Some uses of the symmetric Lanczos algorithm - and why it works!

Conference · Tue Dec 31 04:00:00 EST 1996

OSTI ID:440686

Druskin, V L ^[1]; Greenbaum, A ^[2]; Knizhnerman, L A ^[3]

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1. Schlumberger-Doll Research, Ridgefield, CT (United States)
2. Courant Institute of Mathematical Sciences, New York, NY (United States)
3. Central Geophysical Expedition, Moscow (Russian Federation)

The Lanczos algorithm uses a three-term recurrence to construct an orthonormal basis for the Krylov space corresponding to a symmetric matrix A and a starting vector q{sub 1}. The vectors and recurrence coefficients produced by this algorithm can be used for a number of purposes, including solving linear systems Au = {var_phi} and computing the matrix exponential e{sup -tA}{var_phi}. Although the vectors produced in finite precision arithmetic are not orthogonal, we show why they can still be used effectively for these purposes. The reason is that the 2-norm of the residual is essentially determined by the tridiagonal matrix and the next recurrence coefficient produced by the finite precision Lanczos computation. It follows that if the same tridiagonal matrix and recurrence coefficient are produced by the exact Lanczos algorithm applied to some other problem, then exact arithmetic bounds on the residual for that problem will hold for the finite precision computation. In order to establish exact arithmetic bounds for the different problem, it is necessary to have some information about the eigenvalues of the new coefficient matrix. Here we make use of information already established in the literature, and we also prove a new result for indefinite matrices.

• Research Organization: Front Range Scientific Computations, Inc., Lakewood, CO (United States)
• OSTI ID: 440686
• Report Number(s): CONF-9604167--Vol.2; ON: DE96015307
• Country of Publication: United States
• Language: English

Related Subjects

99 GENERAL AND MISCELLANEOUS
ALGORITHMS
EIGENVALUES
ITERATIVE METHODS
MATRICES
VECTORS