A concept of nonlinear block diagonal dominance. (English) Zbl 0886.65052

A new concept of diagonal dominance, which generalizes the strictly diagonally dominance for nonlinear functions, called block diagonal dominance, is introduced. Sufficient conditions for a function to be block strictly diagonally dominant as well as applications to block asynchronous iterative methods for approximating fixed points, are also given.
Generalized diagonally dominant mappings were introduced and studied by A. Frommer [J. Comput. Appl. Math. 38, No. 1-3, 105-124 (1991; Zbl 0746.65046)].


65H10 Numerical computation of solutions to systems of equations


Zbl 0746.65046
Full Text: DOI


[1] Baudet, G., Asynchronous iterative methods for multiprocessors, J. Assoc. Comput. Mach., 25, 226-244 (1978) · Zbl 0372.68015
[2] Bhaya, A.; Kaszkurewicz, E.; Mota, F., Asynchronous block — iterative methods for almost linear equations, Linear Algebra Appl., 154-156, 487-508 (1991) · Zbl 0729.65033
[3] Chazan, D.; Miranker, W., Chaotic relaxation, Linear Algebra Appl., 2, 199-222 (1969) · Zbl 0225.65043
[4] EI-Baz, D., M-functions and parallel asynchronous algorithms, SIAM J. Numer. Anal., 27, 136-140 (1990) · Zbl 0701.65040
[5] EI-Tarazi, M. N., Some convergence results for asynchronous algorithms, Numer. Math., 39, 325-340 (1982) · Zbl 0479.65030
[6] Feingold, D. G.; Varga, R. S., Block diagonally dominant matrices and generalization of Gerschgorin circle theorem, Pacific J. Math., 4, 1241-1250 (1962) · Zbl 0109.24802
[7] Fischer, H.; Ritter, K., An asynchronous parallel Newton method, Math. Programming, 42, 363-374 (1988) · Zbl 0665.90080
[8] Frommer, A., Generalized nonlinear diagonal dominance and applications to asynchronous iterative methods, J. Comput. Appl. Math., 38, 105-124 (1991) · Zbl 0746.65046
[9] Moré, J., Nonlinear generalizations of matrix diagonal dominance with application to Gauss-Seidel iterations, SIAM J. Numer. Anal., 9, 357-378 (1972) · Zbl 0243.65023
[10] Mukai, H., Parallel algorithms for solving systems of nonlinear equations, (Proc. 17th Ann. Allerton Conf. on Communications, Control and Computation (October 10-12, 1979)), 37-46
[11] Ortega, J.; Rheinboldt, W., Iterative Solution of Nonlinear Equations in Several Variables (1970), Academic Press: Academic Press New York · Zbl 0241.65046
[12] Chuanlong, Wang; Zhaoyong, You, An asynchronous inexact-Newton method for nonlinear programming, (Proc. National 4th Optimization Theory and Applications (October 2-6, 1994)), 297-302
[13] Deren, Wang; SunBaoyun, A parallel algorithm for a class of nonlinear equations applicable to MIMD systems, Chinese J. Comput. Math., 13, 297-306 (1991) · Zbl 0850.65096
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.