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  • Lanczos methods  (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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    Paper (German National Licenses)
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  • 2
    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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