How can the largest eigenvalues be obtained from a PCG loop?

klausb · July 16, 2024, 09:20

The paper "AN OPTIMIZED CHEBYSHEV SMOOTHER IN GAMG SOLVER OF OPENFOAM ON SUNWAY TAIHULIGHT SUPERCOMPUTER" states, that "...the PCG loops are used to obtain the largest eigenvalues since it has same upper bound as for the Chebyshev iteration in the symmetric case...".

How can the largest eigenvalues be obtained from a PCG loop?

Link to the paper: https://dcwan.sjtu.edu.cn/OpenFOAM20...W13-25-133.pdf

dlahaye · February 25, 2025, 13:31

Apologies for spotting this month too late.

The CG algorithm can be instructed to compute the Ritz values. These Ritz values approximate the eigenvalues of the matrix. This is known as the Lanczos algorithm, see https://en.wikipedia.org/wiki/Lanczos_algorithm. See also the books the Youssef Saad.

July 16, 2024, 09:20	How can the largest eigenvalues be obtained from a PCG loop?	#1
klausb Senior Member Klaus Join Date: Mar 2009 Posts: 289 Rep Power: 23	The paper "AN OPTIMIZED CHEBYSHEV SMOOTHER IN GAMG SOLVER OF OPENFOAM ON SUNWAY TAIHULIGHT SUPERCOMPUTER" states, that "...the PCG loops are used to obtain the largest eigenvalues since it has same upper bound as for the Chebyshev iteration in the symmetric case...". How can the largest eigenvalues be obtained from a PCG loop? Link to the paper: https://dcwan.sjtu.edu.cn/OpenFOAM20...W13-25-133.pdf

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February 25, 2025, 13:31		#2
dlahaye Senior Member Domenico Lahaye Join Date: Dec 2013 Posts: 836 Blog Entries: 1 Rep Power: 19	Apologies for spotting this month too late. The CG algorithm can be instructed to compute the Ritz values. These Ritz values approximate the eigenvalues of the matrix. This is known as the Lanczos algorithm, see https://en.wikipedia.org/wiki/Lanczos_algorithm. See also the books the Youssef Saad.