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UBC Theses and Dissertations

Determining the complete spectrum for sparse symmetric matrices Cavers, Ian Alfred


This dissertation studies a restricted form of the fundamental algebraic eigenvalue prob lem. From the broad spectrum of eigenvalue problems and solution methods, it focuses upon sequential direct methods for determining moderately large subsets of eigenvalues or the complete spectrum of large sparse symmetric matrices. The thesis uses a combination of theoretical analysis and experimentation with symbolic and numeric implementations to develop generally applicable, reliable, efficient and accurate algorithms that are easily applied by novice and expert practitioners alike. This dissertation’s approach is to reexam- ine eigenvalue methods based on the similarity reduction of matrices to tridiagonal form, developing algorithms that more fully exploit matrix sparsity. Using specially developed sparse reduction tools, the thesis identifies the deficiencies and limitations of existing direct tridiagonalization methods, providing an improved un derstanding of the underlying fill characteristics of sparse reductions. The best previ ously published approach combines a bandwidth reducing preordering with Rutishauser and Schwarz’s O(bn2) band-preserving tridiagonalization algorithm. This approach places complete reliance upon the preordering to exploit sparsity, but it typically leaves the band of the matrix relatively sparse prior to reduction. The thesis presents several novel sparse reduction algorithms, including the hybrid tridiagonalization methods HYBBC and HYB SBC, that rearrange the elimination of nonzero entries to improve band sparsity utilization. HYBBC combines Bandwidth Contraction, a diagonally-oriented sparse reduction, with Rutishauser and Schwarz’s column-oriented tridiagonalization. For a wide range of 70 prac- tical sparse problems the new algorithm reduces CPU requirements by an average of 31%, with reductions as high as 63%. HYBSBC improves upon HYBBC’s successful techniques by substituting the novel Split Bandwidth Contraction algorithm for Bandwidth Contraction. The Split Bandwidth Contraction algorithm takes additional advantage of band sparsity to significantly improve the efficiency of partial bandwidth contractions. In addition, HYB- SBC employs the Z-transition strategy to precisely regulate the transition between its two reduction stages, permitting tridiagonalization in as little as 1/5 the time of Rutishauser and Schwarz. Finally, to demonstrate the relative efficiency of sparse tridiagonalization based eigenvalue methods, the thesis compares variants of the Lanczos algorithm to HYBSBC using theoretical analysis and experimentation with leading Lanczos codes.

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