Variable-step preconditioned conjugate gradient method for partial ...

method can be used instead of the direct eigensolver when a small number of eigenpairs is required,
i.e., p < √ n.
Table 10. Number of iterations for LOBPCG-VS method with AMG preconditioner
N = 32 MG eigensolver LOBPCG-VS method with
p = m = 5 from [19, 20] DIAG ICF AMG
|λh,1 − λ1| 0.15763 ·10 −9 0.208 ·10 −11 0.379 ·10 −4 0.184 ·10 −17
|λh,2 − λ2| 0.82750 ·10 −9 0.544 ·10 −8 0.427 ·10 −9 0.638 ·10 −10
|λh,3 − λ3| 0.82751 ·10 −9 0.103 ·10 −5 0.817 ·10 −9 0.859 ·10 −9
|λh,4 − λ4| 0.82751 ·10 −9 0.537 ·10 −5 0.253 ·10 −5 0.139 ·10 −5
|λh,5 − λ5| 0.23087 ·10 −8 0.516 ·10 −6 0.246 ·10 −6 0.101 ·10 −6
�Av1 − λ1v1� 0.32745 ·10 −4 0.134 ·10 −5 0.004 ·10 −1 0.177 ·10 −8
�Av2 − λ2v2� 0.76292 ·10 −4 0.686 ·10 −4 0.801 ·10 −5 0.851 ·10 −5
�Av3 − λ3v3� 0.76293 ·10 −4 0.870 ·10 −3 0.441 ·10 −4 0.350 ·10 −4
�Av4 − λ4v4� 0.76293 ·10 −4 0.183 ·10 −2 0.136 ·10 −2 0.106 ·10 −2
�Av5 − λ5v5� 0.12983 ·10 −3 0.480 ·10 −3 0.197 ·10 −3 0.268 ·10 −3
The results in Table 10 shows that the present method with muligrid-like preconditioner give
us an optimal eigensolver for finding a few number of smallest eigenpairs of the discretization
matrix. Moreover, the LOBPCG-VS method has a very big adgvantage with respect to other
modern optimal eigensolvers. It is very simple for implementation and, it is more important, one
can use the modern software packages developed for solving linear system of equations to include
them as corresponding variable-step preconditioner for solving the partial eigenproblem.
References
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