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  • 1990-1994  (16)
  • 1
    Electronic Resource
    Electronic Resource
    On the convergence of line iterative methods for cyclically reduced non-symmetrizable linear systems (1994)
    Springer
    Numerische Mathematik 67 (1994), S. 177-190 
    Details
    ISSN: 0945-3245
    Keywords: Mathematics Subject Classification (1991): 65F10
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary. We derive analytic bounds on the convergence factors associated with block relaxation methods for solving the discrete two-dimensional convection-diffusion equation. The analysis applies to the reduced systems derived when one step of block Gaussian elimination is performed on red-black ordered two-cyclic discretizations. We consider the case where centered finite difference discretization is used and one cell Reynolds number is less than one in absolute value and the other is greater than one. It is shown that line ordered relaxation exhibits very fast rates of convergence.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002110050023
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  • 2
    Electronic Resource
    Electronic Resource
    Modified moments for indefinite weight functions (1990)
    Springer
    Numerische Mathematik 57 (1990), S. 607-624 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS) 30E05, 41A21, 65F10
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The problem of generating the recurrence coefficients of orthogonal polynomials from the moments or from modified moments of the weight function is well understood for positive weight distributions. Here we extend this theory and the basic algorithms to the case of an indefinite weight function. While the generic indefinite case is formally not much different from the positive definite case, there exist nongeneric degenerate situations, and these require a different more complicated treatment. The understanding of these degenerate situations makes it possible to construct a stable approximate solution of an ill-conditioned problem. The application to adaptive iterative methods for linear systems of equations is anticipated.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01386431
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  • 3
    Electronic Resource
    Electronic Resource
    Quadratically constrained least squares and quadratic problems (1991)
    Springer
    Numerische Mathematik 59 (1991), S. 561-580 
    Details
    ISSN: 0945-3245
    Keywords: 65F20
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We consider the following problem: Compute a vectorx such that ∥Ax−b∥2=min, subject to the constraint ∥x∥2=α. A new approach to this problem based on Gauss quadrature is given. The method is especially well suited when the dimensions ofA are large and the matrix is sparse. It is also possible to extend this technique to a constrained quadratic form: For a symmetric matrixA we consider the minimization ofx T A x−2b T x subject to the constraint ∥x∥2=α. Some numerical examples are given.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01385796
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  • 4
    Electronic Resource
    Electronic Resource
    Least-squares fitting of circles and ellipses (1994)
    Springer
    BIT 34 (1994), S. 558-578 
    Details
    ISSN: 1572-9125
    Keywords: 65H10 ; 65F20 ; Least squares ; curve fitting ; singular value decomposition
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Fitting circles and ellipses to given points in the plane is a problem that arises in many application areas, e.g., computer graphics, coordinate meteorology, petroleum engineering, statistics. In the past, algorithms have been given which fit circles and ellipses insome least-squares sense without minimizing the geometric distance to the given points. In this paper we present several algorithms which compute the ellipse for which thesum of the squares of the distances to the given points is minimal. These algorithms are compared with classical simple and iterative methods. Circles and ellipses may be represented algebraically, i.e., by an equation of the formF(x)=0. If a point is on the curve, then its coordinates x are a zero of the functionF. Alternatively, curves may be represented in parametric form, which is well suited for minimizing the sum of the squares of the distances.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01934268
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  • 5
    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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  • 6
    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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  • 7
    Electronic Resource
    Electronic Resource
    Estimates in quadratic formulas (1994)
    Springer
    Numerical algorithms 8 (1994), S. 241-268 
    Details
    ISSN: 1572-9265
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract LetA be a real symmetric positive definite matrix. We consider three particular questions, namely estimates for the error in linear systemsAx=b, minimizing quadratic functional minx(x T Ax−2b T x) subject to the constraint ‖x‖=α, α〈‖A −1 b‖, and estimates for the entries of the matrix inverseA −1. All of these questions can be formulated as a problem of finding an estimate or an upper and lower bound onu T F(A)u, whereF(A)=A −1 resp.F(A)=A −2,u is a real vector. This problem can be considered in terms of estimates in the Gauss-type quadrature formulas which can be effectively computed exploiting the underlying Lanczos process. Using this approach, we first recall the exact arithmetic solution of the questions formulated above and then analyze the effect of rounding errors in the quadrature calculations. It is proved that the basic relation between the accuracy of Gauss quadrature forf(λ)=λ−1 and the rate of convergence of the corresponding conjugate gradient process holds true even for finite precision computation. This allows us to explain experimental results observed in quadrature calculations and in physical chemistry and solid state physics computations which are based on continued fraction recurrences.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02142693
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  • 8
    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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  • 9
    Electronic Resource
    Electronic Resource
    A multishift QR iteration without computation of the shifts (1994)
    Springer
    Numerical algorithms 7 (1994), S. 173-181 
    Details
    ISSN: 1572-9265
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Each iteration of the multishift QR algorithm of Bai and Demmel requires the computation of a “shift vector” defined bym shifts of the origin of the spectrum that control the convergence of the process. A common choice of shifts consists of the eigenvalues of the trailing principal submatrix of orderm, and current practice includes the computation of these eigenvalues in the determination of the shift vector. In this paper, we describe an algorithm based on the evaluation of the characteristic polynomial of a Hessenberg matrix, which directly produces the shift vector without computing eigenvalues. This algorithm is stable, more accurate, faster, and simpler than the current alternative. It also allows for a consistent shift strategy with dynamic adjustment of the number of shifts.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02140681
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  • 10
    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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