Analysis of a Classical Matrix Preconditioning Algorithm

We study a classical iterative algorithm for the problem of balancing matrices in the L∞ norm via a scaling transformation. This algorithm, which goes back to Osborne and Parlett & Reinsch in the 1960s, is implemented as a standard preconditioner in many numerical linear algebra packages. Surprisingly, despite its widespread use over several decades, no bounds were known on its rate of convergence. In this paper we prove that, for a large class of irreducible n x n (real or complex) input matrices~$A$, a natural variant of the algorithm converges in O(n3 log(nρ/ε)) elementary balancing operations, where ρ measures the initial imbalance of A and ε is the target imbalance of the output matrix. (The imbalance of A is maxi |log(aiout/aiin)|, where aiout,aiin are the maximum entries in magnitude in the ith row and column respectively.) This bound is tight up to the log n factor. A balancing operation scales the ith row and column so that their maximum entries are equal, and requires O(m/n) arithmetic operations on average, where m is the number of non-zero elements in A. Thus the running time of the iterative algorithm is ~O(n2m). This is the first time bound of any kind on any variant of the Osborne-Parlett-Reinsch algorithm. The class of matrices for which the above analysis holds are those which satisfy a condition we call Unique Balance, meaning that the limit of the iterative balancing process does not depend on the order in which balancing operations are performed. We also prove a combinatorial characterization of the Unique Balance property, which had earlier been conjectured by Chen.