Abstract
A new class of approximate inverses for arrowhead and special tridiagonal linear systems, based on the concept of sparse approximate Choleski-type factorization procedures, are introduced for computing fast explicit approximate inverses. Explicit preconditioned iterative schemes in conjunction with approximate inverse matrix techniques are presented for the efficient solution of symmetric linear systems. A theorem on the rate of convergence of the explicit preconditioned conjugate gradient scheme is given and estimates of the computational complexity are presented. Applications of the proposed method on linear and nonlinear systems are discussed and numerical results are given.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Arbenz, P. and Golub, G.: QR-like algorithms for symmetric arrow matrices, SIAM J. Matrix Anal. Appl. 13 (1992), 655–658.
Fisher, D., Golub, G., Hald, O., Leiva, C. and Widlund, O.: On Fourier–Toeplitz methods for separable elliptic problems, Math. Comp. 28 (1974), 349–368.
Golub, G.: Some modified matrix eigenvalue problems, SIAM Rev. 15 (1973), 318–334.
Gragg, B. and Harrod W.: The numerically stable reconstruction of Jacobi matrix from spectral data, Numer. Math. 44 (1984), 317–355.
Gravvanis, G. A.: A fast explicit preconditioned symmetric finite element scheme, Neural Parallel Sci. Comput. 9(1) (2001), 59–66.
Gravvanis, G. A.: Fast explicit approximate inverses for solving linear and non-linear finite difference equations, Internat. J. Differential Equations Appl. 1(4) (2000), 451–473.
Gravvanis, G. A.: Generalized approximate inverse preconditioning for solving non-linear elliptic boundary-value problems, Internat. J. Appl. Math. 2(11) (2000), 1363–1378.
Gravvanis, G. A.: Parallel preconditioned algorithms for solving special tridiagonal systems, In: Proc. Third Internat. Conf. on Dynamical Systems and Applications, Vol. III, 1999, pp. 241–248.
Gravvanis, G. A.: Approximate inverse banded matrix techniques, Engrg. Comput. 16(3) (1999), 337–346.
Gravvanis, G. A.: The rate of convergence of explicit approximate inverse preconditioning, Internat. J. Comput. Math. 60 (1996), 77–89.
Gravvanis, G. A.: Explicit preconditioned methods for solving 3D boundary value problems by approximate inverse finite element matrix techniques, Internat. J. Comput. Math. 56 (1995), 77–93.
Gravvanis, G. A.: A three dimensional symmetric linear equation solver, Comm. Numer. Methods Engrg. 10(9) (1994), 717–730.
Lipitakis, E. A. and Evans, D. J.: Numerical solution of non-linear elliptic boundary-value problems by isomorphic iterative methods, Interact. J. Comput. Math. 20 (1986), 261–282.
O'Leary, D. and Stewart, G.: Computing the eigenproblem and eigenvectors of arrowhead matrices, J. Comput. Phys. 90 (1990), 497–505.
Parlett, B.: The Symmetric Eigenvalue Problem, Prentice-Hall, Englewood Cilffs, 1980.
Parlett, B. and Nour-Omid, B.: The use of refined error bound when updating eigenvalues of tridiagonals, Linear Algebra Appl. 68 (1985), 179–219.
Rutishauser, H.: On Jacobi rotation patterns, In: Experimental Arithmetic, High Speed Computing and Mathematics, Proc. Sympos. Appl. Math. 15, Amer. Math. Soc., Providence, 1963, pp. 219–239.
Zha, H.: A two-way chasing scheme for reducing a symmetric arrowhead matrix to tridiagonal form, J. Numer. Algebra Appl. 1 (1992), 49–57.
Gravvanis, G. A.: Solving symmetric arrowhead linear systems by fast approximate inverses, In: Proc. Internat. Conf. on Parallel and Distributed Processing Techniques and Applications, Vol.IV, CSREA Press, 2001, pp. 1762–1768.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Gravvanis, G.A. Solving Symmetric Arrowhead and Special Tridiagonal Linear Systems by Fast Approximate Inverse Preconditioning. Journal of Mathematical Modelling and Algorithms 1, 269–282 (2002). https://doi.org/10.1023/A:1021630031889
Issue Date:
DOI: https://doi.org/10.1023/A:1021630031889