On the fast Lanczos method for computation of eigenvalues of Hankel matrices using multiprecision arithmetics

Bangay, Shaun and Beliakov, Gleb 2016, On the fast Lanczos method for computation of eigenvalues of Hankel matrices using multiprecision arithmetics, Numerical linear algebra with applications, vol. 23, no. 3, pp. 485-500, doi: 10.1002/nla.2035.

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Title On the fast Lanczos method for computation of eigenvalues of Hankel matrices using multiprecision arithmetics
Author(s) Bangay, Shaun
Beliakov, GlebORCID iD for Beliakov, Gleb orcid.org/0000-0002-9841-5292
Journal name Numerical linear algebra with applications
Volume number 23
Issue number 3
Start page 485
End page 500
Total pages 16
Publisher John Wiley & Sons
Place of publication Chichester, Eng.
Publication date 2016-05
ISSN 1070-5325
1099-1506
Keyword(s) eigenvalues
Hankel matrix
Toeplitz matrix
Lanczos method
multiprecision arithmetics
Summary The use of the fast Fourier transform (FFT) accelerates Lanczos tridiagonalisation method for Hankel and Toeplitz matrices by reducing the complexity of matrix-vector multiplication. In multiprecision arithmetics, the FFT has overheads that make it less competitive compared with alternative methods when the accuracy is over 10000 decimal places. We studied two alternative Hankel matrix-vector multiplication methods based on multiprecision number decomposition and recursive Karatsuba-like multiplication, respectively. The first method was uncompetitive because of huge precision losses, while the second turned out to be five to 14 times faster than FFT in the ranges of matrix sizes up to n = 8192 and working precision of b = 32768 bits we were interested in. We successfully applied our approach to eigenvalues calculations to studies of spectra of matrices that arise in research on Riemann zeta function. The recursive matrix-vector multiplication significantly outperformed both the FFT and the traditional multiplication in these studies.
Language eng
DOI 10.1002/nla.2035
Field of Research 010301 Numerical Analysis
080205 Numerical Computation
Socio Economic Objective 970101 Expanding Knowledge in the Mathematical Sciences
HERDC Research category C1 Refereed article in a scholarly journal
ERA Research output type C Journal article
Copyright notice ©2016, John Wiley & Sons
Persistent URL http://hdl.handle.net/10536/DRO/DU:30085167

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