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### 5. The Algorithms Implemented in RCWA

This chapter lists brief descriptions of most algorithms and methods implemented in this package. These descriptions are kept very informal and short, and some of them provide only rudimentary information. They are listed in alphabetical order. The word "trivial" as a description means that essentially nothing is done except of storing or recalling one or several values, and "straightforward" means that no sophisticated algorithm is used.

`ActionOnRespectedPartition(G)`

  "Straightforward" after having computed a respected partition by `RespectedPartition`. One only needs to know how to compute images of residue classes under affine mappings.

`Ball(G,g,d)`

  "Straightforward".

`Ball(G,p,d,act)`

  "Straightforward".

`ClassReflection(r,m)`

  "Trivial".

`ClassShift(r,m)`

  "Trivial".

`ClassTransposition(r1,m1,r2,m2)`

  See Remark 2.9.2 in [K05].

`Coefficients(f)`

  "Trivial".

`CommonRightInverse(l,r)`

  (See `RightInverse`.)

`DecreasingOn(f)`

  Form the union of the residue classes which are determined by the coefficients as indicated.

`Determinant(g)`

  Evaluation of the given expression. For the mathematical meaning (epimorphism!), see Theorem 2.11.9 in [K05].

`DirectProduct(G1,G2, ... )`

  Restrict the groups G1, G2, ... to disjoint residue classes. See `Restriction` and Corollary 2.3.3 in [K05].

`Display(f)`

  "Trivial".

`Divisor(f)`

  Lcm of coefficients, as indicated.

`FactorizationIntoCSCRCT(g)`

  This uses a rather sophisticated method which will likely some time be published elsewhere. At the moment termination is not guaranteed, but in case of termination the result is certain. The strategy is roughly first to make the mapping class-wise order-preserving and balanced, and then to remove all prime factors from multiplier and divisor one after the other in decreasing order by dividing by appropriate class transpositions. The remaining integral mapping can be factored almost similarly easily as a permutation of a finite set can be factored into transpositions.

`FactorizationOnConnectedComponents(f,m)`

  Call GRAPE to get the connected components of the transition graph, and then compute a partition of the suitably "blown up" coefficient list corresponding to the connected components.

`FixedPointsOfAffinePartialMappings(f)`

  "Straightforward".

`GuessedDivergence(f)`

  Numerical computation of the limit of some series, which seems to converge "often". Caution!!!

`Image(f)`, `Image(f,S)`

  "Straightforward" if one can compute images of residue classes under affine mappings and unite and intersect residue classes (Chinese Remainder Theorem). See Lemma 1.2.1 in [K05].

`ImageDensity(f)`

  Evaluation of the given expression.

`g in G` (membership test)

  Test whether the mapping g or its inverse is in the list of generators of G. If it is, return `true`. Test whether its prime set is a subset of the prime set of G. If not, return `false`. Test whether the multiplier or the divisor of g has a prime factor which does not divide the multiplier of G. If yes, return `false`. Test if G is class-wise order-preserving, and g is not. If so, return `false`. Test if the sign of g is -1 and all generators of G have sign 1. If yes, return `false`. Test if G is class-wise order-preserving, all generators of G have determinant 0 and g has determinant <> 0. If yes, return `false`. Test whether the support of g is a subset of the support of G. If not, return `false`. Test whether G fixes the nonnegative integers setwisely, but g does not. If yes, return `false`.

  If G is tame, proceed as follows: Test whether the modulus of g divides the modulus of G. If not, return `false`. Test whether G is finite and g has infinite order. If so, return `false`. Test whether g is tame. If not, return `false`. Compute a respected partition `P` of G and the finite permutation group `H` induced by G on it (see `RespectedPartition`). Check whether g permutes `P`. If not, return `false`. Let `h` be the permutation induced by g on `P`. Check whether `h` lies in `H`. If not, return `false`. Compute an element `g1` of G which acts on `P` like g. For this purpose, factor h into generators of `H` using `PreImagesRepresentative`, and compute the corresponding product of generators of G. Let `k := g/g1`. The mapping `k` is always integral. Compute the kernel `K` of the action of G on `P` using `KernelOfActionOnRespectedPartition`. Check whether `k` lies in `K`. This is done using the package Polycyclic [EN03], and uses an isomorphism from a supergroup of `K` which is isomorphic to the `|P|`-fold direct product of the infinite dihedral group and which always contains `k` to a polycyclically presented group. If `k` lies in `K`, return `true`, otherwise return `false`.

  If G is not tame, proceed as follows: Look for finite orbits of G. If some are found, test whether g acts on them, and whether the induced permutations lie in the permutation groups induced by G. If for one of the examined orbits one of the latter two questions has a negative answer, then return `false`. Look for a positive integer m such that g does not leave a partition of Z into unions of residue classes (mod m) invariant which is fixed by G. If successful, return `false`. If not, try to factor g into generators of G using `PreImagesRepresentative`. If successful, return `true`. If g is in G, this terminates after a finite number of steps. Both runtime and memory requirements are exponential in the word length. If g is not in G, the method runs into an infinite loop.

`IncreasingOn(f)`

  Form the union of the residue classes which are determined by the coefficients as indicated.

`Induction(g,f)`

  Compute `f * g * RightInverse(f)`.

`Induction(G,f)`

  Get a set of generators by applying `Induction(g,f)` to the generators g of G.

`InjectiveAsMappingFrom(f)`

  The function starts with the entire source of f as "preimage" `pre` and the empty set as "image" `im`. It loops over the residue classes (mod `Mod(f)`). For any such residue class `cl` the following is done: Firstly, the image of `cl` under f is added to `im`. Secondly, the intersection of the preimage of the intersection of the image of `cl` under f and `im` under f and `cl` is subtracted from `pre`.

`IntegralConjugate(f)`, `IntegralConjugate(G)`

  Uses the algorithm described in the proof of Theorem 2.5.14 in [K05].

`IntegralizingConjugator(f)`, `IntegralizingConjugator(G)`

  Uses the algorithm described in the proof of Theorem 2.5.14 in [K05].

`Inverse(f)`

  Essentially inversion of affine mappings. See Lemma 1.3.1, Part (b) in [K05].

`IsClassWiseOrderPreserving(f)`

  Test whether the first entry of all coefficient triples is positive.

`IsConjugate(RCWA(Integers),f,g)`

  Test whether f and g have the same order, and whether either both or none of them is tame. If not, return `false`.

  If the mappings are wild, use `ShortCycles` to search for finite cycles not belonging to an infinite series, until their numbers for a particular length differ. This may run into an infinite loop. If it terminates, return `false`.

  If the mappings are tame, use the method described in the proof of Theorem 2.5.14 in [K05] to construct integral conjugates of f and g. Then essentially use the algorithm described in the proof of Theorem 2.6.7 in [K05] to compute "standard representatives" of the conjugacy classes which the integral conjugates of f and g belong to. Finally compare these standard representatives, and return `true` if they are equal and `false` if not.

`IsInjective(f)`

  See `Image`.

`IsIntegral(f)`

  "Trivial".

`IsomorphismMatrixGroup(G)`

  Use the algorithm described in the proof of Theorem 2.6.3 in [K05].

`IsomorphismPermGroup(G)`

  If the group G is infinite, there is no isomorphism to a finite permutation group, thus return `fail`. Otherwise use `ActionOnRespectedPartition`.

`IsomorphismRcwaGroup(G)`

  The method for finite groups uses `RcwaMapping`, Part (d).

  The method for free products of finite groups uses the Table-Tennis Lemma (which is also known as Ping-Pong Lemma, cp. e.g. Section II.B. in [H00]). It uses regular permutation representations of the factors G_r (r = 0, dots ,m-1) of the free product on residue classes modulo n_r := |G_r|. The basic idea is that since point stabilizers in regular permutation groups are trivial, all non-identity elements map any of the permuted residue classes into their complements. To get into a situation where the Table-Tennis Lemma is applicable, the method computes conjugates of the images of the mentioned permutation representations under bijective rcwa mappings sigma_r which satisfy 0(n_r)^sigma_r = Z \ r(m).

  The method for free groups uses an adaptation of the construction given on page 27 in [H00] from PSL(2,C) to RCWA(Z). As an equivalent for the closed discs used there, the method takes the residue classes modulo two times the rank of the free group.

`IsSurjective(f)`

  See `Image`.

`IsTame(G)`

  Checks whether the modulus of the group is non-zero.

`IsTame(f)`

  Application of the criteria given in Corollary 2.5.10 and 2.5.12 and Theorem A.8 and A.11 in [K05]. The criterion "surjective, but not injective means wild" (Theorem A.8 in [K05]) is the subject of [K06]. For applying the criterion of the existence of weakly-connected components of the transition graph which are not strongly-connected (Theorem A.11 in [K05]), the package GRAPE is needed.

  In addition, some probabilistic methods are used. If the result depends on one of these, a warning is displayed.

`IsTransitive(G,Integers)`

  Look for finite orbits, using `ShortOrbits` on a couple of intervals. If a finite orbit is found, return `false`. Test if G is finite. If yes, return `false`.

  Search for an element `g` and a residue class r(m) such that the restriction of `g` to r(m) is given by n -> n + m. Then the cyclic group generated by `g` acts transitively on r(m). The element `g` is searched among the generators of G, its powers, its commutators, powers of its commutators and products of few different generators. The search for such an element may run into an infinite loop, as there is no guarantee that the group has a suitable element.

  If suitable `g` and r(m) are found, proceed as follows:

  Set S := r(m). Set S := S cup S^g for all generators g of G, and repeat this until S remains constant. This may run into an infinite loop.

  If it terminates: If S = Z, return `true`, otherwise return `false`.

`KernelOfActionOnRespectedPartition(G)`

  First determine the abelian invariants of the kernel `K`. For this, compute sufficiently many quotients of orders of permutation groups induced by G on refinements of the stored respected partition `P` by the order of the permutation group induced by G on `P` itself. Then use a random walk through the group G. Compute powers of elements encountered along the way which fix `P`. Translate these kernel elements into elements of a polycyclically presented group isomorphic to the `|P|`-fold direct product of the infinite dihedral group (`K` certainly embeds in this group). Use Polycyclic [EN03] to collect independent "nice" generators of `K`. Proceed until the permutation groups induced by `K` on the refined respected partitions all equal the initially stored quotients.

`LargestSourcesOfAffineMappings(f)`

  Form unions of residue classes modulo the modulus of the mapping, whose corresponding coefficient triples are equal.

`LaTeXObj(f)`

  Collect residue classes those corresponding coefficient triples are equal.

`LikelyContractionCentre(f,maxn,bound)`

  Compute trajectories with starting values from a given interval, until a cycle is reached. Abort if the trajectory exceeds the prescribed bound. Form the union of the detected cycles.

`LocalizedRcwaMapping(f,p)`

  "Trivial".

`mKnot(m)`

  "Straightforward", following the definition given in [K99].

`Modulus(G)`

  Searches for a wild element in the group. If unsuccessful, tries to construct a respected partition (see `RespectedPartition`).

`Modulus(f)`

  "Trivial".

`MovedPoints(G)`

  Needs only forming unions of residue classes and determining fixed points of affine mappings.

`Multiplier(f)`

  Lcm of coefficients, as indicated.

`Multpk(f,p,k)`

  Form the union of the residue classes modulo the modulus of the mapping, which are determined by the given divisibility criteria for the coefficients of the corresponding affine mapping.

`NrConjugacyClassesOfRCWAZOfOrder(ord)`

  The class numbers are taken from Corollary 2.7.1 in [K05].

`OrbitsModulo(f,m)`

  Use GRAPE to compute the connected components of the transition graph.

`OrbitsModulo(G,m)`

  "Straightforward".

`Order(f)`

  Test for `IsTame`. If the mapping is not tame, then return `infinity`. Otherwise use Corollary 2.5.10 in [K05].

`PreImage(f,S)`

  See `Image`.

`PreImagesRepresentative(phi,g)`, `PreImagesRepresentatives(phi,g)`

  As indicated in the documentation of these methods. The underlying idea to successively compute two balls around 1 and g until they intersect non-trivially is standard in computational group theory. For rcwa groups it would mean wasting both memory and runtime to actually compute group elements. Thus only images of tuples of points are computed and stored.

`PrimeSet(f)`, `PrimeSet(G)`

  "Straightforward".

`PrimeSwitch(p)`

  Multiplication of rcwa mappings as indicated.

`Print(f)`

  "Trivial".

`f*g`

  Essentially composition of affine mappings. See Lemma 1.3.1, Part (a) in [K05].

`Random(RCWA(Integers))`

  Computes a product of "randomly" chosen class shifts, class reflections and class transpositions. This seems to be suitable for generating reasonably good examples.

`RankOfKernelOfActionOnRespectedPartition(G)`

  This performs basically the first part of the computations done by `KernelOfActionOnRespectedPartition`.

`RCWA(R)`

  Attributes are set according to Theorem 2.1.1, Theorem 2.1.2, Corollary 2.1.6 and Theorem 2.12.8 in [K05].

`RcwaGroupByPermGroup(G)`

  Uses `RcwaMapping`, Part (d).

`RcwaMapping`

  (a)-(c): "trivial", (d): `n^perm - n` for determining the coefficients, (e): "affine mappings by values at two given points", (f) and (g): "trivial", (h) and (i): correspond to Lemma 2.1.4 in [K05].

`RepresentativeAction(G,src,dest,act)`, `RepresentativeActionPreImage`

  As indicated in the documentation of these methods. The underlying idea to successively compute two balls around src and dest until they intersect non-trivially is standard in computational group theory. Words standing for products of generators of G are stored for any image of src or dest.

`RepresentativeAction(G,P1,P2)`

  Arbitrary mapping: see Lemma 2.1.4 in [K05]. Tame mapping: see proof of Theorem 2.8.9 in [K05]. The former is almost trivial, while the latter is a bit complicate and takes usually also much more time.

`RepresentativeAction(RCWA(Integers),f,g)`

  The algorithm used by `IsConjugate` constructs actually also an element `x` such that `f^x = g`.

`RespectedPartition(f)`, `RespectedPartition(G)`

  Uses the algorithm described in the proof of Theorem 2.5.8 in [K05].

`Restriction(g,f)`

  Compute the action of `RightInverse(f) * g * f` on the image of f.

`Restriction(G,f)`

  Get a set of generators by applying `Restriction(g,f)` to the generators g of G.

`RightInverse(f)`

  "Straightforward" if one knows how to compute images of residue classes under affine mappings, and how to compute inverses of affine mappings.

`Root(f,k)`

  If f is bijective, class-wise order-preserving and has finite order:

  Find a conjugate of f which is a product of class transpositions. Slice cycles prod_i=2^l tau_r_1(m_1),r_i(m_i) of f a respected partition mathcalP into cycles prod_i=1^l prod_j=0^k-1 tau_r_1(km_1),r_i+jm_i(km_i) of the k-fold length on the refined partition which one gets from mathcalP by decomposing any r_i(m_i) in mathcalP into residue classes (mod km_i). Finally conjugate the resulting permutation back.

  Other cases seem to be more difficult and are currently not covered.

`SemilocalizedRcwaMapping(f,pi)`

  "Trivial".

`ShortCycles(f,maxlng)`

  Look for fixed points of affine partial mappings of powers of f.

`ShortOrbits(G,S,maxlng)`

  "Straightforward".

`SetOnWhichMappingIsClassWiseOrderPreserving(f)`, etc.

  Form the union of the residue classes modulo the modulus of the mapping, in whose corresponding coefficient triple the first entry is positive, zero resp. negative.

`Sign(g)`

  Evaluation of the given expression. For the mathematical meaning (epimorphism!), see Theorem 2.12.8 in [K05].

`Size(G)`

  Test whether one of the generators of the group G has infinite order. If so, return `infinity`. Test whether the group G is tame. If not, return `infinity`. Test whether `RankOfKernelOfActionOnRespectedPartition(G)` is nonzero. If so, return `infinity`. Otherwise if G is class-wise order-preserving, return the size of the permutation group induced on the stored respected partition. If G is not class-wise order-preserving, return the size of the permutation group induced on the refinement of the stored respected partition which is obtained by splitting each residue class into three residue classes with equal moduli.

`StructureDescription(G)`

  (Not described here.)

`f+g`

  Pointwise addition of affine mappings.

`Trajectory(f,n,...)`

  Iterated application of an rcwa mapping. In the methods computing "accumulated coefficients" additionally composition of affine mappings.

`TransitionGraph(f,m)`

  "Straightforward" -- just check a sufficiently long interval.

`TransitionMatrix(f,m)`

  Evaluation of the given expression.

`ViewObj(f)`

  "Trivial".

`WreathProduct(G,P)`

  Uses `DirectProduct` to embed the `DegreeAction(P)`th direct power of G, and `RcwaMapping`, Part (d) to embed the finite permutation group P.

`WreathProduct(G,Z)`

  Restricts G to the residue class 3(4), and encodes the generator of Z as tau_0(2),1(2) * tau_0(2),1(4). It is used that the images of 3(4) under powers of this mapping are pairwise disjoint residue classes.

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