An extension of the divide-and-conquer method for a class of symmetric block-tridiagonal eigenproblems

A divide-and-conquer method for computing eigenvalues and eigenvectors of a block-tridiagonal matrix with rank-one off-diagonal blocks is presented. The implications of unbalanced merging operations due to unequal block sizes are analyzed and illustrated with numerical examples. It is shown that an unfavorable order for merging blocks in the synthesis phase of the algorithm may lead to a significant increase of the arithmetic complexity. A strategy to determine a good merging order that is at least close to optimal in all cases is given. The method has been implemented and applied to test problems from a quantum chemistry application.

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

[2]  Peter Arbenz,et al.  Parallel divide and conquer algorithms for the symmetric tridiagonal eigenproblem , 1994 .

[3]  Jeffery D. Rutter.  A Serial Implementation of Cuppen''s Divide and Conquer Algorithm , 1991 .

[4]  Peter Arbenz,et al.  Divide and conquer algorithms for the bandsymmetric eigenvalue problem , 1992, Parallel Computing.

[5]  J. Pople,et al.  Approximate Self‐Consistent Molecular‐Orbital Theory. V. Intermediate Neglect of Differential Overlap , 1967 .

[6]  Ren-Cang Li.  Solving secular equations stably and efficiently , 1993 .

[7]  W. Gander,et al.  Restricted rank modification of the symmetric eigenvalue problem: Theoretical considerations , 1988 .

[8]  Jack Dongarra,et al.  LAPACK Users' Guide, 3rd ed. , 1999 .

[9]  J. Pople,et al.  Approximate Self‐Consistent Molecular Orbital Theory. III. CNDO Results for AB2 and AB3 Systems , 1966 .

[10]  Gene H. Golub,et al.  Matrix computations (3rd ed.) , 1996 .

[11]  J. Barlow.  Error analysis of update methods for the symmetric eigenvalue problem , 1993 .

[12]  Jeffery D. Rutter.  LAPACK Working Note 69: A Serial Implementation of Cuppen''s Divide and Conquer Algorithm for the Symmetric Eigenvalue Problem , 1994 .

[13]  J. Pople,et al.  Approximate Self‐Consistent Molecular Orbital Theory. II. Calculations with Complete Neglect of Differential Overlap , 1965 .

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

[15]  D. Sorensen,et al.  On the orthogonality of eigenvectors computed by divide-and-conquer techniques , 1991 .

[16]  David L. Beveridge,et al.  Approximate molecular orbital theory , 1970 .

[17]  G. Golub,et al.  On the spectral decomposition of Hermitian matrices modified by low rank perturbations , 1988 .

[18]  James Demmel,et al.  Applied Numerical Linear Algebra , 1997 .

[19]  Christoph W. Ueberhuber,et al.  A Low-Complexity Divide-and-Conquer Method for Computing Eigenvalues and Eigenvectors of Symmetric Band Matrices , 2001 .

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

[21]  Jack J. Dongarra,et al.  A Parallel Divide and Conquer Algorithm for the Symmetric Eigenvalue Problem on Distributed Memory Architectures , 1999, SIAM J. Sci. Comput..

[22]  J. Bunch,et al.  Rank-one modification of the symmetric eigenproblem , 1978 .

[23]  Ed Anderson,et al.  LAPACK Users' Guide , 1995 .

[24]  J. Pople,et al.  Approximate Self-Consistent Molecular Orbital Theory. I. Invariant Procedures , 1965 .

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