program for bisection method in fortran 77

Program For Bisection Method In Fortran 77 For Free Public Full

A detailed exp osition of (skew-)Hamiltonian eigenv alue problems can be found in the survey Benner et al. 2004. This pap er should b e understo o d as a sequel to Benner et al. 2000, where the implementation of an implicit v ersion of the square-reduced method V an Loan 1984 has b een describ ed.

The implemented algorithms are based on or- thogonal symplectic decompositions, implying numerical backward stability as well as symmetry preservation for the computed eigenvalues.

These algorithms are supplemented with balancing and block algorithms, which can lead to considerable accuracy and performance improvements.

We demonstrate the usefulness of the subroutines for several, practically relevant examples.

Additionally, the subroutines DGESQB, DGESUB, DHAPVB, DHGPQR, DOSGSB, DOSMSB, and DSHPVB depend on the function ILAHAP for determining block parameters.

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Program For Bisection Method In Fortran 77 For Free Public Full

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These algorithms are supplemen ted with balancing and block algorithms, which can lead to considerable accuracy and p erformance improvemen ts.

As a by-product, an ecient implementation for computing symplectic QR decomp ositions is pro vided.

W e demonstrate the usefulness of the subroutines for several, practically relev ant examples.

INTRODUCTION This article describes F ortran 77 subroutines for computing eigen v alues and in v ari- ant subspaces of a Hamiltonian matrix H A G Q A T, G G T, Q Q T (1) where A, G, Q R n n.

Our subroutines are based on a backw ard stable, structure- exploiting method developed by Benner, Mehrmann and Xu 1998; 1997.

W e This work was supported b y the DF G Researc h Center Mathematics for k ey tec hnologies (FZT 86) in Berlin.

T o copy otherwise, to republish, to p ost on servers, or to redistribute to lists requires prior sp ecic p ermission andor a fee.

The Hamiltonian eigenv alue problem has a num b er of applications in systems and control theory, see e.g.

Benner et al. 2000; Benner et al.

Additionally pro vided are subroutines for computing eigen v alues and inv ariant subspaces of a skew-Hamiltonian matrix W A G Q A T, G G T, Q Q T, (2) where again A, G, Q R n n.

They can b e used to address complex Hamiltonian eigenv alue problems, see Benner et al.

Section 2.4, and might be as well of independent in terest, e.g., for solving certain matrix Riccati equations Stefano vski and T ren cevski 1998 and quadratic eigenv alue problems Mehrmann and W atkins 2000; Tisseur and Meerbergen 2001.

A Hamiltonian matrix H is equiv alently dened by the prop erty H J ( H J ) T, where J 0 I n I n 0.

Likewise, a matrix W is skew-Hamiltonian if and only if W J ( W J ) T.

Notably, the eigenv alues of H are symmetric with resp ect to the imaginary axis, and the eigenv alues of W hav e ev en algebraic and geometric m ultiplicities.

In principle, the eigenv alues of H and W could b e obtained by any general- purpose method for computing eigenv alues of general matrices, e.g., the QR algo- rithm Golub and V an Loan 1996, Sect.

As suc h a metho d, ho wev er, does not exploit the structures of H and W it cannot b e exp ected to preserve the spectral properties induced by the structure in nite precision arithmetic.

Skew-)Hamiltonian structures are preserved if symplectic similarity transfor- mations are used.

In the interest of numerical stability the emplo yed similarity transformations should b e or- thogonal as well.

An algorithm solely based on orthogonal and symplectic similarity transformations is str ongly b ackwar d stable Bunc h 1987, i.e., the computed eigen- v alues are the exact eigenv alues of a slightly p erturbed (sk ew-)Hamiltonian matrix.

Such an algorithm is known for computing the eigenv alues of a skew-Hamiltonian matrix but so far no completely satisfactory algorithm has been found for addressing the Hamiltonian eigenv alue problem.

A detailed exp osition of (skew-)Hamiltonian eigenv alue problems can be found in the survey Benner et al.

This pap er should b e understo o d as a sequel to Benner et al.

V an Loan 1984 has b een describ ed.