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Survey of extrapolation methods for linear
algebra problems
Claude Brezinski
University of Lille, France
[email protected]
Keywords: extrapolation, projection, sequence transformation, numerical linear algebra.
In the first part of this talk, we give a general idea about what sequence
transformations are, in particular those for vector sequences. We explain why
these sequence transformations are extrapolation methods [1] and why some of
them are also projection methods [2].
In the second part of the talk, we show how these procedures can be use
for solving a system of linear equations and for computing the eigenvalues of a
matrix. We focus, in particular, on the ε–algorithms [3].
The last part of the talk is devoted to applications of extrapolations methods
in the context of some linear algebra problems. In particular, we show that they
can be used for estimating the norm of the error for the solution of systems
of linear equations [4], and we apply these estimates to the choice of the best
parameter in Tikhonov regularization method [5]. Another application is the
computation of the trace of powers of matrices [6] and another one concerns
web search and the PageRank algorithm [7]. Finally we show how to accelerate
the convergence of Kaczmarz method for linear systems [8]. Numerical examples
illustrate these applications.
This talk presents several results obtained in joint works with M. RedivoZaglia and others researchers.
References
[1] C. Brezinski, M. Redivo–Zaglia, Extrapolation Methods. Theory and Practice,
North–Holland, Amsterdam, 1991.
[2] C. Brezinski, Projection Methods for Systems of Equations, North–Holland, Amsterdam, 1997.
[3] C. Brezinski, M. Redivo–Zaglia, The simplified topological ε–algorithms for accelerating sequences in a vector space, SIAM J. Sci. Comput., 36 (2014) A2227–A2247.
[4] C. Brezinski, Error estimates for the solution of linear systems. SIAM J. Sci. Comput., 21 (1999) 764-781.
[5] C. Brezinski, G. Rodriguez, S. Seatzu, Error estimates for linear systems with
applications to regularization, Numer. Algorithms, 49 (2008) 85-104.
[6] C. Brezinski, P. Fika, M. Mitrouli, Moments of a linear operator on a Hilbert
space, with applications to the trace of the inverse of matrices and the solution of
equations, Numer. Linear Algebra Appl., 19 (2012) 937-953.
[7] C. Brezinski, M. Redivo–Zaglia, The PageRank vector: properties, computation,
approximation, and acceleration, SIAM J. Matrix Anal. Appl., 28 (2006) 551-575.
[8] C. Brezinski, M. Redivo–Zaglia, Convergence acceleration of Kaczmarz’s method,
J. Eng. Math., DOI 10.1007/s10665-013-9656-3.
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