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  • 1985-1989  (4)
  • 1975-1979  (4)
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  • 1
    Electronic Resource
    Electronic Resource
    Information-Based Complexity (1986)
    Palo Alto, Calif. : Annual Reviews
    Annual Review of Computer Science 1 (1986), S. 319-380 
    Details
    ISSN: 8756-7016
    Source: Annual Reviews Electronic Back Volume Collection 1932-2001ff
    Topics: Computer Science
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1146/annurev.cs.01.060186.001535
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    Paper (German National Licenses)
    Fulltext
  • 2
    Electronic Resource
    Electronic Resource
    Is Gauss quadrature optimal for analytic functions? (1985)
    Springer
    Numerische Mathematik 47 (1985), S. 89-98 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65D30 ; CR: G1.4
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We consider the problem of optimal quadratures for integrandsf: [−1,1]→ℝ which have an analytic extension $$\bar f$$ to an open diskD r of radiusr about the origin such that $$\left| {\bar f} \right|$$ ≦1 on $$\bar D_r $$ . Ifr=1, we show that the penalty for sampling the integrand at zeros of the Legendre polynomial of degreen rather than at optimal points, tends to infinity withn. In particular there is an “infinite” penalty for using Gauss quadrature. On the other hand, ifr〉1, Gauss quadrature is almost optimal. These results hold for both the worst-case and asymptotic settings.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01389877
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    Round-off error analysis of iterations for large linear systems (1978)
    Springer
    Numerische Mathematik 30 (1978), S. 301-314 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65F10 ; CR: 5.14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We deal with the rounding error analysis of successive approximation iterations for the solution of large linear systemsA x =b. We prove that Jacobi, Richardson, Gauss-Seidel and SOR iterations arenumerically stable wheneverA=A *〉0 andA has PropertyA. This means that the computed resultx k approximates the exact solution α with relative error of order ζ ‖A‖·‖A −1‖ where ζ is the relative computer precision. However with the exception of Gauss-Seidel iteration the residual vector ‖Ax k −b‖ is of order ζ ‖A‖2 ‖A −1‖ ‖α‖ and hence the remaining three iterations arenot well-behaved.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01411845
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    Paper (German National Licenses)
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  • 4
    Electronic Resource
    Electronic Resource
    Numerical stability of the Chebyshev method for the solution of large linear systems (1977)
    Springer
    Numerische Mathematik 28 (1977), S. 191-209 
    Details
    ISSN: 0945-3245
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary This paper contains the rounding error analysis for the Chebyshev method for the solution of large linear systemsAx+g=0 whereA=A * is positive definite. We prove that the Chebyshev method in floating point arithmetic is numerically stable, which means that the computed sequence {x k} approximates the solution α such that $$\overline {\mathop {\lim }\limits_k } $$ ‖x k −α‖ is of order ζ‖A‖‖A −1‖‖α‖ where ζ is the relative computer precision. We also point out that in general the Chebyshev method is not well-behaved, which means that the computed residualsr k=Ax k+g are of order ζ‖A‖2‖A −1‖‖α‖.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01394452
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    Paper (German National Licenses)
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  • 5
    Electronic Resource
    Electronic Resource
    Numerical stability for solving nonlinear equations (1976)
    Springer
    Numerische Mathematik 27 (1976), S. 373-390 
    Details
    ISSN: 0945-3245
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The concepts of the condition number, numerical stability and well-behavior for solving systems of nonlinear equationsF(x)=0 are introduced. Necessary and sufficient conditions for numerical stability and well-behavior of a stationary are given. We prove numerical stability and well-behavior of the Newton iteration for solving systems of equations and of some variants of secant iteration for solving a single equation under a natural assumption on the computed evaluation ofF. Furthermore we show that the Steffensen iteration is unstable and show how to modify it to have well-behavior and hence stability.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01399601
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    Paper (German National Licenses)
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  • 6
    Electronic Resource
    Electronic Resource
    The accurate solution of certain continuous problems using only single precision arithmetic (1985)
    Springer
    BIT 25 (1985), S. 635-651 
    Details
    ISSN: 1572-9125
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A typical approach for finding the approximate solution of a continuous problem is through discretization with meshsizeh such that the truncation error goes to zero withh. The discretization problem is solved in floating point arithmetic. Rounding-errors spoil the theoretical convergence and the error may even tend to infinity. In this paper we present algorithms of moderate cost which use only single precision and which compute the approximate solution of the integration and elliptic equation problems with full accuracy. These algorithms are based on the modified Gill-Møller algorithm for summation of very many terms, iterative refinement of a linear system with a special algorithm for the computation of residuals in single precision and on a property of floating point subtraction of nearby numbers.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01936142
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    Paper (German National Licenses)
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  • 7
    Electronic Resource
    Electronic Resource
    Iterative refinement implies numerical stability (1977)
    Springer
    BIT 17 (1977), S. 303-311 
    Details
    ISSN: 1572-9125
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Suppose that a method ϕ computes an approximation of the exact solution of a linear systemAx=b with the relative errorq,q〈1. We prove that if all computations are performed in floating point arithmeticfl and single precision, then ϕ with iterative refinement is numerically stable and well-behaved wheneverq∥A∥ ∥A −1∥ is at most of order unity.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01932150
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    Paper (German National Licenses)
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  • 8
    Book
    Book
    Information-based complexity (1988)
    London u.a. :Academic Press,
    Details
    Title: Information-based complexity
    Author: Traub, Joseph Frederick
    Contributer: Wasillkowski, G.W. , Wozniakowski, H.
    Publisher: London u.a. :Academic Press,
    Year of publication: 1988
    Pages: 523 S.
    Series Statement: Computer science and scientific computing
    Type of Medium: Book
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