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A greedy algorithm ...
A greedy algorithm for the optimal basis problem
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- Burdakov, Oleg, 1953- (författare)
- Parallel Algorithms Team, CERFACS, Toulouse Cedex, France
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(creator_code:org_t)
- Springer, 1997
- 1997
- Engelska.
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Ingår i: BIT Numerical Mathematics. - : Springer. - 0006-3835 .- 1572-9125. ; 37:3, s. 591-599
- Relaterad länk:
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https://urn.kb.se/re...
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https://doi.org/10.1...
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Abstract
Ämnesord
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- The following problem is considered. Given m+1 points {x i }0 m in R n which generate an m-dimensional linear manifold, construct for this manifold a maximally linearly independent basis that consists of vectors of the form x i −x j . This problem is present in, e.g., stable variants of the secant and interpolation methods, where it is required to approximate the Jacobian matrix f′ of a nonlinear mappingf 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 the hadamard condition number which is minimized to find an optimal combination of m pairs {x i ,x j }. We show that the problem is not NP-hard, but can be reduced to the minimum spanning tree problem, which is solved by the greedy algorithm in O(m 2) time. The complexity of this reduction is equivalent to one m×n matrix-matrix multiplication, and according to the Coppersmith-Winograd estimate, is below O(n 2.376) for m=n. Applications of the algorithm to interpolation methods are discussed.
Ämnesord
- NATURVETENSKAP -- Matematik -- Beräkningsmatematik (hsv//swe)
- NATURAL SCIENCES -- Mathematics -- Computational Mathematics (hsv//eng)
Nyckelord
- Optimal basis problem; the Hadamard condition number; minimum spanning tree problem; greedy algorithm; secant approximation of derivatives; interpolation methods
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