Difference between revisions of "Category:ApCoCoA-1:Package gbmr"

From ApCoCoAWiki
(New page: The basic idea behind this package is to make Bertini usable in/with ApCoCoA. This is the alpha version of the package bertini, which includes the Betini interface to CoCoA. Essentially, ...)
 
m (Andraschko moved page Category:Package gbmr to Category:ApCoCoA-1:Package gbmr: Clearer page title)
 
(71 intermediate revisions by 6 users not shown)
Line 1: Line 1:
The basic idea behind this package is to make Bertini usable in/with ApCoCoA.
+
The package gbmr contains numbers of functions for basic computations and Groebner basis computations in <em>non-commutative algebras</em>, such as finitely generated free monoid rings (or non-commutative polynomial rings, non-commutative free associative algebras), finitely presented monoid rings, group ring, etc., over the field of rational numbers Q or over finite fields Z/(p) where p is a prime. More precisedly, this package enables us to do computations as addition, subtraction and multiplication of two non-commutative polynomials, getting the leading word and leading coefficient of a non-zero polynomial, computing the normal remainder of a polynomial w.r.t. a list of polynomials, interreducing a lists of polynomials, enumerating (reduced) (partial) Groebner bases of finitely generated two-sided ideals, and computing truncated Groebner basis of a finitely and homogeneously generated two-sided ideals, etc. Consequently, this package can be applied to many algebraic applications, for instance, enumerating a Macaulay's basis and the values of the Hilbert function of a finitely generated K-algbera, computing leading word ideals, intersections of ideals, and kernels of K-algebra homomorphisms, and so on.
  
This is the alpha version of the package bertini, which includes the Betini interface to CoCoA. Essentially, you can call Bertini from with inside CoCoA, using this Package.
 
  
{{ApCoCoAServer}}
+
Generally speaking, a finitely presented monoid ring is defined by <tt>P=K<X|R></tt>, where <tt>K</tt> is a field, <tt>X</tt> is a finite alphabet (or a finite set of indeterminates), and <tt>R</tt> is a finite set of relations. Clearly, we have <tt>P=K<X|R></tt> is isomorphic to <tt>K<X>/<R></tt>, where <tt>K<X></tt> is the free monoid ring generated by <tt>X</tt> over <tt>K</tt> and <tt><R></tt> is the two-sided ideal generated by <tt>R</tt>.
  
'''NUMERICAL ALGEBRAIC GEOMETRY:'''
 
  
Numerical algebraic geometry is the study based on homotopy continuation method and algebraic geometry. It has same relation to algebraic geomertry, as Numerical Linear Algebra to linear algebra. In Numerical Algebraic Geometry we can fine isolated solutions. For positive dimensional systems, we can find out numerical irreducible deocmpostions.
+
<strong>Important issues about this package:</strong>
  
'''Bertini:''' Software for Numerical Algebraic Geometry
+
(a) Predefined alias for this package is as follows.
  
Bertini is a software desgined for computations in Numerical Algebric Geometry, particularly, for solving polynomial systems numerically using homotopy continuation method available at [http://www.nd.edu/~sommese/bertini/]. Its a general-purpose solver, written in C, that was created for research about polynomial continuation. The Key Features of Bertini are:
+
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Alias NCo := $apcocoa/gbmr;
  
* Finds isolated solutions using total-degree start systems, multihomogeneous-degree start systems, and also user defined homotopies.
+
Note that, before ApCoCoA 1.9.0, the alias for this package is NC. However, since ApCoCoA 1.9.0, the alias NC has been used for the ApCoCoA package ncpoly.  
* Implements parameter continuation for families of systems, such as the inverse kinematics of six-revolute serial-link arms, or the forward kinematics of Stewart-Gough parallel-link robots.
 
* Adaptive multiprecision implemented for finding isolated solutions and for the numerical irreducible decomposition.  
 
* Treats positive-dimensional solutions by computing witness sets.
 
* Has automatic differentiation which preserves the straightline quality of an input system.
 
* Uses homogenization to accurately compute solutions at infinity.
 
* Provides a fractional power-series endgame to accurately compute singular roots.
 
* Allows for subfunctions.
 
* Allows for witness set manipulation via both sampling and membership testing.
 
* Accepts square or nonsquare systems.
 
  
  
 +
(b) By default, the field <tt>K</tt> is the field of rational numbers. It can be set to a finite field through the functions
  
[[Category:ApCoCoA_Manual]]
+
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; NCo.SetFp(); and NCo.SetFp(P);
 +
 
 +
where <tt>P</tt> is a prime number. The former sets the finite field to the binary field <tt>{0,1}</tt>, and the latter to the finite field <tt>{0,1,2,...P-1}</tt>. One can reset <tt>K</tt> to rational numbers via the function
 +
 
 +
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; NCo.UnsetFp();
 +
 
 +
 
 +
(c) The alphabet <tt>X</tt> is represented as a STRING of letters. Every letter in <tt>X</tt> must have a unique appearance. The order of letters in <tt>X</tt> is important since it will induce word orderings on the free monoid <tt><X></tt> (see NCo.SetOrdering). The alphabet <tt>X</tt> is set via the function
 +
 
 +
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; NCo.SetX(X);
 +
 
 +
where <tt>X</tt> is a STRING of letters.
 +
 
 +
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (c.1) Each word (term) in the free monoid <tt><X></tt> is represented as a STRING with all letters coming from <tt>X</tt>. For example, the word
 +
 
 +
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <tt>w=ba</tt>
 +
 
 +
is represented as
 +
 
 +
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; W:="ba";
 +
 
 +
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Note that the identity element in <tt><X></tt> is the empty word which is represented as the empty STRING "".
 +
 
 +
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (c.2) Each non-commutative polynomial is represented as a LIST of monomials, and each monomial is represented as a LIST consisting of an element (coefficient) in <tt>K</tt> and a word (term) in <tt><X></tt>. For example the polynomial
 +
 
 +
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <tt>f=ab+2b^2+3</tt>
 +
 
 +
is represented as
 +
 
 +
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;  F := [[1,"ab"],[2,"bb"],[3,""]];
 +
 
 +
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Note that the zero polynomial <tt>0</tt> is represented as the empty LIST [].
 +
 
 +
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (c.3) In the case that <tt>K={0,1}</tt>, every polynomial can be represented as a LIST of words (terms) in <tt><X></tt>. For example, the polynomial
 +
 
 +
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <tt>p=ab+b^2+1</tt>
 +
 
 +
is represented as
 +
 
 +
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; P := ["ab","bb",""];
 +
 
 +
Notice that this representation is ONLY applied to computations in free monoid rings over the binary field <tt>{0,1}</tt>. See functions with the prefix "B" for more details.
 +
 
 +
 
 +
(d) A <em>word ordering</em> on a monoid is a well-ordering that is compatible with multiplication. One can set word orderings via the function
 +
 
 +
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; NCo.SetOrdering(Ordering);
 +
 
 +
where <tt>Ordering</tt> is a STRING indicating which ordering we are going to work with. Currently, the package only supports the length-lexicographic ordering ("LLEX"), an elimination ordering ("ELIM") and the length-reverse-lexicographic ordering ("LRLEX"). We refer to NCo.SetOrdering for the definitions of these orderings. The default ordering is "LLEX". Note that word orderings are induced by the order of letters in <tt>X</tt>. For example, X:="abc"; Ordering:="LLEX"; means the length-lexicographic word ordering induced by <tt>a>b>c</tt>.
 +
 
 +
(e) For a finitely presented monoid ring <tt>P=K<X|R></tt>, the set <tt>R</tt> of relations is represented as a LIST. and each relation in <tt>R</tt> is represented as a LIST composed of two words in <tt><X></tt>. For example, the relations
 +
 
 +
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <tt>R={ba=ab, ca=ac, cb=bc}</tt>
 +
 
 +
is represented as
 +
 
 +
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; R:= [["ba","ab"], ["ca","ac"], ["cb","bc"]];
 +
 
 +
The relations can be set via the function
 +
 
 +
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; NCo.SetRelations(R);
 +
 
 +
where <tt>R</tt> is a LIST of properly represented relations. One can set the relations to empty via the function
 +
 
 +
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; NCo.UnsetRelations();
 +
 
 +
 
 +
(f) Th following function gives basic information on the working ring.
 +
 
 +
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; NCo.RingEnv();
 +
 
 +
 
 +
(g) For most computations, there are three different functions having the same functionality but under different settings. Let us take addition as an example. There are three functions, namely MRAdd(X,Ordering,Relations,F1,F2), Add(F1,F2) and BAdd(F1,F2), doing addition over monoid rings, free monoid rings and free monoid rings over the binary field, respectively. For details about how to use each of them, please check relevant functions.
 +
 
 +
[[Category:ApCoCoA-1 Manual]]

Latest revision as of 15:17, 2 October 2020

The package gbmr contains numbers of functions for basic computations and Groebner basis computations in non-commutative algebras, such as finitely generated free monoid rings (or non-commutative polynomial rings, non-commutative free associative algebras), finitely presented monoid rings, group ring, etc., over the field of rational numbers Q or over finite fields Z/(p) where p is a prime. More precisedly, this package enables us to do computations as addition, subtraction and multiplication of two non-commutative polynomials, getting the leading word and leading coefficient of a non-zero polynomial, computing the normal remainder of a polynomial w.r.t. a list of polynomials, interreducing a lists of polynomials, enumerating (reduced) (partial) Groebner bases of finitely generated two-sided ideals, and computing truncated Groebner basis of a finitely and homogeneously generated two-sided ideals, etc. Consequently, this package can be applied to many algebraic applications, for instance, enumerating a Macaulay's basis and the values of the Hilbert function of a finitely generated K-algbera, computing leading word ideals, intersections of ideals, and kernels of K-algebra homomorphisms, and so on.


Generally speaking, a finitely presented monoid ring is defined by P=K<X|R>, where K is a field, X is a finite alphabet (or a finite set of indeterminates), and R is a finite set of relations. Clearly, we have P=K<X|R> is isomorphic to K<X>/<R>, where K<X> is the free monoid ring generated by X over K and <R> is the two-sided ideal generated by R.


Important issues about this package:

(a) Predefined alias for this package is as follows.

             Alias NCo := $apcocoa/gbmr;

Note that, before ApCoCoA 1.9.0, the alias for this package is NC. However, since ApCoCoA 1.9.0, the alias NC has been used for the ApCoCoA package ncpoly.


(b) By default, the field K is the field of rational numbers. It can be set to a finite field through the functions

             NCo.SetFp(); and NCo.SetFp(P);

where P is a prime number. The former sets the finite field to the binary field {0,1}, and the latter to the finite field {0,1,2,...P-1}. One can reset K to rational numbers via the function

             NCo.UnsetFp();


(c) The alphabet X is represented as a STRING of letters. Every letter in X must have a unique appearance. The order of letters in X is important since it will induce word orderings on the free monoid <X> (see NCo.SetOrdering). The alphabet X is set via the function

             NCo.SetX(X);

where X is a STRING of letters.

       (c.1) Each word (term) in the free monoid <X> is represented as a STRING with all letters coming from X. For example, the word

             w=ba

is represented as

             W:="ba";

      Note that the identity element in <X> is the empty word which is represented as the empty STRING "".

       (c.2) Each non-commutative polynomial is represented as a LIST of monomials, and each monomial is represented as a LIST consisting of an element (coefficient) in K and a word (term) in <X>. For example the polynomial

             f=ab+2b^2+3

is represented as

             F := [[1,"ab"],[2,"bb"],[3,""]];

      Note that the zero polynomial 0 is represented as the empty LIST [].

       (c.3) In the case that K={0,1}, every polynomial can be represented as a LIST of words (terms) in <X>. For example, the polynomial

             p=ab+b^2+1

is represented as

             P := ["ab","bb",""];

Notice that this representation is ONLY applied to computations in free monoid rings over the binary field {0,1}. See functions with the prefix "B" for more details.


(d) A word ordering on a monoid is a well-ordering that is compatible with multiplication. One can set word orderings via the function

             NCo.SetOrdering(Ordering);

where Ordering is a STRING indicating which ordering we are going to work with. Currently, the package only supports the length-lexicographic ordering ("LLEX"), an elimination ordering ("ELIM") and the length-reverse-lexicographic ordering ("LRLEX"). We refer to NCo.SetOrdering for the definitions of these orderings. The default ordering is "LLEX". Note that word orderings are induced by the order of letters in X. For example, X:="abc"; Ordering:="LLEX"; means the length-lexicographic word ordering induced by a>b>c.

(e) For a finitely presented monoid ring P=K<X|R>, the set R of relations is represented as a LIST. and each relation in R is represented as a LIST composed of two words in <X>. For example, the relations

             R={ba=ab, ca=ac, cb=bc}

is represented as

             R:= [["ba","ab"], ["ca","ac"], ["cb","bc"]];

The relations can be set via the function

             NCo.SetRelations(R);

where R is a LIST of properly represented relations. One can set the relations to empty via the function

             NCo.UnsetRelations();


(f) Th following function gives basic information on the working ring.

             NCo.RingEnv();


(g) For most computations, there are three different functions having the same functionality but under different settings. Let us take addition as an example. There are three functions, namely MRAdd(X,Ordering,Relations,F1,F2), Add(F1,F2) and BAdd(F1,F2), doing addition over monoid rings, free monoid rings and free monoid rings over the binary field, respectively. For details about how to use each of them, please check relevant functions.