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  • Engineering General  (2)
  • Orthogonal polynomials  (2)
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  • 1
    Electronic Resource
    Electronic Resource
    Jacobi matrices for sums of weight functions (1992)
    Springer
    BIT 32 (1992), S. 143-166 
    Details
    ISSN: 1572-9125
    Keywords: primary 65F30 ; secondary 65D10 ; 65D15 ; 65D30 ; 65D32 ; Orthogonal polynomials ; Jacobi matrices ; orthogonal methods ; Lanczos methods ; sums of weight functions ; updating ; downdating
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Orthogonal polynomials are conveniently represented by the tridiagonal Jacobi matrix of coefficients of the recurrence relation which they satisfy. LetJ 1 andJ 2 be finite Jacobi matrices for the weight functionsw 1 andw 2, resp. Is it possible to determine a Jacobi matrix $$\tilde J$$ , corresponding to the weight functions $$\tilde w$$ =w 1+w 2 using onlyJ 1 andJ 2 and if so, what can be said about its dimension? Thus, it is important to clarify the connection between a finite Jacobi matrix and its corresponding weight function(s). This leads to the need for stable numerical processes that evaluate such matrices. Three newO(n 2) methods are derived that “merge” Jacobi matrices directly without using any information about the corresponding weight functions. The first can be implemented using any of the updating techniques developed earlier by the authors. The second new method, based on rotations, is the most stable. The third new method is closely related to the modified Chebyshev algorithm and, although it is the most economical of the three, suffers from instability for certain kinds of data. The concepts and the methods are illustrated by small numerical examples, the algorithms are outlined and the results of numerical tests are reported.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01995114
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  • 2
    Electronic Resource
    Electronic Resource
    Nonsymmetric Lanczos and finding orthogonal polynomials associated with indefinite weights (1991)
    Springer
    Numerical algorithms 1 (1991), S. 21-43 
    Details
    ISSN: 1572-9265
    Keywords: AMS (MOS) ; 42C05 ; 65F30 ; 68M15 ; 65D99 ; Orthogonal polynomials ; modified Chebyshev algorithm ; nonsymmetric Lanczos algorithm ; based fault tolerance
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract The nonsymmetric Lanczos algorithm reduces a general matrix to tridiagonal by generating two sequences of vectors which satisfy a mutual bi-orthogonality property. The process can proceed as long as the two vectors generated at each stage are not mutually orthogonal, otherwise the process breaks down. In this paper, we propose a variant that does not break down by grouping the vectors into clusters and enforcing the bi-orthogonality property only between different clusters, but relaxing the property within clusters. We show how this variant of the matrix Lanczos algorithm applies directly to a problem of computing a set of orthogonal polynomials and associated indefinite weights with respect to an indefinite inner product, given the associated moments. We discuss the close relationship between the modified Lanczos algorithm and the modified Chebyshev algorithm. We further show the connection between this last problem and checksum-based error correction schemes for fault-tolerant computing.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02145581
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  • 3
    Electronic Resource
    Electronic Resource
    Matrix shapes invariant under the symmetric QR algorithm (1995)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 2 (1995), S. 87-93 
    Details
    ISSN: 1070-5325
    Keywords: QR algorithm ; zero pattern ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The QR algorithm is a basic algorithm for computing the eigenvalues of dense matrices. For efficiency reasons it is prerequisite that the algorithm is applied only after the original matrix has been reduced to a matrix of a particular shape, most notably Hessenberg and tridiagonal, which is preserved during the iterative process. In certain circumstances a reduction to another matrix shape may be advantageous. In this paper, we identify which zero patterns of symmetric matrices are preserved under the QR algorithm.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nla.1680020203
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  • 4
    Electronic Resource
    Electronic Resource
    A Lanczos-based method for structural dynamic reanalysis problems (1994)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 37 (1994), S. 2857-2883 
    Details
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: In this paper, a new solution method for the modified eigenvalue problem with specific application to structural dynamic reanalysis is presented. The method, which is based on the block Lanczos algorithm, is developed for multiple low rank modifications to a system and calculates a few selected eigenpairs. Given the solution to the original system Ax = λx, procedures are developed for the modified standard eigenvalue Problem (A + ΔA)x̄ = λx̄, where 1ΔA = ΣjBSjBT, where Sj = SjT ∊ Rp × p, p ≪ n and B ∊ Rn × p is constant for all the perturbations Sj.2ΔA = ΣiΣj BiSjBiT, where Bi ∊ Rn × p may vary with the pertubations Sj.The procedures are then extended for the reciprocal and generalized eigenvalue problems so that they are directly applicable to the structural dynamic reanalysis problem. Numerical examples are given to demonstrate the applications of the method.
    Additional Material: 5 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nme.1620371610
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