جستجوی مقالات مرتبط با کلیدواژه
lanczos algorithm
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تکرار جستجوی کلیدواژه lanczos algorithm در مقالات مجلات علمی
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In this paper, we introduce a new algorithm for constructing a symmetric pentadiagonal matrix by using three interlacing spectrum, say $(\lambda_i)_{i=1}^n$, $(\mu_i)_{i=1}^n$ and $(\nu_i)_{i=1}^n$ such that\begin{eqnarray*}0<\lambda_1<\mu_1<\lambda_2<\mu_2<...<\lambda_n<\mu_n,\\\mu_1<\nu_1<\mu_2<\nu_2<...<\mu_n<\nu_n,\end{eqnarray*}where $(\lambda_i)_{i=1}^n$ are the eigenvalues of pentadiagonal matrix $A$, $(\mu_i)_{i=1}^n$ are the eigenvalues of $A^*$ (the matrix $A^*$ differs from $A$ only in the $(1,1)$ entry) and $(\nu_i)_{i=1}^n$ are the eigenvalues of $A^{**}$ (the matrix $A^{**}$ differs from $A^*$ only in the $(2,2)$ entry). From theinterlacing spectrum, we find the first and second columns of eigenvectors. Sufficient conditions for the solvability of the problem are given. Then we construct the pentadiagonal matrix $A$ from these eigenvectors and given eigenvalues by using the block Lanczos algorithm. We also give an example to demonstrate the efficiency of the algorithm.Keywords: Inverse eigenvalue problem, Pentadiagonal matrix, Interlacing property, Lanczos algorithm
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Iranian Journal of science and Technology (A: Siences), Volume:37 Issue: 3, Summer 2013, PP 349 -358Lanczos-type algorithms are well known for their inherent instability. They typically breakdown occurs when relevant orthogonal polynomials do not exist. Current approaches to curing breakdown rely on jumping over the non-existent polynomials to resume computation. This may have to be used many times during the solution process. We suggest an alternative to jumping, which consists of restarting the algorithms that fail. Three different strategies can be taken: (ST1) Restarting following breakdown of the algorithm in use; (ST2) pre-emptive restarting after a fixed number of iterations; (ST3) restarting when near breakdown is detected through monitoring. We describe a restarting framework with a generic algorithm that invokes one or the other of the three strategies suggested. Four of the most prominent recently developed Lanczos-type algorithms namely, and will be presented and then deployed in the restarting framework. However, we will only report on results obtained with strategy ST2 as it is the only viable one at the moment.Keywords: Lanczos algorithm, Systems of Linear Equations, Formal Orthogonal Polynomials, Restarting, Switching, Breakdown
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