Shift-and-invert Lanczos algorithm for ideal MHD stability analysis Original Research Article

CAS3D and TERPSICHORE have been designed to analyze the global ideal MHD stability of 3-D equilibria. Their critical part is to solve for the smallest eigenvalue and its corresponding eigenvector of a large but sparse, real symmetric band matrix. This matrix is usually indefinite. In CAS3D inverse iteration has been applied to do this, where the spectral shift is given by the EISPACK eigenvalue solver. It has been shown that the application of such kind of software becomes very expensive in the sense of flops and storage when matrix order and bandwidth become very large. Here, this problem is resolved by using the Lanczos algorithm with no reorthogonalization which is economical in flops and storage. A shift is applied to transform the matrix into a positive definite one. So an efficient Cholesky factorization can be constructed in invert Lanczos recursion. The version of CAS3D2MN with the shift-and-invert Lanczos algorithm is called CAS3D2MNvl. Practical calculations in CAS3D2MNvl indicate that the shift-and-invert strategy is reliable and efficient. An increase in memory has usually been kept below 0.8%, and only 15 ∼ 20 steps are needed to obtain the smallest eigenvalue. As compared with the EISPACK subroutine, the memory requirement is much smaller and CPU time is saved significantly by factors of 50 ∼ 100. Finally finite-mode-number ballooning modes in 3-D MHD equilibria have been discussed briefly.