Arne Storjohann

Near optimal algorithms for computing Smith normal forms of integer matrices

We present new algorithms for computing Smith normal forms of matrices over the integers and over the integers modulo d For the case of matrices over Z d we present an algorithm that computes the Smith form S of an A Z n m d in only O n m operations from Z d Here is the ex ponent for matrix multiplication over rings two n n matrices over a ring R can be multiplied in O n opera tions from R We apply our algorithm for matrices over Z d to get an algorithm for computing the Smith form S of an A Z n m in O n m M n log jjAjj bit opera tions where jjAjj max jAi jj and M t bounds the cost of multiplying two dte bit integers These complexity re sults improve signi cantly on the complexity of previously best known Smith form algorithms both deterministic and probabilistic which guarantee correctness

Asymptotically fast computation of Hermite normal forms of integer matrices

This paper presents a new algorithm for computing the Hermite normal form H of an A Z n m of rank m to gether with a unimodular pre multiplier matrix U such that UA H Our algorithm requires O m nM m log jjAjj bit operations to produce both H and a candidate for U Here jjAjj maxij jAijj M t bit operations are su cient to multiply two dte bit integers and is the exponent for matrix multiplication over rings two m m matrices over a ring R can be multiplied in O m ring operations from R The previously fastest algorithm of Hafner McCurley re quires O m nM m log jjAjj bit operations to produce H but does not produce a candidate for U Previous methods require on the order of O n M m log jjAjj bit operations to produce a candidate for U our algorithm improves on this signi cantly in both a theoretical and practical sense

High-order lifting and integrality certification

Reductions to polynomial matrix multiplication are given for some classical problems involving a nonsingular input matrix over the ring of univariate polynomials with coefficients from a field. High-order lifting is used to compute the determinant, the Smith form, and a rational system solution with about the same number of field operations as required to multiply together two matrices having the same dimension and degree as the input matrix. Integrality certification is used to verify correctness of the output. The algorithms are space efficient.

The shifted number system for fast linear algebra on integer matrices

The shifted number system is presented: a method for detecting and avoiding error producing carries during approximate computations with truncated expansions of rational numbers. Using the shifted number system the high-order lifting and integrality certification techniques of Storjohann 2003 for polynomial matrices are extended to the integer case. Las Vegas reductions to integer matrix multiplication are given for some problems involving integer matrices: the determinant and a solution of a linear system can be computed with about the same number of bit operations as required to multiply together two matrices having the same dimension and size of entries as the input matrix. The algorithms are space efficient.

A BLAS based C library for exact linear algebra on integer matrices

Algorithms for solving linear systems of equations over the integers are designed and implemented. The implementations are based on the highly optimized and portable ATLAS/BLAS library for numerical linear algebra and the GNU Multiple Precision library (GMP) for large integer arithmetic.

Diophantine linear system solving

A sinq)lo randoruixed algorithnl is given for fillding an integer solubion to a s:-stclu of linear Di01)lmnt,ine equations. Givcu as input a s?;stcrrl which admits an int.cger solul.ion, the idgorithru can 1~ used t,o find such a. solution wit,h probabilit~~ ilt least l/2. The running time (nunlber of bit operations) is esscntiall~ cubic in the dimemion of the s?;stem. The malogous result is prcsentctl for 1inca.r spst.cnls over the ring of polqonlials wit.li coefficients from a field.

An O ( n 3 ) algorithm for the Frobenius normal form

We describe an O(n) eld operations algorithm for computing the Frobenius normal form of an n n matrix. As applications we get O(n) algorithms for two other classical problems: computing the minimal polynomial of a matrix and testing two matrices for similarity. Assuming standard matrix multiplication, the previously best known deterministic complexity bound for all three problems is O(n).

Computing the rank and a small nullspace basis of a polynomial matrix

We reduce the problem of computing the rank and a null-space basis of a univariate polynomial matrix to polynomial matrix multiplication. For an input <i>n</i> x <i>n</i> matrix of degree, <i>d</i> over a field <b>K</b> we give a rank and nullspace algorithm using about the same number of operations as for multiplying two matrices of dimension, <i>n</i> and degree, <i>d</i>. If the latter multiplication is done in <b>MM</b>(<i>n,d</i>)= <i>O</i>^~(<i>n</i>^ω<i>d</i> operations, with ω the exponent of matrix multiplication over <b>K</b>, then the algorithm uses <i>O</i>^~<b>MM</b>(<i>n,d</i>) operations in, <b>K</b>. For <i>m</i> x <i>n</i> matrices of rank <i>r</i> and degree <i>d</i>, the cost expression is <i>O</i>(<i>nmr</i> ^ω-2<i>d</i>). The soft-O notation <i>O</i>^~ indicates some missing logarithmic factors. The method is randomized with Las Vegas certification. We achieve our results in part through a combination of matrix Hensel high-order lifting and matrix minimal fraction reconstruction, and through the computation of minimal or small degree vectors in the nullspace seen as a <b>K</b>[<i>x</i>]-module.

Solving sparse rational linear systems

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.

Certified dense linear system solving

Abstract A randomized algorithm is given for solving a system of linear equations over a principal ideal domain. The algorithm returns a solution vector which has minimal denominator. A certificate of minimality is also computed. A given system has a Diophantine solution precisely when the minimal denominator is one. Cost estimates are given for systems over the ring of integers and ring of polynomials with coefficients from a field.