On Lanczos' algorithm for tri-diagonalization
TL;DR: It will be shown that there exists a vector x such that the algorithm starting from xι = jι = x can be continued so that "unlucky" case may not occur, and one of the initial vectors can be chosen arbitrarily to avoid this case.
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Abstract: The Lanczos algorithm transforming a given matrix into a tri-diagonal form is well known in numerical analysis and is discussed in many literatures. The possibility of this algorithm is shown in Rutishauser's excellent paper [ΊΓ]. However it seems to the author that no further theoretical consideration has been made since then. The process starts from a pair of trial vectors x\\ and yλ. A pair of the ί-th iterated vectors x{ and y{ can be constructed successively if γj*χjφθ ( l ^ y ^ i —1). Hence, if yp+ι*xp+ι = 0 for somep<Ln — 1, we must modify the algorithm so as to continue. This is possible in case where xp+ί = 0 or 7̂ +1 = 0, while any method of modification is not known in case where Λ ^ + I ^ O and yP+ιφΰ. We shall call the former case \"lucky\" and the latter \"unlucky\". The only thing for us to do in \"unlucky\" case is to choose new starting vectors xu j i and begin again in the hope that this case will not happen later. Rutishauser's result (Q8[] Satz 1) guarantees this possibility. In practical computation, however, \"unlucky\" case may occur after repeated modifications in \"lucky\" cases. Once we encountered with \"unlucky\" case, we have to abandon all the efforts made before and start again with new trial vectors (if we stick to the old knowledge). Then a question arises naturally: Is it actually necessary to go back to the first step? In this paper we shall treat this problem. Roughly speaking, the answer is as follows: It is sufficient to go back to the latest modification. As a special case of this result, we can show that one of the initial vectors can be chosen arbitrarily to avoid \"unlucky\" case. Further it will be shown that there exists a vector x such that the algorithm starting from xι = jι = x can be continued so that \"unlucky\" case may not occur. These results will be stated in Theorems 3-6 of §2 and a new procedure will be proposed at p. 279. Finally, in connection with the Lanczos algorithm, we shall give, in Appendix, some properties concerning the location of the eigenvalues of tri-diagonal matrices.
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Citations
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TL;DR: In this article, a review of the essential points of Lanczos's orthogonalization procedure is presented, which is of great importance for the determination of the eigenvalues of a real, but otherwise general matrix.
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The method of Lanczos for calculating the characteristic roots and vectors of a real symmetric matrix
R.A. Brooker,F.H. Sumner +1 more
- 01 Apr 1956
TL;DR: A method suggested by Lanczos for arriving at the reduced characteristic function of a matrix is drawn attention and it is shown how it may be used with an electronic computer to evaluate automatically the characteristic roots and vectors of a real symmetric matrix.
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