A Stable Divide and Conquer Algorithm for the Unitary Eigenproblem

We present a divide and conquer algorithm for computing the eigendecomposition of a unitary upper Hessenberg matrix H. Previous divide and conquer approaches suffer a potential loss of orthogonality among the computed eigenvectors of H. Using a backward stable method based on previous work by Gu and Eisenstat in the rank-one modification of the symmetric eigenproblem, our algorithm provides a backward stable method for computing the eigenvectors. The method also compares well against the efficiency of other available methods.

[1]  S. Eisenstat,et al.  A Stable and Efficient Algorithm for the Rank-One Modification of the Symmetric Eigenproblem , 1994, SIAM J. Matrix Anal. Appl..

[2]  Douglas M. Priest,et al.  Algorithms for arbitrary precision floating point arithmetic , 1991, [1991] Proceedings 10th IEEE Symposium on Computer Arithmetic.

[3]  Danny C. Sorensen,et al.  An implementation of a divide and conquer algorithm for the unitary eigen problem , 1992, TOMS.

[4]  Ji-Guang Sun.  Residual Bounds on Approximate Solutions for the Unitary Eigenproblem , 1996, SIAM J. Matrix Anal. Appl..

[5]  William B. Gragg,et al.  The QR algorithm for unitary Hessenberg matrices , 1986 .

[6]  J. Cuppen.  A divide and conquer method for the symmetric tridiagonal eigenproblem , 1980 .

[7]  L. Reichel,et al.  On the eigenproblem for orthogonal matrices , 1986, 1986 25th IEEE Conference on Decision and Control.

[8]  James Demmel,et al.  LAPACK Users' Guide, Third Edition , 1999, Software, Environments and Tools.

[9]  L. Reichel,et al.  A divide and conquer method for unitary and orthogonal eigenproblems , 1990 .

[10]  William B. Gragg,et al.  Constructing a Unitary Hessenberg Matrix from Spectral Data , 1991 .

[11]  Jack J. Dongarra,et al.  A fully parallel algorithm for the symmetric eigenvalue problem , 1985, PPSC.

[12]  Lothar Reichel,et al.  Determination Of Pisarenko Frequency Estimates As Eigenvalues Of An Orthogonal Matrix , 1988, Optics & Photonics.

[13]  Gene H. Golub,et al.  Matrix computations , 1983 .

[14]  Stanley C. Eisenstat,et al.  A Divide-and-Conquer Algorithm for the Symmetric Tridiagonal Eigenproblem , 1995, SIAM J. Matrix Anal. Appl..