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On the fast Lanczos method for computation of eigenvalues of Hankel matrices using multiprecision arithmetics
journal contribution
posted on 2016-05-01, 00:00 authored by Shaun BangayShaun Bangay, Gleb BeliakovGleb BeliakovThe 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.
History
Journal
Numerical linear algebra with applicationsVolume
23Issue
3Pagination
485 - 500Publisher
John Wiley & SonsLocation
Chichester, Eng.Publisher DOI
ISSN
1070-5325eISSN
1099-1506Language
engPublication classification
C Journal article; C1 Refereed article in a scholarly journalCopyright notice
2016, John Wiley & SonsUsage metrics
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