SVDComplexes

Macaulay2 package SVDComplexes -- support for computing homology, ranks and SVD complexes, for a chain complex over the real numbers. This package implements the algorithms in the paper ”Singular value decomposition of complexes”, by D. Brake, J. Hauenstein, F. Schreyer, A. Sommese, and M. Stillman, https://arxiv.org/abs/1804.09838. Singular value decompositions of matrices are extremely useful in practice. In particular, an SVD can often reveal the rank (numeric rank) of a matrix. In the above paper, we extend the notion of singular value decomposition from matrices over the reals or complexes to a complex of matrices over the reals or complexes. For some applications, one obtains a complex over the (approximate) reals, and one would like to know what the ranks of the matrices are, and therefore the ranks of the homology groups. One way to do this would be to compute the SVD of each matrix separately, often revealing the desired ranks. This is less than satisfactory however, as it ignores the fact that this sequence is an approximation of a complex, i.e. each two consecutive matrices multiply to zero. In this package, and the referenced paper, we give 2 algorithms for computing the SVD of a complex, and the resulting putative ranks of the matrices or ranks of the homology groups

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