Geodesic
Improving backward stability of Sakurai-Sugiura method with balancing technique in polynomial eigenvalue problem
Applications of Mathematics, Tome 62 (2017) no. 4, pp. 357-375

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One of the most efficient methods for solving the polynomial eigenvalue problem (PEP) is the Sakurai-Sugiura method with Rayleigh-Ritz projection (SS-RR), which finds the eigenvalues contained in a certain domain using the contour integral. The SS-RR method converts the original PEP to a small projected PEP using the Rayleigh-Ritz projection. However, the SS-RR method suffers from backward instability when the norms of the coefficient matrices of the projected PEP vary widely. To improve the backward stability of the SS-RR method, we combine it with a balancing technique for solving a small projected PEP. We then analyze the backward stability of the SS-RR method. Several numerical examples demonstrate that the SS-RR method with the balancing technique reduces the backward error of eigenpairs of PEP.
One of the most efficient methods for solving the polynomial eigenvalue problem (PEP) is the Sakurai-Sugiura method with Rayleigh-Ritz projection (SS-RR), which finds the eigenvalues contained in a certain domain using the contour integral. The SS-RR method converts the original PEP to a small projected PEP using the Rayleigh-Ritz projection. However, the SS-RR method suffers from backward instability when the norms of the coefficient matrices of the projected PEP vary widely. To improve the backward stability of the SS-RR method, we combine it with a balancing technique for solving a small projected PEP. We then analyze the backward stability of the SS-RR method. Several numerical examples demonstrate that the SS-RR method with the balancing technique reduces the backward error of eigenpairs of PEP.
DOI : 10.21136/AM.2017.0016-17
Classification : 15A18, 65F15
Keywords: SS-RR method; polynomial eigenvalue problem; balancing technique 
Chen, Hongjia; Imakura, Akira; Sakurai, Tetsuya. Improving backward stability of Sakurai-Sugiura method with balancing technique in polynomial eigenvalue problem. Applications of Mathematics, Tome 62 (2017) no. 4, pp. 357-375. doi: 10.21136/AM.2017.0016-17
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     title = {Improving backward stability of {Sakurai-Sugiura} method with balancing technique in polynomial eigenvalue problem},
     journal = {Applications of Mathematics},
     pages = {357--375},
     year = {2017},
     volume = {62},
     number = {4},
     doi = {10.21136/AM.2017.0016-17},
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     zbl = {06770049},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/AM.2017.0016-17/}
}
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