Computing Hermite and Smith normal forms of triangular integer matrices

This paper considers the problem of transforming a triangular integer input matrix to canonical Hermite and Smith normal form. We provide algorithms and prove deterministic running times for both transformation problems that are optimal in the matrix dimension. The algorithms are easily implemented, assume standard integer arithmetic, and admit excellent performance in practice. The results presented here lead to a faster algorithm Ibr computing the Smith normal form of an arbitrary (i.e. non-triangular) input matrix. Keywordw Hermite normal form: Smith normal form; Integer matrix I . I n t r o d u c t i o n It follows from Hermitc [!] that any m × n rank n integer matrix d can be transformed by applying a sequence of integer row operations to an upper triangular matrix H that has off-diagonal entries nonnegative and with magnitude smaller than the positive diagonal entry in the same column. The triangularization H called the Hermite normal form ofA always exists and is unique. In this paper we consider the problem of computing the Hermite normal form H i This work has been supported by grants from the Swiss Federal Office for Education and Science in conjunction with partial support by ESPRIT LTR Project no. 20244 ALCOM-IT. 2 E-mail: [email protected]. 0024-37951981519.00 © 1998 Elsevier Science Inc. All rights reserved. PII: S 0 0 2 4 3 7 9 5 ( 9 8 ) 1 00 ! 2-5 26 A. Storjohann I Linear Algebra and its Applications 282 (1998) 25-45 of an n x n non-singular matrix T that is already upper triangular and has offdiagonal entries bounded in magnitude by the product D of the diagonal entries. We get the following transformation diagram T "hi tl2 t13 h2 t23 h3