Wayne Eberly

On computing the determinant and Smith form of an integer matrix

A probabilistic algorithm is presented to find the determinant of a nonsingular, integer matrix. For a matrix A/spl isin/Z/sup n/spl times/n/ the algorithm requires O(n/sup 3.5/(log n)/sup 4.5/) bit operations (assuming for now that entries in A have constant size) using standard matrix and integer arithmetic. Using asymptotically fast matrix arithmetic, a variant is described which requires O(n/sup 2+/spl theta//2//spl middot/log/sup 2/nloglogn) bit operations, where n/spl times/n matrices can be multiplied with O(n/sup /spl theta//) operations. The determinant is found by computing the Smith form of the integer matrix an extremely useful canonical form in itself. Our algorithm is probabilistic of the Monte Carlo type. That is, it assumes a source of random bits and on any invocation of the algorithm there is a small probability of error.

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 randomized Lanczos algorithms

Las Vegas algorithms that are based on Lanczos’s method for solving symmetric linear systems are presented and analyzed. These are compared to a similar randomized Lanczos algorithm that has been used for integer factorization, and to the (provably reliable) algorithm of Wiedemann. The analysis suggests that our Lanczos algorithms are preferable to several versions of Wiedemann’s method for computations over large fields, especially for certain symmetric matrix computations.

Very fast parallel polynomial arithmetic

Parallel algorithms for polynomial arithmetic—multiplication of n polynomials of degree m, polynomial division with remainder, and polynomial interpolation—are presented. These algorithms can be implemented using polynomial time constructible families of Boolean circuits of polynomial size and optimal order depth, or log space constructible families of polynomial size and near optimal depth, for computations over

Solving sparse rational linear systems

\mathbb{Z},\mathbb{Q}

Faster inversion and other black box matrix computations using efficient block projections

, finite fields, and several other domains. Arithmetic circuits of polynomial size and optimal order depth are obtained for polynomial arithmetic over arbitrary fields.

Size-depth tradeoffs for algebraic formulae

We propose a new algorithm to find a rational solution to a sparse system of linear equations over the integers. This algorithm is based on a p-adic lifting technique combined with the use of block matrices with structured blocks. It achieves a sub-cubic complexity in terms of machine operations subject to a conjecture on the effectiveness of certain sparse projections. A LinBox-based implementation of this algorithm is demonstrated, and emphasizes the practical benefits of this new method over the previous state of the art.

Efficient Decomposition of Associative Algebras over Finite Fields

Efficient block projections of non-singular matrices have recently been used by the authors in [10] to obtain an efficient algorithm to find rational solutions for sparse systems of linear equations. In particular a bound ofO~(n2.5) machine operations is presented for this computation assuming that the input matrix can be multiplied by a vector with constant-sized entries using O~(n) machine operations. Somewhat more general bounds for black-box matrix computations are also derived. Unfortunately, the correctness of this algorithm depends on the existence of efficient block projections of non-singular matrices, and this was only conjectured. In this paper we establish the correctness of the algorithm from [10] by proving the existence of efficient block projections for arbitrary non-singular matrices over sufficiently large fields. We further demonstrate the usefulness of these projections by incorporating them into existing black-box matrix algorithms to derive improved bounds for the cost of several matrix problems. We consider, in particular, matrices that can be multiplied by a vector using O~(n) field operations: We show how to compute the inverse of any such non-singular matrix over any field using an expected number of O~(n2.27) operations in that field. A basis for the null space of such a matrix, and a certification of its rank, are obtained at the same cost. An application of this technique to Kaltofen and Villards Baby-Steps/Giant-Steps algorithms for the determinant and Smith Form of an integer matrix is also sketched, yielding algorithms requiring O~(n2.66) machine operations. More general bounds involving the number of black-box matrix operations to be used are also obtained. The derived algorithms are all probabilistic of the Las Vegas type. They are assumed to be able to generate random elements - bits or field elements - at unit cost, and always output the correct answer in the expected time given.

Efficient decomposition of separable algebras

Some tradeoffs between the size and depth of algebraic formulas are proved. It is shown that, for any fixed in >0, any algebraic formula of size S can be converted into an equivalent formula of depth O(log S) and size O(S/sup 1+ in /). This result is an improvement over previously known results where, to obtain the same depth bound, the formula size is Omega (S/sup alpha /), with alpha >or=2.<<ETX>>

Long-Lived, Fast, Waitfree Renaming with Optimal Name Space and High Throughput

We present new, efficient algorithms for some fundamental computations with finite-dimensional (but not necessarily commutative) associative algebras over finite fields. For a semisimple algebra A we show how to compute a complete Wedderburn decomposition of A as a direct sum of simple algebras, an isomorphism between each simple component and a full matrix algebra, and a basis for the centre of A. If A is given by a generating set of matrices inFm?m, then our algorithm requires aboutO (m3) operations inF, in addition to the cost of factoring a polynomial inFx ] of degree O(m), and the cost of generating a small number of random elements from A. We also show how to compute a complete set of orthogonal primitive idempotents in any associative algebra over a finite field in this same time.