Abstract
Computation of approximate factors for the inverse constitutes an algebraic approach to preconditioning large and sparse linear systems. In this paper, the aim is to combine standard preconditioning ideas with sparse approximate inverse approximation, to have dense approximate inverse approximations (implicitly). For optimality, the approximate factoring problem is associated with a minimization problem involving two matrix subspaces. This task can be converted into an eigenvalue problem for a Hermitian positive semidefinite operator whose smallest eigenpairs are of interest. Because of storage and complexity constraints, the power method appears to be the only admissible algorithm for devising sparse–sparse iterations. The subtle issue of choosing the matrix subspaces is addressed. Numerical experiments are presented.
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References
Adams, J.F., Lax, P., Phillips, R.: On matrices whose real linear combinations are nonsingular. Proc. Am. Math. Soc. 16, 318–322 (1965). Correction: Proc. Am. Math. Soc. 17, 945–947 (1966)
Bai Z., Fahey G., Golub G.: Some large-scale matrix computation problems. J. Comput. Appl. Math. 74, 71–89 (1996)
Benzi M.: Preconditioning techniques for large linear systems: a survey. J. Comput. Phys. 182(2), 418–477 (2002)
Benzi M., Gander M., Golub G.H.: Optimization of the Hermitian and skew-Hermitian splitting iteration for saddle-point problems. BIT 43(suppl.), 881–900 (2003)
Benzi M., Haws J.C., Tuma M.: Preconditioning highly indefinite and nonsymmetric matrices. SIAM J. Sci. Comput. 22, 1333–1353 (2000)
Benzi M., Tuma M.: Numerical experiments with two approximate inverse preconditioners. BIT 38, 234–241 (1998)
Benzi M., Tuma M.: A sparse approximate inverse preconditioner for nonsymmetric linear systems. SIAM J. Sci. Comput. 19, 968–994 (1998)
Bhatia, R.: Matrix factorizations and their perturbations. Linear Algebra Appl. 197/198, 245–276 (1994)
Chow E.: A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM J. Sci. Comput. 21, 1804–1822 (2000)
Chow E., Saad Y.: Approximate inverse preconditioners via sparse-sparse iterations. SIAM J. Sci. Comput. 19, 995–1023 (1998)
Duff I.S., Erisman A.M., Reid J.K.: Direct Methods for Sparse Matrices. Oxford University Press, London (1989)
Eckmann, B.: Hurwitz–Radon matrices revisited: from effective solution of the Hurwitz matrix equations to Bott periodicity. In: Mathematical Survey Lectures 1943–2004, CRM Proceedings and Lecture Notes, vol. 6, pp. 23–35. Springer, Berlin (1994)
Golub G.H., van Loan C.F.: Matrix Computations, 3rd edn. The John Hopkins University Press, Baltimore (1996)
Grote M., Huckle T.: Parallel preconditioning with sparse approximate inverses. SIAM J. Sci. Comput. 18, 838–853 (1997)
Higham N.: Functions of matrices: theory and computation. SIAM, Philadelphia (2008)
Holtz O., Mehrmann V., Schneider H.: Potter, Wielandt, and Drazin on the matrix equation AB = ω BA: new answers to old questions. Am. Math. Mon. 111(8), 655–667 (2004)
Horn R.A., Johnson C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1987)
Huhtanen M.: Factoring matrices into the product of two matrices. BIT 47, 793–808 (2007)
Huhtanen M.: Matrix subspaces and determinantal hypersurfaces. Ark. Mat. 48, 57–77 (2010)
Huhtanen, M.: Differential geometry of matrix inversion. Math. Scand. (to appear)
Hurwitz A.: Über die Komposition der quadratischen Formen. Math. Ann. 88, 1–25 (1923)
Matrix Market.: National Institute of Standards and Technology. http://math.nist.gov/MatrixMarket/
Parlett, B.: The symmetric eigenvalue problem. In: Classics in Applied Mathematics, vol. 20. SIAM, Philadelphia (1997)
Radon J.: Lineare Scharen orthogonaler Matrizen. Abh. Sem. Hamburg 1, 1–14 (1922)
Saad, Y.: Iterative Methods for Sparse Linear Systems, 2nd edn. SIAM, Philadelphia (2003)
van der Vorst, H.: Iterative Krylov methods for large linear systems. In: Cambridge Monographs on Applied and Computational Mathematics, vol. 13. Cambridge University Press, Cambridge (2003)
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M. Byckling and M. Huhtanen were supported by the Academy of Finland.