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

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**reviewer:** M. Znojil
**reviewernum:** 9689

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

**zblno:** DE015329997

**author:** Luk, Franklin T.; Qiao, Sanzheng:

**shorttitle:** A fast algorithm for Hankel matrices

**source:** Lin. Algebra Appl. 316, No. 1-3, 171-182 (2000)

**rpclass:** 65F15

**rsclass:** 15A23; 65T50

**keywords:** Hankel complex matrices; Lanczos tridiagonalization; complex orthogonal transformation; eigenvalues

**revtext:** A pioneering transfer of experience from the real symetric
matrices to their complex symmetric counterparts. NB: The idea
could find a broad applicability (cf. PT symmetric quantum
mechanics models in F. M. Fernandez et al, J. Phys. A: Math. Gen.
31 (1998) 10105 for a sample). For Hankel matrices of this type
the authors develop a fast algorithm for the determination of all
the eigenvalues. Starting from the concept of a complex
orthogonality and from a fast Hankel-vector multiplication the
paper suggests a combination of the Lanczos tridiagonalization
with QR diagonalization. Numerical illustrations are encouraging.
Several theoretical questions (cf. the loss of the
complex-orthogonality of the Lanczos vectors) remain open.

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