Rational Lanczos approximations to the matrix square root and related functions

Numerical Linear Algebra with Applications

Volumn 16, Issue 6, 2009, Pages 431-445

  Moret, Igor ^a  

^a UNIVERSITY OF TRIESTE   (Italy)

Author keywords

Functions of matrices; Krylov subspace methods; Lanczos method

Indexed keywords

EID: 67649442860     PISSN: 10705325     EISSN: None     Source Type: Journal    
DOI: 10.1002/nla.625     Document Type: Article

References (21)

1. Druskin V, Knizhnerman L. Extended Krylov subspaces: approximation of the matrix square root and related functions. SIAM Journal on Matrix Analysis and Applications 1998; 19(3):755-771.
2. Allen EJ, Baglama J, Boyd SK. Numerical approximation of the product of the square root of a matrix with a vector. Linear Algebra and its Applications 2000; 310:167-181.
3. Ilić M, Turner IW, Pettitt AN. Bayesian computations and efficient algorithms for computing functions of large, sparse matrices. ANZIAM Journal 2004; 45(C):504-518.
4. Druskin V, Knizhnerman L. Two polynomial methods for calculating functions of symmetric matrices. Computational Mathematics and Mathematical Physics 1989; 29(6):112-121.
5. Alpak F, Torres-Verdin C, Sepehrnoori K, Fang S, Knizhnerman L. An extended Krylov subspace method to simulate single-phase fluid flow phenomena in axisymmetiric and anisotropic porous media. Journal of Petroleum Science and Engineering 2003; 40:121-144.
6. Knizhnerman L, Simoncini V. A new investigation of the extended Krylov subspace method for matrix function evaluations. 2008; submitted.
7. Moret I, Novati P. RD-rational approximations of the matrix exponential. BIT 2004; 44(3):595-615.
8. Van den Eshof J, Hochbruck M. Preconditioning Lanczos approximations to the matrix exponential. SIAM Journal on Scientific Computing 2005; 27:1438-1457.
9. Moret I. On RD-rational Krylov approximations to the core-functions of exponential integrators. Numerical Linear Algebra with Applications 2007; 14:445-457.
10. Novati P. On the construction of Restricted-Denominator Exponential W-methods. Journal of Computational and Applied Mathematics 2008; 212(1):86-101.
11. Popolizio M, Simoncini V. Acceleration techniques for approximating the matrix exponential operator. SIAM Journal on Matrix Analysis and Applications 2008; 30(2):657-683.
12. Cheney EW. Introduction to Approximation Theory. McGraw Hill: New York, 1966.
13. Bhatia R. Matrix Analysis. Springer: Berlin, 1996.
14. Druskin V, Knizhnerman L. Gaussian spectral rules for the three-point second differences: I. A two-point positive definite problem in a semi-infinite domain. SIAM Journal on Numerical Analysis 1998; 37(2):403-422.
15. Bromwich TJ. Introduction to the Theory of Infinite Series. Macmillan Co.: London, 1926.
16. Nevanlinna O. Convergence of Iterations for Linear Equations. Birkhäuser: Basel, 1993.
17. Greenbaum A, Druskin V, Knizhnerman L. On solving indefinite linear systems by means of the Lanczos method. Computational Mathematics and Mathematical Physics 1999; 39(3):371-377.
18. Knizhnerman L. On Adaptation of the Lanczos method to the spectrum. Research Note, Schlumberger-Doll Research, Ridgefield, 1995.
19. Knizhnerman L. Adaptation of the Lanczos and Arnoldi methods to the spectrum, or why the two Krylov subspace methods are powerful. Chebyshev Digest 2002; 3(2):141-164.
20. Beckermann B, Kuijlaars ABJ. Superlinear convergence of conjugate gradients. SIAM Journal on Numerical Analysis 2001; 39(1):300-329.
21. Frommer A, Simoncini V. Stopping criteria for rational matrix functions of Hermitian and symmetric matrices. SIAM Journal on Scientific Computing 2008; 30(3):1387-1412.