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Lanczos algorithm
The Lanczos algorithm is an iterative algorithm devised by Cornelius Lanczos〔Lanczos, C. "An iteration method for the solution of the eigenvalue problem of linear differential and integral operators", J. Res. Nat’l Bur. Std. 45, 225-282 (1950).〕 that is an adaptation of power methods to find the most useful eigenvalues and eigenvectors of an order linear system with a limited number of operations, , where is much smaller than . Although computationally efficient in principle, the method as initially formulated was not useful, due to its numerical instability. In 1970, Ojalvo and Newman 〔Ojalvo, I.U. and Newman, M.,"Vibration modes of large structures by an automatic matrix-reduction method", AIAA J., 8 (7), 1234-1239 (1970).〕 showed how to make the method numerically stable and applied it to the solution of very large engineering structures subjected to dynamic loading. This was achieved using a method for purifying the vectors to any degree of accuracy, which when not performed, produced a series of vectors that were highly contaminated by those associated with the lowest natural frequencies. In their original work, these authors also suggested how to select a starting vector (i.e. use a random number generator to select each element of the starting vector) and suggested an empirically determined method for determining , the reduced number of vectors (i.e. it should be selected to be approximately 1 ½ times the number of accurate eigenvalues desired). Soon thereafter their work was followed by Paige 〔Paige, C.C., "The computation of eigenvalues and eigenvectors of very large sparse matrices", the U. of London Ph.D. thesis (1971).〕〔Paige, C.C., "Computational variants of the Lanczos method for the eigenproblem", J. Inst. Maths Applics 10, 373-381 (1972).〕 who also provided an error analysis. In 1988, Ojalvo 〔Ojalvo, I.U., "Origins and advantages of Lanczos vectors for large dynamic systems", Proc. 6th Modal Analysis Conference (IMAC), Kissimmee, FL, 489-494 (1988).〕 produced a more detailed history of this algorithm and an efficient eigenvalue error test. Currently, the method is widely used in a variety of technical fields and has seen a number of variations.
==Power method for finding eigenvalues==
(詳細はeigendecomposition of , then . As gets very large, the diagonal matrix of eigenvalues will be dominated by whichever eigenvalue is largest (neglecting the case of two or more equally large eigenvalues, of course). As this happens, will converge to the largest eigenvalue and to the associated eigenvector. If the largest eigenvalue is multiple, then will converge to a vector in the subspace spanned by the eigenvectors associated with those largest eigenvalues. Having found the first eigenvector/value, one can then successively restrict the algorithm to the null space of the known eigenvectors to get the second largest eigenvector/values and so on.
In practice, this simple algorithm does not work very well for computing very many of the eigenvectors because any round-off error will tend to introduce slight components of the more significant eigenvectors back into the computation, degrading the accuracy of the computation. Pure power methods also can converge slowly, even for the first eigenvector.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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