Goto Chapter: Top 1 2 Bib Ind
 [Top of Book]  [Contents]   [Previous Chapter]   [Next Chapter] 

2 Main function
 2.1 Functions concerning the presentation
 2.2 General tools
 2.3 Computing Hall-polynomials
 2.4 Methods for storing things
 2.5 Runtime test

2 Main function

In this chapter we describe the main function of the HallPoly package.

2.1 Functions concerning the presentation

2.1-1 GroupByVector
‣ GroupByVector( n, t )( function )

This function returns the group of Hirsch-length n with presentation G(t). t∈ ℤ^{n choose 3} has to describe a consistent presentation. t needs to be an array so that t[i][j][k] = t_i,j,k.

2.1-2 VectorByGroupPcp
‣ VectorByGroupPcp( G )( function )

This function returns parameters t∈ ℤ^{n choose 3} (where n is the Hirsch-length of G) so that G has the presentation G(t). G has to be a T-group given by generators that form a T-basis.

2.1-3 VectorByGroupCS
‣ VectorByGroupCS( G )( function )

This function returns parameters t∈ ℤ^{n choose 3} (where n is the Hirsch-length of G) so that G has the presentation G(t). G has to be a T-group.

2.1-4 HallRecByVector
‣ HallRecByVector( t )( function )

This function returns a record containing the Hall polynomials for the group G(t), where t ∈ ℤ^{n choose 3}. t needs to be an array so that t[i][j][k] = t_i,j,k ∈ ℤ. The parameterised Hall polynomials for the Hirsch lengths up to n have to be known.

2.2 General tools

2.2-1 SolveRec
‣ SolveRec( g, v )( function )

This function returns the unique rational polynomial f(v) that satifies f(0) = 0 and f(v+1) = f(v) + g(v) for every positive integer v. v is an indeterminate and g contains the coefficients of g(v) (i.e. g(v) = ∑_i=0^mg[i+1]v^i).

gap> v := Indeterminate(Rationals, "v");;
gap> g := [1,1];;
gap> SolveRec(g, v);
1/2*v^2+1/2*v

2.2-2 PolyDecomp
‣ PolyDecomp( f )( function )

This function returns the degree and the number of monomials of f, where f is a polynomial in the variables A[1],...,A[n],B[1],...,B[n],x with parameters T[i][j][k].

The degree is defined as the maximal degree of a monomial in f and the degree of a monomial is the sum of the exponents of its indeterminates.

2.3 Computing Hall-polynomials

2.3-1 HallConstruction
‣ HallConstruction( n, gb )( function )

This calls the algorithm that computes new parameterized Hall-polynomials for Hirsch-length n. Parameterized Hall-polynomials up to Hirsch-length n-1 have to be stored in the record HallRec. The result is stored in HallRec. gb is a boolean that specifies, whether Groebner bases should be used for reducing the computed polynomials.

gap> HallConstruction(5,true);
 - Step 5: compute mult-polynomial 
    compute r[i][j] for 1 by 2
    compute r[i][j] for 1 by 3
    compute r[i][j] for 1 by 4
    compute r[i][j] for 1 by 5
    compute r[i][j] for 2 by 3
    compute r[i][j] for 2 by 4
    compute r[i][j] for 2 by 5
    compute r[i][j] for 3 by 4
    compute r[i][j] for 3 by 5
    compute r[i][j] for 4 by 5
    starting symbolic collection 
    got poly with 69 summands
 - Step 5: reduce mult-polynomial via inherit 
    mult poly has 44 summands
 - Step 5: compute new consistency relations 
 get Groebner in HallConsistency 
 ... done 
 - Step 5: reduce mult polynomial via consistency
    mult poly has 44 summands
 - Step 5: compute power polynomial 
    extract terms  
    solve recursion  
 - Step 5: reduce power polynomial 
    power poly has 84 summands
 - Step 5: done 

2.3-2 HallMult
‣ HallMult( t, a, b )( function )

If t is a vector of parameters (either with a consistent presentation or indeterminates) of length n and a and b are vectors of length n (either integer values or indeterminates), then this function returns the multiplication polynomials for that case.

The parameterised Hall-polynomials for Hirsch length n have to be known.

2.3-3 HallPower
‣ HallPower( t, a, x )( function )

If t is a vector of parameters (either with a consistent presentation or indeterminates) of length n and a is a vector of length n (either integer values or indeterminates) and x is an integer or an indeterminate, then this function returns the power polynomials for that case.

The parameterised Hall-polynomials for Hirsch length n have to be known.

2.4 Methods for storing things

2.4-1 SaveHallPolynomials
‣ SaveHallPolynomials( n )( function )

This function stores the multiplication and power polynomials plus the consistency relations for Hirsch-length n (stored in HallRec[n]) in the file hallpoly/lib/hrln.gi. If the file already exists or the Hall-polynomials for Hirsch-length n haven't been computed yet, an error message is returned.

gap> SaveHallPolynomials(6);
true
gap> SaveHallPolynomials(8);
There are no Hall-polynomials to store.
false

2.4-2 SaveParameters
‣ SaveParameters( t, file )( function )

This function stores the parameters t in a file named file in the directory hallpoly/lib. t has to be an array of size n × n × n.

gap> t := GenT(6);;
gap> SaveParameters(t,"test");
true

2.4-3 CheckFileExists
‣ CheckFileExists( n )( function )

This function returns, whether the file for the Hall-polynomials of Hirsch-length n in hallpoly/lib already exists.

gap> CheckFileExists(6);
true

2.5 Runtime test

2.5-1 RuntimeTestHallColl
‣ RuntimeTestHallColl( a, b )( function )

This function compares the runtime of evaluating Hall polynomials and the standard collection algorithm. To do so, the function computes a random subgroups in the group of 6 × 6 unitriangular matrices over ℤ with Hirsch length le 7 and b random products in each subgroup.

Returns the overall runtimes in milliseconds.

gap> RuntimeTestHallColl(2,4);
 Looking for subgroup 1...
 Found a subgroup of Hirsch length 7.
  Hall polynomials: 0ms, Collection: 12400ms
  Hall polynomials: 4ms, Collection: 2992ms
  Hall polynomials: 4ms, Collection: 93384ms
  Hall polynomials: 4ms, Collection: 608ms

 Looking for subgroup 2...
 Found a subgroup of Hirsch length 6.
  Hall polynomials: 0ms, Collection: 36ms
  Hall polynomials: 0ms, Collection: 8ms
  Hall polynomials: 0ms, Collection: 100ms
  Hall polynomials: 0ms, Collection: 36ms

 Overall runtime: Hall polynomials:  0:00:00.012, Collection:  0:01:49.564
[ 12, 109564 ]
 [Top of Book]  [Contents]   [Previous Chapter]   [Next Chapter] 
Goto Chapter: Top 1 2 Bib Ind

generated by GAPDoc2HTML