B. David Saunders

On Wiedemann's Method of Solving Sparse Linear Systems

Douglas Wiedemann’s (1986) landmark approach to solving sparse linear systems over finite fields provides the symbolic counterpart to non-combinatorial numerical methods for solving sparse linear systems, such as the Lanczos or conjugate gradient method (see Golub and van Loan (1983)). The problem is to solve a sparse linear system, when the individual entries lie in a generic field, and the only operations possible are field arithmetic; the solution is to be exact. Such is the situation, for instance, if one works in a finite field. Wiedemann bases his approach on Krylov subspaces, but projects further to a sequence of individual field elements. By making a link to the Berlekamp/Massey problem from coding theory — the coordinate recurrences — and by using randomization an algorithm is obtained with the following property. On input of an n×n coefficient matrix A given by a so-called black box, which is a program that can multiply the matrix by a vector (see Figure 1), and of a vector b, the algorithm finds, with high probability in case the system is solvable, a random solution vector x with Ax = b. It is assumed that the field has sufficiently many elements, say no less than 50n log(n), otherwise one goes to a finite algebraic extension. The complexity of the method is in the general singular case O(n log(n)) calls to the black box for A and an additional O(n log(n)) field arithmetic operations. Note that the black box model for matrix sparsity is a significant abstraction. For a matrix that has an abundance of zero entries, multiplying the matrix by a vector may cost no more than O(n) field operations, in which case the algorithm becomes almost quadratic. However, the model also applies to structured matrices with few or no zero entries, such as Toeplitzand Vandermonde-like matrices, or matrices that correspond to resultants (Canny

Efficient matrix preconditioners for black box linear algebra

Abstract The main idea of the “black box” approach in exact linear algebra is to reduce matrix problems to the computation of minimum polynomials. In most cases preconditioning is necessary to obtain the desired result. Here good preconditioners will be used to ensure geometrical/algebraic properties on matrices, rather than numerical ones, so we do not address a condition number. We offer a review of problems for which (algebraic) preconditioning is used, provide a bestiary of preconditioning problems, and discuss several preconditioner types to solve these problems. We present new conditioners, including conditioners to preserve low displacement rank for Toeplitz-like matrices. We also provide new analyses of preconditioner performance and results on the relations among preconditioning problems and with linear algebra problems. Thus, improvements are offered for the efficiency and applicability of preconditioners. The focus is on linear algebra problems over finite fields, but most results are valid for entries from arbitrary fields.

On Efficient Sparse Integer Matrix Smith Normal Form Computations

We present a new algorithm to compute the Integer Smith normal form of large sparse matrices. We reduce the computation of the Smith form to independent, and therefore parallel, computations modulo powers of word-size primes. Consequently, the algorithm does not suffer from coefficient growth. We have implemented several variants of this algorithm (elimination and/or black box techniques) since practical performance depends strongly on the memory available. Our method has proven useful in algebraic topology for the computation of the homology of some large simplicial complexes.

Computing Simplicial Homology Based on Efficient Smith Normal Form Algorithms

We recall that the calculation of homology with integer coefficients of a simplicial complex reduces to the calculation of the Smith Normal Form of the boundary matrices which in general are sparse. We provide a review of several algorithms for the calculation of Smith Normal Form of sparse matrices and compare their running times for actual boundary matrices. Then we describe alternative approaches to the calculation of simplicial homology. The last section then describes motivating examples and actual experiments with the GAP package that was implemented by the authors. These examples also include as an example of other homology theories some calculations of Lie algebra homology.

Sparse Polynomial Interpolation in Nonstandard Bases

In this paper, we consider the problem of interpolating univariate polynomials over a field of characteristic zero that are sparse in (a) the Pochhammer basis or, (b) the Chebyshev basis. The polynomials are assumed to be given by black boxes, i.e., one can obtain the value of a polynomial at any point by querying its black box. We describe efficient new algorithms for these problems. Our algorithms may be regarded as generalizations of Ben-Or and Tiwaris (1988) algorithm (based on the BCH decoding algorithm) for interpolating polynomials that are sparse in the standard basis. The arithmetic complexity of the algorithms is

FLOWS ON GRAPHS APPLIED TO DIAGONAL SIMILARITY AND DIAGONAL EQUIVALENCE FOR MATRICES

O(t^2 + t\log d)

Transitive closure and related semiring properties via eliminants

which is also the complexity of the univariate version of the Ben-Or and Tiwari algorithm. That algorithm and those presented here also share the requirement of

Efficient matrix rank computation with application to the study of strongly regular graphs

2t

Integer Smith form via the valence: experience with large sparse matrices from homology

evaluation points.

Quadratic-time certificates in linear algebra

Abstract Three equivalence relations are considered on the set of n × n matrices with elements in F 0 , an abelian group with absorbing zero adjoined. They are the relations of diagonal similarity, diagonal equivalence, and restricted diagonal equivalence. These relations are usually considered for matrices with elements in a field. But only multiplication is involved. Thus our formulation in terms of an abelian group with o is natural. Moreover, if F is chosen to be an additive group, diagonal similarity is characterized in terms of flows on the pattern graph of the matrices and diagonal equivalence in terms of flows on the bipartie graph of the matrices. For restricted diagonal equivalence a pseudo-diagonal of the graph must also be considered. When no pseudo-diagonal is present, the divisibility properties of the group F play a role. We show that the three relations are characterized by cyclic, polygonal, and pseudo-diagonal products for multiplicative F . Thus, our method of reducing propositions concerning the three equivalence relations to propositions concerning flows on graphs, provides a unified approach to problems previously considered independently, and yields some n, w or improved results. Our consideration of cycles rather than circuits eliminates certain restrictions (e.g., the complete reducibility of the matrices) which have previously been imposed. Our results extend theorems in Engel and Schneider [5], where however the group F is permitted to be non-commutative.