Get The Lanczos Method: Evolution and Application PDF

By Louis Komzsik

ISBN-10: 0898715377

ISBN-13: 9780898715378

The Lanczos approach: Evolution and alertness is split into specific elements. the 1st half stories the evolution of 1 of the main regular numerical concepts within the undefined. the improvement of the strategy, because it turned extra powerful, is established via easy-to-understand algorithms. the second one half includes commercial purposes drawn from the author’s event. those chapters offer a special interplay among the numerical algorithms and their engineering functions. this can be a worthy reference for numerical analysts and engineers and will be used as a supplementary or reference textual content on the graduate point. Readers can be conversant in simple linear algebra and numerical research.

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Example text

Due to the finite precision arithmetic, after some number of steps the orthogonality among the Lanczos vectors was lost. This phenomenon may be avoided by additional work to maintain the orthogonality. The method of monitoring the convergence and the assessment of the accuracy of the solution, as well as the most important issue of the breakdown of the process, are discussed in this chapter. 14) are subject to round-off error: where fx, fy are the round-off errors accumulated in the step. 18). This ultimately may result in a breakdown of the process when 8k becomes or approaches zero.

For more details on this issue, including an algorithm to monitor the loss of orthogonality, see [20]; for the unsymmetric generalization, see [12]. 2 is recommended. 3 Reduction of block tridiagonal form Before we can turn the problem over to the tridiagonal solution phase, the block tridiagonal TJ matrix must be transformed to a scalar tridiagonal Tj form. The problem is to eliminate terms in the subdiagonal blocks 5, and in the diagonal blocks A{ below the subdiagonal. The method of Householder [22] or Givens [17] is well suited for this.

Block real symmetric Lanczos method Introducing the notation W j,k = Q Qj, reordering, and performing some algebra yields where G i,k = Qj Fk — Q F represents the local round-off error. The Wi+l,k term represents the global loss of orthogonality dependent on the losses having already occurred at previous steps that are contained in the other {W} terms on the right-hand side of the equation. A bound for the global loss is measured by where the right-hand side terms are available from earlier steps of the recurrence.

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The Lanczos Method: Evolution and Application by Louis Komzsik

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