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On using the minimum spanning tree algorithm for optimal secant approximation of derivatives
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- Burdakov, Oleg, 1953- (author)
- CERFACS, Parallel Algorithms Team, Toulouse, France
- (creator_code:org_t)
- Wiley, 1996
- 1996
- English.
- In: Zeitschrift für angewandte Mathematik und Mechanik. - : Wiley. - 0044-2267 .- 1521-4001. ; 76:S3, s. 389-390
- Journal article (peer-reviewed)
Abstract
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- The following problem is considered. Given m + 1 points {xi}0m in Rn which generate an m-dimensional linear manifold, construct for this manifold a maximally linearly independent basis that consists of vectors of the form xi - xj. This problem is present in, e.g., stable variants of the secant method, where it is required to approximate the Jacobian matrix f' of a nonlinear mapping f by using values off computed at m + 1 points. In this case, it is also desirable to have a combination of finite differences with maximal linear independence. As a natural measure of linear independence, we consider a functional which is maximized to find an optimal combination of m pairs {xi, xj}. We show that the problem is not of combinatorial complexity but can be reduced to the minimum spanning tree (MST) problem, which is solved by an MST-type algorithm in O(m2n) time.
Subject headings
- NATURVETENSKAP -- Matematik -- Beräkningsmatematik (hsv//swe)
- NATURAL SCIENCES -- Mathematics -- Computational Mathematics (hsv//eng)
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