Matrix Balancing in Lp Norms: Bounding the Convergence Rate of Osborne's Iteration
Rafail Ostrovsky, Yuval Rabani, Arman YousefiAbstract:
W study an iterative matrixConditioning due to Osbome(1960). The goal of the algorithm is to convert a square matrix into a balanced matrix here 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 liner 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 Osbome's algorithm that uses the L∞ 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,j)in,j=1 resolving in particular a main open problem mentioned by Schulman and Sinclair.
comment: SODA 2017: 154-169
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