Abstract
We propose a new algorithm, the DomEig algorithm, for obtaining the line-sum-symmetric similarity-scaling of a given irreducible, essentially nonnegative matrix A. It is based on results concerning the minimum dominant eigenvalue of an essentially nonnegative matrix under trace-preserving perturbations of its diagonal. In this note we relate the minimum dominant eigenvalue problem to the problem of determining the diagonal scaling matrix for line-sum-symmetry. We present the DomEig algorithm, prove its convergence, and discuss briefly the results of a comparison of this algorithm with another algorithm, the DSS algorithm, often used for line-sum symmetry. The experiments suggest that, for matrices of order greater than 50, the convergence rate, measured either in flop counts or CPU time, is significantly greater for DomEig than for DSS, with the improvement in rate increasing as the order increases. The algorithm may be useful in such applications as the scaling of large social accounting matrices.
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References
B.C. Eaves, A.J. Hoffman, U.G. Rothblum, and H. Schneider, “Line-sum-symmetric scalings of square nonnegative matrices,” Mathematical Programming Study, vol. 25, pp. 124-141, 1985.
B. Gilbert, “Summary of research on the DomEig and DSS algorithms,” Interim Report to the Department of Computer Science, University of Victoria, 1993.
R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge University Press, New York, 1985.
R.A. Horn and C.R. Johnson, Topics in Matrix Analysis, Cambridge University Press, New York, 1991.
C.R. Johnson, D.D. Olesky, D.P. Stanford, and P. van den Driessche, “Dominant eigenvalues under trace-preserving diagonal perturbations,” Linear Algebra Appl., vol. 212/213, pp. 415-435, 1994.
B. Kalantari, L. Khachiyan, and A. Shokoufandeh, On the complexity of matrix balancing, SIAM J. Matrix Anal. Appl., vol. 18, no. 2, pp. 450-463, 1997.
D.G. Luenberger, Linear and Nonlinear Programming, 2nd ed., Addison Wesley, New York, 1984.
E.E. Osborne, On pre-conditioning of matrices, J. Assoc. Comput. Mach., vol. 7, pp. 338-345, 1960.
J. Pitkin, “A geometric interpretation of the dominant eigenvalue problem and the analysis of related algorithms,” William and Mary Mathematics Department, Technical Report TR98.01.
M.H. Schneider and S.A. Zenios, A comparative study of algorithms for matrix balancing, Operations Research, vol. 38, no. 3, pp. 439-455, 1990.
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Johnson, C.R., Pitkin, J. & Stanford, D.P. Line-Sum Symmetry via the DomEig Algorithm. Computational Optimization and Applications 17, 5–10 (2000). https://doi.org/10.1023/A:1008754123750
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DOI: https://doi.org/10.1023/A:1008754123750