Reduction of functions of some partitioned matrices

Authors:
Victor Lovass-Nagy and David L. Powers

Journal:
Math. Comp. **23** (1969), 127-133

MSC:
Primary 65.35

MathSciNet review:
0238480

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Abstract: In the numerical analysis of physical problems, there often arise large matrices which exhibit certain kinds of block-symmetry when partitioned appropriately. In this article, the structures of the frequently-occurring hyper-circulant and hyper-Jacobi matrices are examined, and it is shown how the calculation of any analytic function of such matrices may be reduced to the calculation of functions of the submatrices. Examples drawn from current engineering literature are given as well as small illustrative examples.

**[1]**S. Charmonman, ``An efficient algorithm for inverting a block-symmetric matrix,''*Math. Comp.*, v. 21, 1967, pp. 715-717.**[2]**E. Egerváry,*On hypermatrices whose blocks are commutable in pairs and their application in lattice-dynamics*, Acta Sci. Math. Szeged**15**(1954), 211–222. MR**0064736****[3]**Bernard Friedman,*Report on a conference on dynamics of ionized media*, Research Rep. No. EM-30, New York University, Washington Square College, Research Group, 1951. MR**0042341****[4]**Emilie V. Haynsworth,*Special types of partitioned matrices*, J. Res. Nat. Bur. Standards Sect. B**65B**(1961), 7–12. MR**0150162****[5]**R. W. Hockney,*A fast direct solution of Poisson’s equation using Fourier analysis*, J. Assoc. Comput. Mach.**12**(1965), 95–113. MR**0213048****[6]**Elliott W. Montroll,*Markoff chains and excluded volume effect in polymer chains*, J. Chem. Phys.**18**(1950), 734–743. MR**0036468****[7]**F. E. Steidler & H. H. Horovitz, ``The calculated load-carrying ability of non-Newtonian lubricants in hydro-dynamic bearings,''*Chemical Engineering Progress Symposium Series*, no. 42, Vol. 59, 1963, pp. 99-107.**[8]**John Williamson,*The latent roots of a matrix of special type*, Bull. Amer. Math. Soc.**37**(1931), no. 8, 585–590. MR**1562202**, 10.1090/S0002-9904-1931-05215-0

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DOI:
http://dx.doi.org/10.1090/S0025-5718-1969-0238480-8

Article copyright:
© Copyright 1969
American Mathematical Society