Asymptotically fast computation of Hermite normal forms of integer matrices

Abstract

This paper presents a new algorithm for computing the Hermite normal form H of an A c Z "m of rank m together with a unimodular pre-multiplier matrix U such that UA = H. Our algorithm requires O-(m '-lnM(mlog[lAl\)) bit oper@ions to produce both H and U. Here, IIAII = max,j lAij [, M(t) bit operations are sufficient to multiply two (t]-bit integers, and 0 is the exponent for matrix multiplication over rings: two m x m matrices over a ring R can be multiplied in O(me) ring operations from R, The previously fastest algorithm of Hafner & McCurley requires 0-(m2nM (m log I IAI [)) bit operations to produce H, but does not produce a unimodular matrix U which satisfies UA = H. Previous methods require on the order of 0-(n3M(m log I[Al 1)) bit operations to produce a U-our algorithm improves on this significantly in both a theoretical and practical sense.

An LLL-reduction algorithm with quasi-linear time complexity

“…Using [17,31], it can be computed in time O(d ω+1+ε β 1+ε ), where the input matrix B ∈ Z d×d satisfies max b i ≤ 2 β .…”

A Family of Sparse Polynomial Systems Arising in Chemical Reaction Systems

“…For computation of a good module basis we use the Hermite normal form (Adkins and Weintraub, 1992, p. 301). For its efficient implementation see Storjohann (1998) and Storjohann and Labahn (1996). A hint to the following lemma may be found in Feinberg (1987Feinberg ( , p. 2262).…”

Computation of Storage Requirements for Multi-Dimensional Signal Processing Applications

“…Table I summarizes the results of our experiments [48] carried out on a PC with a 1.85-GHz Athlon XP processor and 512-MB memory. The benchmarks used are: 1) a motion detection algorithm used in the transmission of real-time video signals on data networks [1]; 2) a real-time regularity detection algorithm used in robot vision; 3) Durbin's algorithm for solving Toeplitz systems with unknowns; 4) the kernel of a motion estimation algorithm for moving objects (MPEG-4); 5) a singular value decomposition (SVD) updating algorithm [49] used in spatial division multiplex access (SDMA) modulation in mobile commu- 8 For instance, the Storjohann-Labahn algorithm [45] requires O (n mB(n log A)) bit operations, where B(t) is the number of bit operations to multiply two dte-bit integers, and is the exponent for matrix multiplications over rings ( = 3 for standard multiplication, but = 2:38 for the best known multiplication algorithm). Here, O is the soft-Oh notation: f = O (g) if and only if f = O(g 1 log g) for some constant c > 0. nication receivers, in beamforming, and Kalman filtering; and 6) the kernel of a voice coding application-component of a mobile radio terminal.…”