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  • Engineering General  (2)
  • Lanczos methods  (2)
  • 65F15  (1)
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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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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    Estimating the largest singular values of large sparse matrices via modified moments (1991)
    Springer
    Numerical algorithms 1 (1991), S. 353-373 
    Details
    ISSN: 1572-9265
    Keywords: 65D32 ; 65F15 ; 65F20 ; 65F50 ; 65Y05 ; Chebyshev ; modified moments ; singular value decomposition ; sparse
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We describe a procedure for determining a few of the largest singular values of a large sparse matrix. The method by Golub and Kent which uses the method of modified moments for estimating the eigenvalues of operators used in iterative methods for the solution of linear systems of equations is appropriately modified in order to generate a sequence of bidiagonal matrices whose singular values approximate those of the original sparse matrix. A simple Lanczos recursion is proposed for determining the corresponding left and right singular vectors. The potential asynchronous computation of the bidiagonal matrices using modified moments with the iterations of an adapted Chebyshev semi-iterative (CSI) method is an attractive feature for parallel computers. Comparisons in efficiency and accuracy with an appropriate Lanczos algorithm (with selective re-orthogonalization) are presented on large sparse (rectangular) matrices arising from applications such as information retrieval and seismic reflection tomography. This procedure is essentially motivated by the theory of moments and Gauss quadrature.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02142380
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    Adaptive Lanczos methods for recursive condition estimation (1991)
    Springer
    Numerical algorithms 1 (1991), S. 1-19 
    Details
    ISSN: 1572-9265
    Keywords: AMS(MOS) ; 15A18 ; 65F10 ; 65F20 ; 65F35 ; Adaptive methods ; condition estimation ; control ; downdating ; eigenvalues ; Lanczos methods ; matrix modifications ; recursive least squares ; signal processing ; singular values ; updating
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Estimates for the condition number of a matrix are useful in many areas of scientific computing, including: recursive least squares computations, optimization, eigenanalysis, and general nonlinear problems solved by linearization techniques where matrix modification techniques are used. The purpose of this paper is to propose anadaptiveLanczosestimator scheme, which we callale, for tracking the condition number of the modified matrix over time. Applications to recursive least squares (RLS) computations using the covariance method with sliding data windows are considered.ale is fast for relatively smalln-parameter problems arising in RLS methods in control and signal processing, and is adaptive over time, i.e., estimates at timet are used to produce estimates at timet+1. Comparisons are made with other adaptive and non-adaptive condition estimators for recursive least squares problems. Numerical experiments are reported indicating thatale yields a very accurate recursive condition estimator.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02145580
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    Paper (German National Licenses)
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  • 4
    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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    Paper (German National Licenses)
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  • 5
    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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    Paper (German National Licenses)
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