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Improving backward stability of Sakurai-Sugiura method with balancing technique in polynomial eigenvalue problem

Hongjia Chen;  Akira Imakura;  Tetsuya Sakurai — 2017

Applications of Mathematics

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...

A real-valued block conjugate gradient type method for solving complex symmetric linear systems with multiple right-hand sides

Yasunori Futamura;  Takahiro Yano;  Akira Imakura;  Tetsuya Sakurai — 2017

Applications of Mathematics

We consider solving complex symmetric linear systems with multiple right-hand sides. We assume that the coefficient matrix has indefinite real part and positive definite imaginary part. We propose a new block conjugate gradient type method based on the Schur complement of a certain 2-by-2 real block form. The algorithm of the proposed method consists of building blocks that involve only real arithmetic with real symmetric matrices of the original size. We also present the convergence property of...

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