Matrix balancing in Lp norms: Bounding the convergence rate of osborne's iteration

We study an iterative matrix conditioning algorithm due to Osborne (1960). The goal of the algorithm is to convert a square matrix into a balanced matrix where every row and corresponding column have the same norm. The original algorithm was proposed for balancing rows and columns in the L2 norm, and it works by iterating over balancing a row-column pair in fixed round-robin order. Variants of the algorithm for other norms have been heavily studied and are implemented as standard preconditioners in many numerical linear algebra packages. Recently, Schulman and Sinclair (2015), in a first result of its kind for any norm, analyzed the rate of convergence of a variant of Osborne's algorithm that uses the L1 norm and a different order of choosing row-column pairs. In this paper we study matrix balancing in the L1 norm and other Lp norms. We show the following results for any matrix A = (a) =1, resolving in particular a main open problem mentioned by Schulman and Sinclair. 1. We analyze the iteration for the L1 norm un- der a greedy order of balancing. We show that it converges to an -balanced matrix in K = O(minf log w; n3=2 log(w=)g) iterations that cost a total of O(m + Kn log n) arithmetic op- erations over O(n log(w=))-bit numbers. Here m Pis the number of non-zero entries of A, and w = ij jaij j=amin with amin = minfjaij j : 6= 0g. 2. We show that the original round-robin implementation converges to an -balanced matrix in O(2n2 log w) it- erations totaling O(2mnlog w) arithmetic operations over O(n log(w=))-bit numbers. 3. We show that a random implementation of the iteration converges to an-balanced matrix in O(2 log w) iterations using O(m + 2n log w) arithmetic operations over O(log(wn))-bit numbers. 4. We demonstrate a lower bound of (1= p ) on the convergence rate of any implementation of the iteration. 5. We observe, through a known trivial reduction, that our results for L1 balancing apply to any Lp norm for all finite p, at the cost of increasing the number of iterations by only a factor of p.