%0 Journal Article %1 Ericsson_1980 %A Ericsson, Thomas %A Ruhe, Axel %D 1980 %I American Mathematical Society (AMS) %J Mathematics of Computation %K 15a18-eigenvalues-singular-values-and-eigenvectors 65f15-numerical-eigenvalues-eigenvectors 65n30-pdes-bvps-finite-elements %N 152 %P 1251–1268 %R 10.1090/s0025-5718-1980-0583502-2 %T The spectral transformation Lánczos method for the numerical solution of large sparse generalized symmetric eigenvalue problems %U https://www.ams.org/journals/mcom/1980-35-152/S0025-5718-1980-0583502-2/ %V 35 %X A new algorithm is developed which computes a specified number of eigenvalues in any part of the spectrum of a generalized symmetric matrix eigenvalue problem. It uses a linear system routine (factorization and solution) as a tool for applying the Lanczos algorithm to a shifted and inverted problem. The algorithm determines a sequence of shifts and checks that all eigenvalues get computed in the intervals between them. It is shown that for each shift several eigenvectors will converge after very few steps of the Lanczos algorithm, and the most effective combination of shifts and Lanczos runs is determined for different sizes and sparsity properties of the matrices. For large problems the operation counts are about five times smaller than for traditional subspace iteration methods. Tests on a numerical example, arising from a finite element computation of a nuclear power piping system, are reported, and it is shown how the performance predicted bears out in a practical situation.

@article{Ericsson_1980, abstract = {A new algorithm is developed which computes a specified number of eigenvalues in any part of the spectrum of a generalized symmetric matrix eigenvalue problem. It uses a linear system routine (factorization and solution) as a tool for applying the Lanczos algorithm to a shifted and inverted problem. The algorithm determines a sequence of shifts and checks that all eigenvalues get computed in the intervals between them. It is shown that for each shift several eigenvectors will converge after very few steps of the Lanczos algorithm, and the most effective combination of shifts and Lanczos runs is determined for different sizes and sparsity properties of the matrices. For large problems the operation counts are about five times smaller than for traditional subspace iteration methods. Tests on a numerical example, arising from a finite element computation of a nuclear power piping system, are reported, and it is shown how the performance predicted bears out in a practical situation. }, added-at = {2020-02-13T06:18:47.000+0100}, author = {Ericsson, Thomas and Ruhe, Axel}, biburl = {https://www.bibsonomy.org/bibtex/2046de70ce7f52ff92e370f9a00d73b82/gdmcbain}, doi = {10.1090/s0025-5718-1980-0583502-2}, interhash = {b441378ac01bf3e1827e7397ffce2181}, intrahash = {046de70ce7f52ff92e370f9a00d73b82}, journal = {Mathematics of Computation}, keywords = {15a18-eigenvalues-singular-values-and-eigenvectors 65f15-numerical-eigenvalues-eigenvectors 65n30-pdes-bvps-finite-elements}, number = 152, pages = {1251–1268}, publisher = {American Mathematical Society ({AMS})}, timestamp = {2020-02-13T06:18:47.000+0100}, title = {The spectral transformation Lánczos method for the numerical solution of large sparse generalized symmetric eigenvalue problems}, url = {https://www.ams.org/journals/mcom/1980-35-152/S0025-5718-1980-0583502-2/}, volume = 35, year = 1980 }