## Efficient reduction of banded Hermitian positive definite generalized eigenvalue problems to banded standard eigenvalue problems.(English)Zbl 1455.65055

The author proposes an algorithm for the reduction of the banded positive definite generalized eigenvalue problem $$Ax = Bx\lambda$$ to an equivalent standard eigenvalue problem, such that the banded structure is preserved. The given method combines ideas of an algorithm proposed by C. R. Crawford in [Commun. ACM 16, 41–44 (1973; Zbl 0247.65025)] and LAPACK’s reduction routines _{SY,HE}GST and retains their respective advantages. In addition, it includes two algorithmic parameters (block size, $$n_b$$, and width of blocked orthogonal transformations, $$w$$) that can be adjusted to optimize performance. A heuristic for automatically determining suitable values for these parameters was also presented. Numerical experiments confirm the efficiency of the new method.

### MSC:

 65F15 Numerical computation of eigenvalues and eigenvectors of matrices 65-04 Software, source code, etc. for problems pertaining to numerical analysis

Zbl 0247.65025

### Software:

LINPACK; LAPACK; Algorithm 679; BLAS; ELPA; SBR Toolbox
Full Text:

### References:

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