bysuborbit.xml

<!-- 

         bysuborbit.xml            orb package documentation             
                                                               Juergen Mueller
                                                               Max Neunhoeffer
                                                                  Felix Noeske

         Copyright (C) 2005-2008 by the authors

This chapter explains functionality for enumeration by suborbits.

-->

<Chapter Label="bysuborbit">
<Heading>Orbit enumeration by suborbits</Heading>

The code described in this chapter is quite complicated and one has
to understand quite a lot of theory to use it. The reason for this is
that a lot of preparatory data has to be found and supplied by the user
in order for this code to run at all. Also the situations in which it
can be used are quite special. However, in such a situation, the user
is rewarded with impressive performance.

<P/> The main reference for the theory is <Cite Key="MNW"/>. We briefly
recall the basic setup: Let <M>G</M> be a group acting from the right
on some set <M>X</M>. Let <M>k</M> be a natural number, set 
<M>X_{{k+1}} := X</M>, and let
<Display> U_1 &lt; U_2 &lt; \ldots &lt; U_k &lt; U_{{k+1}} = G </Display>
be a chain of <Q>helper</Q> subgroups. Further, for <M>1 \leq i \leq k</M>
let <M>X_i</M> be a <M>U_i</M> set and let <M>\pi_i : X_{{i+1}} \to X_i</M>
be a homomorphism of <M>U_i</M>-sets.

<P/> This chapter starts with a section about the main orbit enumeration
function and the corresponding preparation functions. It then proceeds
with a section on the used data structures, which will necessarily be
rather technical. Finally, the chapter concludes with a section
on higher level data structures like lists of orbit-by-suborbit
objects and their administration. Note that there are quite a few
examples in Chapter <Ref Chap="examples"/>.

<Section>
<Heading><C>OrbitBySuborbits</C> and its resulting objects</Heading>

<ManSection>
<Func Name="OrbitBySuborbit" Arg="setup, p, j, l, i, percentage"/>
<Returns> an orbit-by-suborbit object </Returns>
<Description>
    This is the main function in the whole business. All notations
    from the beginning of this Chapter <Ref Chap="bysuborbit"/> remain
    in place. The argument <A>setup</A>
    must be a setup record lying in the filter 
    <Ref Filt="IsOrbitBySuborbitSetup"/> described in detail in 
    Section <Ref Sect="obsodatastrucs"/> and produced for example by
    <Ref Func="OrbitBySuborbitBootstrapForVectors"/> or
    <Ref Func="OrbitBySuborbitBootstrapForLines"/> described below.
    In particular, it contains all the generators for <M>G</M> and the
    helper subgroups acting on the various sets. The argument <A>p</A>
    must be the starting point of the orbit. Note that the function
    possibly does not take <A>p</A> itself as starting point but rather
    its <M>U_k</M>-minimalisation, which is a point in the same 
    <M>U_k</M>-orbit as <A>p</A>. This information is important
    for the resulting stabiliser and words representing the 
    <M>U_k</M>-suborbits.
    <P/>
    The integers <A>j</A>, <A>l</A>, and <A>i</A>, for which
    <M>k+1 \ge <A>j</A> \ge <A>l</A> &gt; <A>i</A> \ge 1</M> must hold,
    determine the running mode. <A>j</A> indicates in which set <M>X_j</M>
    the point <A>p</A> lies and thus in which set the orbit enumeration
    takes place, with <M>j=k+1</M> indicating the original set <M>X</M>. 
    The value <A>l</A> indicates which group to use for orbit enumeration.
    So the result will be a <M>U_l</M> orbit, with <M><A>l</A>=<A>k</A>+1</M>
    indicating a <M>G</M>-orbit. Finally, the value <A>i</A> indicates
    which group to use for the <Q>by suborbit</Q> part, that is, the orbit
    will be enumerated <Q>by <M>U_{<A>i</A>}</M>-orbits</Q>. Note that
    nearly all possible combinations of these parameters actually occur,
    because this function is also used in the <Q>on-the-fly</Q> 
    precomputation happening behind the scenes. The most common usage
    of this function for the user is <M><A>j</A>=<A>l</A>=<A>k</A>+1</M>
    and <M><A>i</A>=k</M>.
    <P/>
    Finally, the integer <A>percentage</A> says, how much of the full
    orbit should be enumerated, the value is in percent, thus <M>100</M>
    means the full orbit. Usually, only values greater than <M>50</M>
    are sensible, because one can only prove the size of the orbit
    after enumerating at least half of it.
    <P/>
    The result is an <Q>orbit-by-suborbit</Q> object. For such an object
    in particular the operations <Ref Meth="Size" Label="fororb"/>, 
    <Ref Oper="Seed"/>,
    <Ref Oper="SuborbitsDb"/>,
    <Ref Oper="WordsToSuborbits"/>, <Ref Oper="Memory" Label="forob"/>,
    <Ref Meth="Stabilizer" Label="obso"/>, and 
    <Ref Oper="Seed"/> are defined, see below.
</Description>
</ManSection>

<ManSection>
<Func Name="OrbitBySuborbitKnownSize" 
      Arg="setup, p, j, l, i, percentage, knownsize"/>
<Returns> an orbit-by-suborbit object </Returns>
<Description>
    Basically does the same as <Ref Func="OrbitBySuborbit"/> but does
    not compute the stabiliser by evaluating Schreier words. Instead,
    the size of the orbit to enumerate must already be known and be
    given in the argument <A>knownsize</A>. The other arguments are as
    for the function <Ref Func="OrbitBySuborbit"/>.
</Description>
</ManSection>

<ManSection>
<Meth Name="Size" Arg="orb" Label="fororb"/>
<Returns> an integer </Returns>
<Description>
Returns the number of points in the orbit-by-suborbit <A>orb</A>.
</Description>
</ManSection>

<ManSection>
<Meth Name="Seed" Arg="orb"/>
<Returns> a point in the orbit </Returns>
<Description>
Returns the starting point of the orbit-by-suborbit <A>orb</A>. It is
the <M>U_i</M>-minimalisation of the starting point given to
<Ref Func="OrbitBySuborbit"/>.
</Description>
</ManSection>

<ManSection>
<Oper Name="SuborbitsDb" Arg="orb"/>
<Returns> a database of suborbits </Returns>
<Description>
Returns the data base of suborbits of the orbit-by-suborbit object
<A>orb</A>. In particular, such a database object has methods
for the operations
<Ref Oper="Memory" Label="forob"/>, <Ref Oper="TotalLength"
    Label="fordb"/>, and
<Ref Oper="Representatives"/>. For descriptions see below.
</Description>
</ManSection>

<ManSection>
<Oper Name="WordsToSuborbits" Arg="orb"/>
<Returns> a list of words </Returns>
<Description>
Returns a list of words in the groups <M>U_*</M> reaching each of the
suborbits in the orbit-by-suborbit <A>orb</A>. Here a word is
a list of integers. Positive numbers index generators in following
numbering: The first few numbers are numbers of generators of <M>U_1</M>
the next few adjacent numbers index the generators of <M>U_2</M> and
so on until the generators of <M>G</M> in the end. Negative numbers
indicate the corresponding inverses of these generators.
<P/>
Note that <Ref Func="OrbitBySuborbit"/> takes the <M>U_i</M>-minimalisation
of the starting point as its starting point and the words here are all
relative to this new starting point.
</Description>
</ManSection>

<ManSection>
<Oper Name="Memory" Arg="ob" Label="forob"/>
<Returns> an integer </Returns>
<Description>
Returns the amount of memory needed by the object <A>ob</A>, which can
be either an orbit-by-suborbit object, a suborbit database object,
or an object in the filter <Ref Filt="IsOrbitBySuborbitSetup"/>.
The amount of memory used is given in bytes. Note that this includes
all hashes, databases, and preparatory data of substantial size.
For orbit-by-suborbits the memory needed for the precomputation is
not included, ask the setup object for that.
</Description>
</ManSection>

<ManSection>
<Meth Name="Stabilizer" Arg="orb" Label="obso"/>
<Returns> a permutation group </Returns>
<Description>
Returns the stabiliser of the starting point of the orbit-by-suborbit
in <A>orb</A> in form of a permutation group, using the given
faithful permutation representation in the setup record.
<P/>
Note that <Ref Func="OrbitBySuborbit"/> takes the <M>U_i</M>-minimalisation
of the starting point as its starting point and the stabiliser returned
here is the one of this new starting point.
</Description>
</ManSection>

<ManSection>
<Oper Name="StabWords" Arg="orb"/>
<Returns> a list of words </Returns>
<Description>
Returns generators for the stabiliser of the starting point of the 
orbit-by-suborbit
in <A>orb</A> in form of words as described with the operation <Ref
Oper="WordsToSuborbits"/>. Note again that <Ref Func="OrbitBySuborbit"/>
takes the <M>U_i</M>-minimalisation of the starting point as its
starting point and the stabiliser returned here is the one of this new
starting point.
</Description>
</ManSection>

<ManSection>
<Oper Name="SavingFactor" Arg="orb" Label="fororb"/>
<Returns> an integer </Returns>
<Description>
Returns the quotient of the total number of points stored in the
orbit-by-suborbit <A>orb</A> and the total number of <M>U</M>-minimal points
stored. Note that the memory for the precomputations is not considered
here!
</Description>
</ManSection>

The following operations apply to orbit-by-suborbit database objects:

<ManSection>
<Oper Name="TotalLength" Arg="db" Label="fordb"/>
<Returns> an integer </Returns>
<Description>
Returns the total number of points stored in all suborbits in the 
orbit-by-suborbit database <A>db</A>.
</Description>
</ManSection>

<ManSection>
<Oper Name="Representatives" Arg="db"/>
<Returns> a list of points </Returns>
<Description>
Returns a list of representatives of the suborbits stored in the
orbit-by-suborbit database <A>db</A>.
</Description>
</ManSection>

<ManSection>
<Oper Name="SavingFactor" Arg="db" Label="fordb"/>
<Returns> an integer </Returns>
<Description>
Returns the quotient of the total number of points stored in the
suborbit database <A>db</A> and the total number of <M>U</M>-minimal points
stored. Note that the memory for the precomputations is not considered
here!
</Description>
</ManSection>

<ManSection>
<Oper Name="OrigSeed" Arg="orb"/>
<Returns> a point </Returns>
<Description>
    Returns the original starting point for the orbit, not yet
    minimalised.
</Description>
</ManSection>

</Section>


<Section>
<Heading>Preparation functions for <Ref Func="OrbitBySuborbit"/></Heading>

<ManSection>
<Func Name="OrbitBySuborbitBootstrapForVectors" 
      Arg="gens, permgens, sizes, codims, opt"/>
  <Returns> a setup record in the filter <Ref Filt="IsOrbitBySuborbitSetup"/>
  </Returns>
<Description>
All notations from the beginning of this Chapter <Ref Chap="bysuborbit"/> 
remain in place.
This function is for the action of matrices on row vectors, so all generators
must be matrices. The set <M>X</M> thus is a row space usually over a 
finite field and the sets <M>X_i</M> are quotient spaces. The matrix
generators for the various groups have to be adjusted with a base change,
such that the canonical projection onto <M>X_i</M> is just to take the
first few entries in a vector, which means, that the submodules divided
out are generated by the last standard basis vectors.
<P/>
The first argument <A>gens</A> must be a list of lists of generators. 
The outer list must have length <M>k+1</M> with entry <M>i</M> being a list 
of matrices generating <M>U_i</M>, given in the action on <M>X=X_{{k+1}}</M>.
The above mentioned base change must have been done. The second
argument <A>permgens</A> must be an analogous list with generator lists
for the <M>U_i</M>. 
These representations are used to compute membership and
group orders of stabilisers. In its simplest form, <A>permgens</A> is a 
list of permutation representations of the same degree, giving a set of 
generators for each individual group <M>U_i</M>. Alternatively, if for some 
<M>U_i</M>, <M>i > 1</M>, it is required that the stabilizer of its action 
is to be calculated as a matrix group, generators of <M>U_i</M> in some 
matrix representation may be supplied. However, it is then mandatory that 
for all <M>1 &lt; i \leq k+1</M> the generator lists have the following format:
The <M>i</M>-th entry of <A>permgens</A> is 
a list concatenating the generator lists of <M>U_1</M> up to <M>U_i</M> (in
this order) all of whose elements are in either some permutation or some matrix 
representation. Note that currently, the generators of <M>U_1</M> need
to be always given in a permutation representation.
The argument <A>sizes</A> must be a list of
length <M>k+1</M> and entry <M>i</M> must be the group order of <M>U_i</M>
(again with <M>U_{{k+1}}</M> being <M>G</M>). Finally, the argument
<A>codims</A> must be a list of length <M>k</M> containing integers
with the <M>i</M>th entry being the codimension of the <M>U_i</M>-invariant
subspace <M>Y_i</M> of <M>X</M> with <M>X_i = X/Y_i</M>. These
codimensions must not decrease for obvious reasons, but some of them
may be equal. The last argument <A>opt</A> is an options record.
See below for possible entries.
<P/>
The function does all necessary steps to fill a setup record
(see <Ref Sect="obsodatastrucs"/>) to be used with 
<Ref Func="OrbitBySuborbit"/>. For details see the code.
<P/>
Currently, the following components in the options record <A>opt</A>
have a meaning:
<List>
<Mark><C>regvecfachints</C></Mark>
<Item>If bound it must be a list. In position <M>i</M> for <M>i&gt;1</M>
there may be a list of vectors in the <M>i</M>-th quotient space <M>X_i</M>
that can be used to distinguish the left <M>U_{{i-1}}</M> cosets in
<M>U_i</M>. All vectors in this list are tried and the first one that
actually works is used.
</Item>
<Mark><C>regvecfullhints</C></Mark>
<Item>If bound it must be a list. In position <M>i</M> for <M>i&gt;1</M>
there may be a list of vectors in the full space <M>X</M>
that can be used to distinguish the left <M>U_{{i-1}}</M> cosets in
<M>U_i</M>. All vectors in this list are tried and the first one that
actually works is used.
</Item>
<Mark><C>stabchainrandom</C></Mark>
<Item>If bound the value is copied into the <C>stabchainrandom</C>
    component of the setup record.</Item>
<Mark><C>nostabchainfullgroup</C></Mark>
<Item>If bound it must be <K>true</K> or <K>false</K>. If it is 
    unbound or set to <K>true</K>, no stabilizer chain is computed for the
    group <M>U_{k+1}</M>. Its default value is <K>false</K>.</Item>
</List>
</Description>
</ManSection>
    
<ManSection>
<Func Name="OrbitBySuborbitBootstrapForLines" 
      Arg="gens, permgens, sizes, codims, opt"/>
  <Returns> a setup record in the filter <Ref Filt="IsOrbitBySuborbitSetup"/>
  </Returns>
<Description>
All notations from the beginning of this Chapter <Ref Chap="bysuborbit"/> 
remain in place. This does exactly the same as 
<Ref Func="OrbitBySuborbitBootstrapForVectors"/> except that it handles
the case of matrices acting on one-dimensional subspaces. Those 
one-dimensional subspaces are represented by normalised vectors,
where a vector is normalised if its first non-vanishing entry is equal
to <M>1</M>.
</Description>
</ManSection>

<ManSection>
<Func Name="OrbitBySuborbitBootstrapForSpaces" 
      Arg="gens, permgens, sizes, codims, spcdim, opt"/>
  <Returns> a setup record in the filter <Ref Filt="IsOrbitBySuborbitSetup"/>
  </Returns>
<Description>
All notations from the beginning of this Chapter <Ref Chap="bysuborbit"/> 
remain in place. This does exactly the same as 
<Ref Func="OrbitBySuborbitBootstrapForVectors"/> except that it handles
the case of matrices acting on <A>spcdim</A>-dimensional subspaces. Those 
subspaces are represented by fully echelonised bases.
</Description>
</ManSection>
</Section>

<Section Label="obsodatastrucs">
<Heading>Data structures for orbit-by-suborbits</Heading>

The description in this section is necessarily technical. It is meant
more as extended annotations to the source code than as user documentation.
Usually it should not be necessary for the user to know the details
presented here.

The function <Ref Func="OrbitBySuborbit"/> needs an information record
of the following form:

<ManSection>
<Filt Name="IsOrbitBySuborbitSetup" Arg="ob" Type="Category"/>
<Returns> <K>true</K> or <K>false</K> </Returns>
<Description>
Objects in this category are also in <C>IsComponentObjRep</C>. We describe
the components, refering to the setup at the beginning of this
Chapter <Ref Chap="bysuborbit"/>.
<List>
<Mark><C>k</C></Mark>
<Item> The number of helper subgroups.  </Item>
<Mark><C>size</C></Mark>
<Item> A list of length <M>k+1</M> containing the orders of the groups
    <M>U_i</M>, including <M>U_{{k+1}} = G</M>.
</Item>
<Mark><C>index</C></Mark>
<Item> A list of length <M>k</M> with the index <M>[U_i:U_{{i-1}}]</M>
    in position <M>i</M> (<M>U_0 = \{1\}</M>).
</Item>
<Mark><C>els</C></Mark>
<Item> A list of length <M>k+1</M> containing generators of the groups
in their action on various sets. In position <M>i</M> we store all the
generators for all groups acting on <M>X_i</M>, that is for the groups
<M>U_1, \ldots, U_i</M> (where position <M>k+1</M> includes the generators
for <M>G</M>. In each position the generators of all those groups
are concatentated starting with <M>U_1</M> and ending with <M>U_i</M>.
</Item>
<Mark><C>elsinv</C></Mark>
<Item> The inverses of all the elements in the <C>els</C> component in the
same arrangement.
</Item>
<Mark><C>trans</C></Mark>
<Item> A list of length <M>k</M> in which position <M>i</M> for <M>i>1</M>
contains 
a list of words in the generators for a transversal of <M>U_{{i-1}}</M>
in <M>U_i</M> (with <M>U_0 = 1</M>).
</Item>
<Mark><C>pifunc</C></Mark>
<Item> Projection functions. This is a list of length <M>k+1</M> containing
in position <M>j</M> a list of length <M>j-1</M> containing in position
<M>i</M> a &GAP; function doing the projection <M>X_j \to X_i</M>.
These &GAP; functions take two arguments, namely the point to map
and secondly the value of the <C>pi</C> component at positions
<C>[j][i]</C>. Usually <C>pifunc</C> is just the slicing operator
in &GAP; and <C>pi</C> contains the components to project onto
as a range object.
</Item>
<Mark><C>pi</C></Mark>
<Item> See the description of the <C>pifunc</C> component.
</Item>
<Mark><C>op</C></Mark>
<Item> A list of <M>k+1</M> &GAP; operation functions, each taking a point
<M>p</M> and a generator <M>g</M> in the action given by the index and 
returning <M>pg</M>.
</Item>
<Mark><C>info</C></Mark>
<Item> 
A list of length <M>k</M> containing a hash table with the minimalisation
lookup data. These hash tables grow during orbit enumerations as
precomputations are done behind the scenes.
<P/>
<C>info[1]</C> contains precomputation data for <M>X_1</M>. Assume
<M>x \in X_1</M> to be <M>U_1</M>-minimal. For all <M>z \in xU_1</M>
with <M>z \neq x</M> we store the number of an element in the
<C>wordcache</C> mapping <M>z</M> to <M>x</M>. For <M>z=x</M> we
store a record with two components <C>gens</C> and <C>size</C>, 
where <C>gens</C> stores generators for the stabiliser
Stab<M>_{{U_1}}(x)</M> as words in the group generators and
<C>size</C> stores the size of that stabiliser.
<P/>
<C>info[i]</C> for <M>i>1</M> contains precomputation data for
<M>X_i</M>. Assume <M>x \in X_i</M> to be <M>U_i</M>-minimal.
For all <M>U_{{i-1}}</M>-minimal <M>z \in xU_i \setminus xU_{{i-1}}</M> 
we store the number
of an element in <C>trans[i]</C> mapping <M>z</M> into <M>xU_{{i-1}}</M>.
For all <M>U_{{i-1}}</M>-minimal <M>z \in xU_{{i-1}}</M> with
<M>z \neq x</M> 
we store the negative of the number of a word in <C>wordcache</C> that
is in the generators of <M>U_{{i-1}}</M> and maps <M>z</M> to <M>x</M>.
For <M>z=x</M> we store the stabiliser information as in the case
<M>i=1</M>.
<P/>
This information together with the information in the following
componente allows the minimalisation function to do its job.
</Item>
<Mark><C>cosetrecog</C></Mark>
<Item> 
A list of length <M>k</M> beginning with the index <M>1</M>. The
entry at position <M>i</M> is bound to a function taking <M>3</M> arguments, 
namely <M>i</M> itself, a word in the group generators of 
<M>U_1, \ldots, U_k</M> which lies in <M>U_i</M>,
and the setup record. The function computes the number <C>j</C> of an element
in <C>trans[i]</C>, such that the element of <M>U_i</M> described by
the word lies in <C>trans[i][j] U_{{i-1}}</C>.
</Item>
<Mark><C>cosetinfo</C></Mark>
<Item> 
A list of things that can be used by the functions in
<C>cosetrecog</C>.
</Item>
<Mark><C>suborbnr</C></Mark>
<Item> 
A list of length <M>k</M> that contains in position <M>i</M> the number
of <M>U_i</M>-orbits in <M>X_i</M> archived in <C>info[i]</C> during
precomputation.
</Item>
<Mark><C>sumstabl</C></Mark>
<Item> 
A list of length <M>k</M> that contains in position <M>i</M> the sum
of the point stabiliser sizes of all <M>U_i</M>-orbits <M>X_i</M>
archived in <C>info[i]</C> during precomputation.
</Item>
<Mark><C>permgens</C></Mark>
<Item> 
A list of length <M>k+1</M> containing in position <M>i</M> generators
for <M>U_1, \ldots, U_i</M> in a faithful permutation representation
of <M>U_i</M>. Generators fit to the generators in <C>els</C>.
For the variant <Ref Func="OrbitBySuborbitKnownSize"/> the <M>k+1</M>
entry can be unbound.
</Item>
<Mark><C>permgensinv</C></Mark>
<Item> 
The inverses of the generators in <C>permgens</C> in the same arrangement.
</Item>
<Mark><C>sample</C></Mark>
<Item> 
A list of length <M>k+1</M> containing sample points in the sets
<M>X_i</M>.
</Item>
<Mark><C>stabchainrandom</C></Mark>
<Item> 
The value is used as the value for the <C>random</C> option
for <C>StabChain</C> calculations to determine stabiliser sizes.
Note that the algorithms are randomized if you use this feature with
a value smaller than <M>1000</M>.
</Item>
<Mark><C>wordhash</C></Mark>
<Item> 
A hash to quickly recognise already used words. For every word in the
hash the position of that word in the <C>wordcache</C> list is stored
as value in the hash.
</Item>
<Mark><C>wordcache</C></Mark>
<Item> 
A list of words in the wordcache for indexing purposes.
</Item>
<Mark><C>hashlen</C></Mark>
<Item> 
Initial length of hash tables used for the enumeration of lists
of <M>U_i</M>-minimal points.
</Item>
<Mark><C>staborblenlimit</C></Mark>
<Item> 
This contains the limit, up to which orbits of stabilisers are computed
using word action. After this limit, the stabiliser elements are actually
evaluated in the group.
</Item>
<Mark><C>stabsizelimitnostore</C></Mark>
<Item>
    If the stabiliser in the quotient is larger than this limit, the
    suborbit is not stored.
</Item>
<Mark><C>cache</C></Mark>
<Item>A linked list cache object (see <Ref Oper="LinkedListCache"/>)
    to store already computed transversal elements. The cache nodes
    are referenced in the <C>transcache</C> component and are stored
    in the cache <C>cache</C>.
</Item>
<Mark><C>transcache</C></Mark>
<Item>This is a list of lists of weak pointer objects. The weak
    pointer object at position <C>[i][j]</C> holds references to
    cache nodes of transversal elements of <M>U_{i-1}</M> in <M>U_i</M> in
    representation <M>j</M>. 
</Item>
</List>
</Description>
</ManSection>

<Subsection>
    <Heading>The global record <C>ORB</C></Heading>

In this section we describe the global record <C>ORB</C>, which
contains some entries that can tune the behaviour of the
orbit-by-suborbit functions. The record has the following components:

<List>
    <Mark><C>MINSHASHLEN</C></Mark>
    <Item>This positive integer is the initial value of the hash size
        when enumerating orbits of stored stabilisers to find all or 
        search through <M>U_{{i-1}}</M>-minimal vectors in an
        <M>U_i</M>-orbit. The default value is <M>1000</M>.</Item>
    <Mark><C>ORBITBYSUBORBITDEPTH</C></Mark>
    <Item>This integer indicates how many recursive calls to
        <C>OrbitBySubOrbitInner</C> have been done. The initial value
        is <M>0</M> to indicate that no such call has happened. This
    variable is necessary since the minimalisation routine sometimes
    uses <C>OrbitBySubOrbitInner</C> recursively to complete some
    precomputation <Q>on the fly</Q> during some other
    orbit-by-suborbit enumeration. This
    component is always set to <M>0</M> automatically when calling
    <Ref Func="OrbitBySuborbit"/> or <Ref
        Func="OrbitBySuborbitKnownSize"/> so the user should usually
    not have to worry about it at all.
    </Item>
    <Mark><C>PATIENCEFORSTAB</C></Mark>
    <Item>This integer indicates how many Schreier generators for the
    stabiliser are tried before assuming that the stabiliser is
    complete. Whenever a new generator for the stabiliser is found
    that increases the size of the currently known stabiliser, the
    count is reset to <M>0</M> that is, only when
    <C>ORB.PATIENCEFORSTAB</C> unsuccessful Schreier generators have
    been tried no more Schreier generators are created. The default
    value for this component is <M>1000</M>. This feature is purely
    heuristical and therefore this value has to be adjusted for some
    orbit enumerations.
    </Item>
    <Mark><C>PLEASEEXITNOW</C></Mark>
    <Item>This value is usually set to <K>false</K>. Setting it to
        <K>true</K> in a break loop tells the orbit-by-suborbit
        routines to exit gracefully at the next possible time. Simply
        leaving such a break loop with <K>quit;</K> is not safe, since
    the routines might be in the process of updating precomputation
    data and the data structures might be left corrupt. Always use
    this component to leave an orbit enumeration prematurely.
    </Item>
    <Mark><C>REPORTSUBORBITS</C></Mark>
    <Item>This positive integer governs how often information messages
        about newly found suborbits are printed. The default value is
        <M>1000</M> saying that after every <M>1000</M> suborbits a message 
        is printed, if the info level is at its default value
        <M>1</M>. If the info level is increased, then this component
        does no longer affect the printing and all found suborbits
        are reported.
    </Item>
    <Mark><C>TRIESINQUOTIENT</C> and <C>TRIESINWHOLESPACE</C></Mark>
    <Item>The bootstrap routines 
        <Ref Func="OrbitBySuborbitBootstrapForVectors"/>,
        <Ref Func="OrbitBySuborbitBootstrapForLines"/> and
        <Ref Func="OrbitBySuborbitBootstrapForSpaces"/> all need to
        compute transversals of one helper subgroup in the next one.
        They use orbit enumerations in various spaces to achieve this.
        The component <C>TRIESINQUOTIENT</C> must be a non-negative
        integer and indicates how often a random vector in the
        corresponding quotient space is tried to find an orbit that
        can distinguish between cosets. The other component
        <C>TRIESINWHOLESPACE</C> also must be a non-negative integer
        and indicates how often a random vector in the whole space
        is tried. The default values are <M>3</M> and <M>20</M>
        resepectively.
        </Item>
</List>

</Subsection>

</Section>

<Section>
<Heading>Lists of orbit-by-suborbit objects</Heading>

There are a few functions that help to administrate lists of
orbit-by-suborbits.

<ManSection>
<Func Name="InitOrbitBySuborbitList" Arg="setup, nrrandels"/>
<Returns> a list of orbit-by-suborbits object </Returns>
<Description>
Creates an object that stores a list of orbit-by-suborbits. The
argument <A>setup</A> must be an orbit-by-suborbit setup record
and <A>nrrandels</A> must be an integer. It indicates how many random
elements in <M>G</M> should be used to do a probabilistic check for
membership in case an orbit-by-suborbit is only partially known.
</Description>
</ManSection>

<ManSection>
<Func Name="IsVectorInOrbitBySuborbitList" Arg="v, obsol"/>
<Returns> <K>fail</K> or an integer </Returns>
<Description>
Checks probabilistically, if the element <A>v</A> lies in one of
the partially enumerated orbit-by-suborbits in the orbit-by-suborbit list
object <A>obsol</A>. If yes, the number of that orbit-by-suborbit is
returned and the answer is guaranteed to be correct. If the
answer is <K>fail</K> there is a small probability that the point
actually lies in one of the orbits but this could not be shown.
</Description>
</ManSection>

<ManSection>
<Func Name="OrbitsFromSeedsToOrbitList" Arg="obsol, li"/>
<Returns> nothing </Returns>
<Description>
Takes the elements in the list <A>li</A> as seeds for orbit-by-suborbits.
For each such seed it is first checked whether it lies in one of the
orbit-by-suborbits in <A>obsol</A>, which must be an orbit-by-suborbit
list object. If not found, 51&percent; of the orbit-by-suborbit of the 
seed is enumerated and added to the list <A>obsol</A>.
<P/>
This function is a good way to quickly enumerate a greater number of 
orbit-by-suborbits.
</Description>
</ManSection>

<ManSection>
<Func Name="VerifyDisjointness" Arg="obsol"/>
<Returns> <K>true</K> or <K>false</K> </Returns>
<Description>
This function checks deterministically, whether the orbit-by-suborbits
in the orbit-by-suborbit list object <A>obsol</A> are disjoint or not
and returns the corresponding boolean value. This is not a 
Monte-Carlo algorithm. If the answer is <K>false</K>, the function
writes out, which orbits are in fact identical.
</Description>
</ManSection>

<ManSection>
<Oper Name="Memory" Arg="obsol" Label="forobsol"/>
<Returns> an integer </Returns>
<Description>
    Returns the total memory used for all orbit-by-suborbits in the 
    orbit-by-suborbit-list <A>obsol</A>.
    Precomputation data is not included, ask the setup object instead.
</Description>
</ManSection>

<ManSection>
<Oper Name="TotalLength" Arg="obsol" Label="forobsol"/>
<Returns> an integer </Returns>
<Description>
Returns the total number of points stored in all orbit-by-suborbits in the 
orbit-by-suborbit-list <A>obsol</A>.
</Description>
</ManSection>

<ManSection>
<Meth Name="Size" Arg="obsol" Label="forobsol"/>
<Returns> an integer </Returns>
<Description>
Returns the total number of points in the orbit-by-suborbit-list <A>obsol</A>.
</Description>
</ManSection>

<ManSection>
<Oper Name="SavingFactor" Arg="obsol" Label="forobsol"/>
<Returns> an integer </Returns>
<Description>
Returns the quotient of the total number of points stored in all 
orbit-by-suborbits in the orbit-by-suborbit-list <A>obsol</A>
and the total number of <M>U</M>-minimal points
stored, which is the average saving factor considering all orbit-by-suborbits
together. Note that the memory for the precomputations is not considered
here!
</Description>
</ManSection>

</Section>

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</Chapter>