You may also wish to look at the documentation for
the type which represents (explicitly) Finitely Generated Modules.
ModuleElem are closely linked
together (analogously to the triple
module is a reference counting smart pointer to an object of
type derived from
ModuleBase; all concrete types for representing modules
are derived from
ModuleBase. For a library implementor the class
ModuleBase defines the minimal interface which every concrete module
class must offer; indeed the concrete class must be derived from
A user of CoCoALib who does not wish to add to the library need know only what it is in this section.
ModuleElem belongs to some
module. So before you can compute with
ModuleElems you must create the
module(s) which contain them.
To create a
module you must a pseudo-constructor for one of the concrete
module classes (refer to their documentation for details): e.g.
NewFreeModule(R, n) -- create a new FreeModule of n components over R
The functions which one may apply directly to a module are:
NumCompts(M) -- the number of components an element of M has RingOf(M) -- the base ring of M (i.e. M is a module over this ring) gens(M) -- a read only C++ vector containing the generators of M zero(M) -- a read only ModuleElem which is the zero of M M1 == M2 -- are the two modules identical (same repr in memory)? M1 != M2 -- opposite of M1 == M2
As you can see there is not a lot one can do to a module. Primarily
they exist to "give the correct type" to module elements; internally
they play a crucial role in applying operations to module elements. A
C++ value of type
ModuleElem represents an element of some concrete
module. The module to which the value belongs is called the
that value. The owner of an object of type
ModuleElem must be specified
(explicitly or implicitly) when it is created, and cannot be changed
during the lifetime of the object; the value it contains may, however,
be changed (C++ const rules permitting).
v be a non-const
v1, v2 be
const ModuleElems all
belonging to the same concrete module
R be the base ring of
r a const element of
R. Then we summarize the possible operations
using C++ syntax:
owner(v1) // gives the module to which v1 belongs -v1 // Usual arithmetic operations v1 + v2 v1 - v2 // between ModuleElems and r * v1 v1 * r // RingElems. v1 / r v = v1 v += v1 v -= v1 v *= r v /= r v1 == v2 v1 != v2 IsZero(v1) cout << v1 v[pos] // throws if the module is not FGModule
In every case it is an error to combine/compare
ModuleElems belonging to
different modules. As you would expect, instead of multiplying or dividing
RingElem you may also multiply or divide by a machine integer, a
BigInt or a
I shall suppose that the user documentation has already been read and
digested. It could also be helpful to have read the documentation for
ring since the design philosophy here imitates that used for rings.
module is simply a reference counting smart pointer class to a
concrete module (i.e. an object belonging to a class derived from
ModuleElem, like a
RingElem, comprises two components: one specifying
the algebraic structure to which the value belongs, and the other being
an opaque representation of the value which can be correctly interpreted
only by the owning module. The data members are:
module myM; // the module to which the ModuleElem belongs ModuleRawValue myValue; // "opaque" representation of the value, // concrete modules must "import" this value.
The design philosophy for modules follows closely that used for rings. This means
that every operation on
ModuleElems is actually effected by calling
the appropriate member function of the owning
module. These member
functions expect raw values as input. A normal
within itself both the identity of the
module to which it belongs and
its value as an element of that particular module -- we call the first
owner and the second datum the
RawValue. A RawValue can
be correctly interpreted only if supplied as argument to a member
function of the owning module -- calling module member functions for
an incompatible concrete module and RawValue will very likely have
grave consequences (officially stated as undefined behaviour, and
most probably perceived as a program crash).
The member functions of a module do not check their arguments for being sensible. This decision is largely just a design policy imitating that used for rings, but may also lead to some slight beneficial effect on run-time performance. It does naturally imply that the programmer bears a considerable burden of responsibility.
For ring elements (especially those in a small finite field), noticeable speed gains arise from using directly raw values and ring member functions. For modules the analogous effect exists in theory but will likely be negligible in practice. Nevertheless we list here the member functions of a module; this list will be useful to library authors who wish to create their own concrete module classes.
v be a non-const RawValue, and
v1, v2 const RawValues belonging to
r be a RingBase::RawValue belonging to the base ring of
M.myNumCompts() M.myRing() M.myGens() -- returns a const ref to a C++ vector of module:elems M.myZero() -- returns a const ref to a ModuleElem M.myNew(v) -- allocates resources, apply only to uninitialized RawValue M.myNew(v, v1) -- allocates resources, apply only to uninitialized RawValue M.myDelete(v) -- releases resources M.mySwap(v, w) M.myAssign(v, v1) M.myNegate(v, v1) M.myAdd(v, v1, v2) M.mySub(v, v1, v2) M.myMul(v, r, v1) M.myDiv(v, r, v1) -- NOTE funny arg order! M.myOutput(out, v1) M.myOutputSelf(out) M.myIsZero(v1) M.myIsEqual(v1, v2)
This code is too new, largely untried/untested. As soon as it gets some use, there will be some material to put here :-)
The documentation is very incomplete. Will be fixed (eventually). Maintainer documentation is incompleter than user doc.