There are several prominent computational problems for which simple iterative methods are widely preferred in practice despite an absence of runtime or performance analysis (or "worse", actual evidence that more sophisticated methods have superior performance according to the usual criteria). These situations raise interesting challenges for the analysis of algorithms.
We are concerned in this work with one such simple method: a classical iterative algorithm for balancing matrices via scaling transformations. This algorithm, which goes back to Osborne and Parlett & Reinsch in the 1960s, is implemented as a standard preconditioner for eigenvalue computations in many numerical linear algebra packages. Surprisingly, despite its widespread use over several decades, no bounds were known on its rate of convergence.
We prove the first such bound: a natural L_infinity -norm variant of the algorithm converges, on n by n matrices, in O(n^3 log(n)) elementary scaling transformations. This is tight up to the log n factor. We also prove a conjecture of Chen that characterizes those matrices for which the limit of the balancing process is independent of the order in which balancing operations are performed (joint work with Alistair Sinclair).