## Dept. of Mathematics and Computer Science (Berlin)

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**reviewer:** Znojil, Miloslav
**reviewernum:** 9689

**revieweremail:** znojil@ujf.cas.cz

**zblno:** DE015196101

**author:** Chandrasekaran, S.

**shorttitle:** Symmetric definite generalized eigenvalue algorithm

**source:** SIAM J. Matrix Anal. Appl. 21, No. 4, 1202-1228 (2000)

**rpclass:** 65F15

**rsclass:** 65K05; 15A18; 15A22; 15A23

**keywords:** generalized eigenvalues, eigenvectors,error analysis, deflation, perturbation bounds, pencils, stable algorithm

**revtext:** Generalized eigenvalue problem Ax = $\lambda$ Bx is considered
with symmetric A and B and positive definite B. The author
proposes a new algorithm. It is designed to combine the merits of
the strict simultaneous diagonalization of both A and B (often
violated when using the simple-minded factorization of B in finite
precision) and of keeping all the eigenvalues real (violated,
e.g., by QZ algorithm). Its essence lies in a factorization which
admits a deflation of the eigenvectors, roughly speaking, in a
decreasing order of their eigenvalues. This guarantees the
stability for, admittedly, singular A. Proofs are provided via an
extensive analysis of error propagation, incorporating an
interesting perturbation bound. The formalism is presented in
infinite precision, with subsequent demonstration of the
``survival" of the scheme when the (numerically) vanishing
eigenvalues have to be deflated as strict zeros. It is
conjectured that the number of operations (recomputations) grows
logarithmically with the precision. Efficiency and robustness are
demonstrated via numerical tests.

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