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  • 1
    Monograph available for loan
    Monograph available for loan
    Matrix computations (1996)
    Baltimore [u.a.] : Johns Hopkins Univ. Press
    Details Availability
    Call number: 19/M 97.0071
    Type of Medium: Monograph available for loan
    Pages: xxvii, 694 S.
    Edition: 3rd ed.
    ISBN: 0801854148
    Series Statement: John Hopkins studies in the mathematical sciences
    Classification:
    C.1.8.
    Language: English
    Location: Reading room
    Branch Library: GFZ Library
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  • 2
    Unknown
    Recent advances in iterative methods (1994)
    New York [u.a.] : Springer
    Associated volumes
    Details Availability
    Call number: M 94.0390
    In: The IMA volumes in mathematics and its applications
    Pages: XVI, 225 S.
    ISBN: 0387942521
    Series Statement: The IMA volumes in mathematics and its applications 60
    Classification:
    Geodetic Theory and Modeling
    Language: English
    Branch Library: GFZ Library
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  • 3
    Electronic Resource
    Electronic Resource
    Bounds for the round-off errors in the richardson second order method (1962)
    Springer
    BIT 2 (1962), S. 212-223 
    Details
    ISSN: 1572-9125
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The application of the Richardson second order iterative method to positive definite, symmetric linear equations is investigated. Absolute and statistical bounds for the round-off error are derived. The statistical theory agrees well with numerical experiments, until the accumulated round-off error becomes of the order of magnitude of the error in the computed solution. After this point the statistical dependence between the local round-off errors makes the observed variances larger than the theoretical variances.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01940168
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  • 4
    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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  • 5
    Electronic Resource
    Electronic Resource
    Convergence of a two-stage Richardson process for nonlinear equations (1986)
    Springer
    BIT 26 (1986), S. 209-216 
    Details
    ISSN: 1572-9125
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01933747
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  • 6
    Electronic Resource
    Electronic Resource
    Bounds for the Entries of Matrix Functions with Applications to Preconditioning (1999)
    Springer
    BIT 39 (1999), S. 417-438 
    Details
    ISSN: 1572-9125
    Keywords: Matrix functions ; quadrature rules ; Lanczos process ; band matrices ; exponential decay ; preconditioned conjugate gradients
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Let A be a symmetric matrix and let f be a smooth function defined on an interval containing the spectrum of A. Generalizing a well-known result of Demko, Moss and Smith on the decay of the inverse we show that when A is banded, the entries of f(A)are bounded in an exponentially decaying manner away from the main diagonal. Bounds obtained by representing the entries of f(A)in terms of Riemann-Stieltjes integrals and by approximating such integrals by Gaussian quadrature rules are also considered. Applications of these bounds to preconditioning are suggested and illustrated by a few numerical examples.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1022362401426
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  • 7
    Electronic Resource
    Electronic Resource
    Inexact Inverse Iteration for Generalized Eigenvalue Problems (2000)
    Springer
    BIT 40 (2000), S. 671-684 
    Details
    ISSN: 1572-9125
    Keywords: Inverse iteration ; shift-and-invert ; inner-outer iterations
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper, we study an inexact inverse iteration with inner-outer iterations for solving the generalized eigenvalu problem Ax = λBx, and analyze how the accuracy in the inner iterations affects the convergence of the outer iterations. By considering a special stopping criterion depending on a threshold parameter, we show that the outer iteration converges linearly with the inner threshold parameter as the convergence rate. We also discuss the total amount of work and asymptotic equivalence between this stopping criterion and a more standard one. Numerical examples are given to illustrate the theoretical results.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1022388317839
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  • 8
    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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  • 9
    Electronic Resource
    Electronic Resource
    The convergence of inexact Chebyshev and Richardson iterative methods for solving linear systems (1988)
    Springer
    Numerische Mathematik 53 (1988), S. 571-593 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65F10 ; CR: G1.3
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The Chebyshev and second-order Richardson methods are classical iterative schemes for solving linear systems. We consider the convergence analysis of these methods when each step of the iteration is carried out inexactly. This has many applications, since a preconditioned iteration requires, at each step, the solution of a linear system which may be solved inexactly using an “inner” iteration. We derive an error bound which applies to the general nonsymmetric inexact Chebyshev iteration. We show how this simplifies slightly in the case of a symmetric or skew-symmetric iteration, and we consider both the cases of underestimating and overestimating the spectrum. We show that in the symmetric case, it is actually advantageous to underestimate the spectrum when the spectral radius and the degree of inexactness are both large. This is not true in the case of the skew-symmetric iteration. We show how similar results apply to the Richardson iteration. Finally, we describe numerical experiments which illustrate the results and suggest that the Chebyshev and Richardson methods, with reasonable parameter choices, may be more effective than the conjugate gradient method in the presence of inexactness.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01397553
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  • 10
    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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