Summary
We study block matricesA=[Aij], where every blockA ij ∈ℂk,k is Hermitian andA ii is positive definite. We call such a matrix a generalized H-matrix if its block comparison matrix is a generalized M-matrix. These matrices arise in the numerical solution of Euler equations in fluid flow computations and in the study of invariant tori of dynamical systems. We discuss properties of these matrices and we give some equivalent conditions for a matrix to be a generalized H-matrix.
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References
[BH] Bramble, J.H., Hubbard, B.E. (1964): On a finite difference analogue of an elliptic boundary value problem which is neither diagonally dominant nor of nonnegative type. J. Math. Phys.43, 117–132
[BP] Berman, A., Plemmons, R.J. (1979): Nonnegative Matrices in the Mathematical Sciences. Academic Press, New York
[DLi] Dick, E., Linden, J. (1989): A multigrid flux-difference splitting method for steady incompressible Navier-Stokes equations. Proceedings of the GAMM Conference on Numerical Methods in Fluid Mechanics, Delft
[DLo] Dieci, L., Lorenz, J. Block M-Matrices and Computation of Invariant Tori. To appear
[EM] Elsner, L., Mehrmann, V. (1991): Convergence of block iterative methods for linear systems arising in the numerical solution of Euler equations. Numer. Math.59, 541–559
[GvL] Golub, G.H., van Loan, C.F. (1989): Matrix Computations, 2nd edn. The Johns Hopkins University Press, Baltimore, London 1989
[Ha] Hackbusch, W. (1985): Multigrid methods and Applications. Springer, Berlin Heidelberg, New York
[He] Heller, D. (1976): Some Aspects of the Cyclic Reduction Algorithm for Block Tridiagonal Linear Systems. SIAM J. Numer. Anal.13, 484–496
[HJ] Horn, R.A., Johnson, C.A. (1985): Matrix Analysis. Cambridge University Press, Cambridge
[HS] Hemker, P.W., Spekreijse, S.P. (1986): Multiple Grid and Osher's Scheme for the efficient solution of the steady Euler equations. Appl. Numer. Math.2, 475–493
[MV] Meijerink, J.A., van der Vorst, H.A. (1977): An Iterative Solution Method for Linear Systems of Which the Coefficient Matrix is a Symmetric M-Matrix. Math. Comput.31, 148–162
[OP] Ortega, J.M., Plemmons, R.J. (1987): Extensions of the Ostrowski-Reich Theorem for SOR iterations. Linear Algebra Appl.88/89 559–573
[R] Robert, Y. (1982): Regular incomplete factorisations of real positive definite matrices. Linear Algebra Appl.48, 105–117
[V1] Varga, R.S. (1976): On recurring theorems on Diagonal Dominance. Linear Algebra Appl.13, 1–9
[V2] Varga, R.S. (1962): Matrix Iterative Analysis. Prentice-Hall, Engelwood Cliffs, N.J.
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Research supported by the Graduiertenkolleg mathematik der Universität Bielefeld