vector

vector, an Octave code which considers a problem involving vectors, which can be considered to belong to equivalence classes, for which an arbitrary collection of coefficients must be gathered, averaged, and then scattered again.

Consider vectors X of dimension D, whose entries are integers between 1 and B.

For instance, if D = 3 and B = 3, our set of vectors includes the 27 triples:

      111, 112, 113, 121, 122, 123, 131, 132, 133,
      211, 212, 213, 221, 222, 223, 231, 232, 233,
      311, 312, 313, 321, 322, 323, 331, 332, 333

We will say two vectors are equivalent if one is simply a permutation of the other. Thus, for instance, in the above example, the six vectors 123, 132, 213, 231, 312 and 321 are equivalent.

Now we suppose that every vector X has an associated weight or coefficient C. We wish to modify the set of coefficients so that each vector in an equivalence class is assigned a coefficient computed as the average of the coefficients of all the members of that class.

To do this efficiently, we need to be able to generate each vector and its original coefficient, identify the corresponding equivalence class and ``gather'' the coefficient to an ongoing sum, then normalize that sum by the size of the equivalence class, and ``scatter'' the new coefficients back to the vectors.

The software supplied here is constructed in such a way as to enable the smooth carrying out of this process. In particular, the equivalence classes are identified by ``monotone vectors'', that is, the unique entry of each equivalence class whose elements are in ascending order.

Licensing:

The computer code and data files described and made available on this web page are distributed under the MIT license

Languages:

vector is available in a MATLAB version and an Octave version.

Related Data and Programs:

vector_test

combo, an Octave code which contains many combinatorial routines.

monomial, an Octave code which enumerates, lists, ranks, unranks and randomizes multivariate monomials in a space of m dimensions, with total degree less than n, equal to n, or lying within a given range.

subset, an Octave code which generates, ranks and unranks various combinatorial objects.

Source Code:

• monotone_vector_from_vector.m, returns the monotone vector corresponding to a vector.
• monotone_vector_next.m, returns the next monotone vector.
• monotone_vector_random.m, returns a random monotone vector.
• vector_lex_next.m, generates vectors in lexicographic order.
• vector_lex_rank.m, ranks a vector using lexicographic order.
• vector_lex_unrank.m, unranks a vector using lexicographic order.
• vector_multiplicity.m, computes the multiplicity of a vector.
• vector_next_equivalent.m, generates the next equivalent vector.
• vector_random.m, generates a random vector.