available in the .spad files) to a web representation.I wanted to convert quadratics p(x)->Q(x') by finding n,m: x'=m*x+n; so I put. p(x)-q(x')=0 x'-m*x-n=0 and got an answer in terms f(n,m)x^2+g(n,m)*x+h(n,m)=0 so then I had to set f(n,m)=0 g(n,m)=0 h(n,m)=0 as a separate step. Which worked fine to give me triangular output. I am questioning the second step; not that it's wrong, but IMO it shouldn't be necessary. Is there a technique to avoid that?Hmmm... in the first step you get a condition for m and n that still involves x. You get rid of the x by requiring that each coefficient is zero. Then you get 3 equations for m and n in the coefficients of the original quadratic polynomials that must simultaneously equate to zero. No you solve these equations in whatever way and express m and n in terms of the original coefficients. If you don't do the trick with removing the x and getting 3 equations, it would still be there. So why do you think there is another method for solving that problem? Ralf Well you do seem to have a good grasp of the situation so I think you are more than a "translator" (forgive me as I didn't "google" you). In any case: I found the answer and posted it to mathhelpforum. I will post it below but the core idea is to treat the polynomials in a matrix form that separates the coefficients into different slots; then forming the groebner input out of the resulting array of coefficient polynomials. realize that crossposting is frowned upon but seeing you maintain some information on groebner programs was to much to pass up. This is part of an effort to systematize orthogonal polynomials; right now maping a lot (all?) of finite interval types to the Gauss Hypergeometric differential equation. Then the terms for all of that type can be written down without further ado; and then most relationships/properties are simple applications of combinatorial analysis and such properties. Ray Here is the solution I came up with. It is constructed to be built upon for other results without manual interference when applied. Most of the following is probably boring to most people but since I asked for help I feel obligated to be explicit in the answer. Use the matrix form of the polynomials. C, C' the coefficient arrays: i.e. C=[c, b, a], C'=[c', b', a'] in the Gauss Hypergeometric case of the leading coefficient C'=[0 1 -1] X,X' the basis arrays; i.e. X=[1,x,x^2]', X'=[1, x', x'^2] M a multiplication matrix: M = [1 0 0][0 m 0][0 0 m^2] The polynomials we are interested in are the same but shifted and scaled so that: C . X=C' . X' The transform is basically from basis X to basis X' keeping the valuations constant. Now a basis/ coefficient transform on coefficient arrays is equal to P(n) Pascal's matrix P(n)=[1 0 0][n 1 0][n^2 2*n 1] (For an explanation see Aceto references below) Thus for x'-n=m*x we want: P(-n) . X' =M*X or X'=P(n) . M . X So we have C . X = C' . P(n) . M . X We expand X to a Vandermonde form to enable cancellation. C = C' . P(n) . M or C- C' . P(n) . M =0 name the equation D Then D is the array of the polynomials we want to feed to the groebnerFactorize (Axiom) routine to get a triangular array of equations for n, m. The reason for groebnerFactorize is that while groebner itself generates a lot of possible solutions; some of them are ones we don't want. This can be resolved by more restrictions but groebnerFactorize is more convenient because it allows one to specify certain phrases as forbidden in the second argument. This can be used to have the program winnow out bad solution sets. This capability is essential in other cases where page after page of alternatives appear and you don't want to continuously print them out to find the one (or more) expressions you don't want in the solution. For instance the trivial (zero) set of solutions. One final note is that I had to make a coefficient alteration when I solved the equations earlier; that is C=[d*c, d*b, d*a]. I needed the extra degree of freedom (d) in order to accommodate a sign change (if needed). References: Aceto: The Matrices of Pascal and Other Greats - L Aceto, D Trigiante - American Mathematical Monthly, 2001 JSTOR: An Error Occurred Setting Your User Cookie Which unfortunately is behind a firewall but the following is just as good for this not. A unified matrix approach to the representation of Appell polynomials Lidia Acetoa, Helmuth R. Malonekb, and Gra¸ca Tomazc http://arxiv.org/pdf/1406.1444v1.pdf _______________________________________________ Axiom-math mailing list [hidden email] https://lists.nongnu.org/mailman/listinfo/axiom-math |
> Here is the solution I came up with.
Is that the solution to this problem? http://axiom-wiki.newsynthesis.org/ExampleGroebner Actually, I wanted to computed a Groebner basis for that case, but that simply give [1], saying, that for generic a, b, c, u, v, w, such m and n cannot be found. So I conclude that your problem is most probably something else. Can you please edit the above site and state your problem as clearly as possible? Thanks Ralf _______________________________________________ Axiom-math mailing list [hidden email] https://lists.nongnu.org/mailman/listinfo/axiom-math |
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