Algorithm 800
swMATH ID:  4405 
Software Authors:  Benner, Peter; Byers, Ralph; Barth, Eric 
Description:  Algorithm 800. Fortran 77 subroutines for computing the eigenvalues of Hamiltonian matrices. I: The squarereduced method. The authors describe a set of LAPACKbased Fortran 77 subroutines for reducing a Hamiltonian matrix to squarereduced form using orthogonal symplectic transformations and for approximating all of its eigenvalues using an implicit version of Van Loan’s method. This approach requires less than half as many FLOPS as are needed by the general QR algorithm. Routines are provided for computing the squarereduced form, computing the eigenvalues using the output from the first routine, computing the eigenvalues (this routine is easy to use), and symplectic and norm scaling. There is a comprehensive discussion of the numerical tests used, which include several applications in control theory. Tests on randomly generated matrices confirm that the reduction in FLOPS is reflected in a similar reduction in execution times compared to the general QR routine, DGEEVX, from the LAPACK library. The authors’ software also preserves the plusminus eigenvalue pairs of the Hamiltonian, which are destroyed by rounding errors using DGEEVX. The eigenvalues of larger magnitude are computed to an accuracy in line with their condition numbers; very small eigenvalues may be perturbed by up to the square root of the unit rounding error. Such a loss of accuracy may sometimes (but not always) be avoided by scaling. Of the three scaling strategies mentioned (norm, symplectic, and Hessenberg), norm scaling was found to be the least effective in practice. (review ACM) 
Homepage:  http://dl.acm.org/citation.cfm?id=347837.347852 
Programming Languages:  Fortran 77 
Keywords:  algorithms; documentation; performance 
Related Software:  LAPACK; SLICOT; Algorithm 854; CAREX; mctoolbox; ARPACK; EISPACK; Matlab; Yambo; mftoolbox; ELPA; BerkeleyGW; DMSRIC; DGRSVX; DSUBSP; HQR3; JDQZ; CTDSX; JDQR; BLAS 
Cited in:  11 Publications 
Standard Articles
1 Publication describing the Software, including 1 Publication in zbMATH  Year 

Algorithm 800. Fortran 77 subroutines for computing the eigenvalues of Hamiltonian matrices. I: The squarereduced method. Zbl 1137.65338 Benner, Peter; Byers, Ralph; Barth, Eric 
2000

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Cited by 12 Authors
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Cited in 6 Serials
Cited in 5 Fields
11  Numerical analysis (65XX) 
2  History and biography (01XX) 
2  Linear and multilinear algebra; matrix theory (15XX) 
2  Computer science (68XX) 
1  Systems theory; control (93XX) 