© 2005,2014 John Abbott
GNU Free Documentation License, Version 1.2

CoCoALib Documentation Index

User documentation for OrdvArith

OrdvArith objects are "low level" values, and thus probably of little interest to most users of CoCoALib. They perform arithmetic operations on OrdvElem values, i.e. compressed vectors of non-negative small integers (which represent "order vectors" of power products). The main aim is fast multiplication and comparison of two power products (using a specified PP ordering -- see PPOrdering).

All operations on OrdvElem values must be effected through an explicit OrdvArith member function call; this design is similar to that of rings and RingElems. The main design aim was speed rather than convenience; as a consequence the member fns listed below expect the caller to have allocated the memory used to contain the results of computations (e.g. in the parameter ordv).

Initializers and Converters for OrdvElem

These fns are all member fns of OrdvArith.

Note: the two functions which convert between expv and ordv representations might be quite slow, especially if a general ordering is used. Even with the simplest ordering (i.e. lex) the conversion is not instant because order vectors are held in a packed representation.

Arithmetic operations on OrdvElem

These fns are all member fns of OrdvArith.

Note: since order vectors are linearly related to exponent vectors, the functions myMul and myDiv actually compute the sum and difference of the order vectors. No check is made for over-/under-flow!

Other operations on OrdvElem

These fns are all member fns of OrdvArith.

Background about matrices and PP orderings

This section is for the curious.

To better understand the what an OrdvArith object does, let us begin by setting the scene. We recall that for all practical purposes an arithmetic ordering on power products can be specified by a matrix of integers M as follows: Let t1 = x_1^e_1 * x_2^e_2 * ... * x_n^e_n be a power product, and t2 = x_1^f_1 * x_2^f_2 * ... * x_n^f_n be another. Then we call (e_1, e_2,..., e_n) the exponent vector for t1, and similarly for t2. For brevity we shall write expv(t1), etc.

The matrix M determines the ordering thus: we say that t1 < t2 iff M*expv(t1) comes before M*expv(t2) in lex ordering. We call the product M*expv(t1) the order vector for t1, and for brevity we shall write ordv(t1) to denote it; similarly for t2.

Typically the matrix M is subject to some suitability criteria, e.g. M should be square and invertible. We shall assume henceforth that M has been chosen so that all order vectors contain only non-negative entries. While reading the rest of these notes it may be convenient to think of M as being non-singular, so that there is a 1-1 correspondence between power products and their order vectors.

Now the scene has been set, we can explain what an OrdvArith object does. It can effect the conversion from exponent vector to order vector, and vice versa. It can also operate directly on order vectors. Certain special orderings are recognized, so that special relationships between the exponent vector and order vector can be exploited to enable faster computation.

Maintainer documentation for OrdvArith

See subsection below about thread-safety!

The base class OrdvArith::base just contains several handy values related to the number of indets and the packing mechanism. The ctor does some sanity checking on the supplied parameters, and computes some handy values for packing/unpacking vectors.

Mem fns myMul, myDiv and myCmp are inline for speed. Recall that myMul and myDiv do not check for over-/under-flow (for speed).

The mem fns myCompress and myDecompress have to check whether myPackingDensity is 1 because C++ shift operators work "strangely" if the shift size equals the wordsize.

There are several derived classes which supply efficient "short-cut" impls for some operations when specific knowledge of the ordering permits this.

Data member myNumIndets is required when dealing with exponent vectors (since C vectors do not record their own length). It is the number of valid entries in a C vector representing an exponent vector.

Data member myGradingDim specifies how many initial components of an order vector comprise the grading. It is needed in myWDeg.

Data member myOrdvWords is used only to supply the return value to the friend function OrdvWords. This value is needed so that a caller can allocate the correct amount of space in which to build a new order vector value. By default this is initialized to a huge value, so that it will quickly become evident at run-time if it hasn't been initialized to a sane value.

Data member myOrdvWordsForCmp is used in myMul, myDiv and myCmp to choose between an inline function and a virtual call. Its value may be non-zero and different from myOrdvWords if a redundant representation is being used (e.g. for a StdDegRevLex ordering). By default this is initialized to a huge value, so that it will quickly become evident at run-time if it hasn't been initialized to a sane value.

The member functions myMul, myDiv, and myCmp are non-virtual so that the compiler can implement them inline: at run-tme they check the data member myOrdvWordsForCmp to decide whether to the use the inline function or delegate to a "shadow" virtual function. This rather ugly arrangement was necessary to achieve acceptable run-time performance.

The member function myMulIndetPower is not pure because a reasonable generic implementation exists. Similarly, myOutput(OMOut, ordv) is not pure.

Threadsafety and CoCoA_THREADSAFE_HACK

The code contains some #if blocks to distinguish between single-threaded and multi-threaded run-time environments. In a single-threaded environment the base class contains two "global" buffers used when converting between exponent vectors and compressed order vectors; in a multi-threaded environment these buffers are not used, but each function needing to do such conversions creates appropriate buffers in local variables (so there are lots of #if directives).

Bugs, Shortcomings and other ideas

In some ways, myCmp could simply be operator(); thus calls would look like ord(ordv1, ordv2) where ord is an object of type PPOrdering.

We need a way to handle order vectors which have large integer entries! (also ordering matrices with large integer entries). Recall that some ordvs may involve mpz_t integers! Note that the polynomial type needs to know how big an ordv can be: that's what the OrdvWords member function is for.

Should StdDegRevLex actually store an extra component so that deg(...,x[0]) can be calculated easily? Do we really need this to be quick? It would be needed for computing GCDs, testing divisibility etc, but these operations would normally be done only on "rich PP" objects -- talk to Anna!

The restriction to order compatible gradings may not be wholly necessary. The PPs in a polynomial homogeneous with respect to a k-dimensional grading are completely specified by n-k of the entries in the order vector, though precisely which entries must be retained depends on the grading and the ordering. Thus a later generalization to non order compatible gradings may not be too painful.

ANNA: must add a section about modular order matrix JOHN: yes, you must! Where does 46336 come from???

The default implementation of myIsIndet is not very efficient, but is it really worth writing many different (efficient) implementations?