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  • 1
    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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  • 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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  • 4
    Electronic Resource
    Electronic Resource
    Can adaption help on the average? (1984)
    Springer
    Numerische Mathematik 44 (1984), S. 169-190 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 68C25 ; CR: F2.1
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We study adaptive information for approximation of linear problems in a separable Hilbert space equipped with a probability measure μ. It is known that adaption does not help in the worst case for linear problems. We prove that adaption also doesnot help on the average. That is, there exists nonadaptive information which is as powerful as adaptive information. This result holds for “orthogonally invariant” measures. We provide necessary and sufficient conditions for a measure to be orthogonally invariant. Examples of orthogonally invariant measures include Gaussian measures and, in the finite dimensional case, weighted Lebesgue measures.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01410103
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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
    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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  • 7
    Electronic Resource
    Electronic Resource
    Optimal radius of convergence of interpolatory iterations for operator equations (1980)
    Springer
    Aequationes mathematicae 21 (1980), S. 159-172 
    Details
    ISSN: 1420-8903
    Keywords: Primary 47H10, 47H15, 65J05, 68A20
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The convergence of the class of direct interpolatory iterationsI n for a simple zero of a non-linear operatorF in a Banach space of finite or infinite dimension is studied. A general convergence result is established and used to show that ifF is entire the “radius of convergence” goes to infinity withn while ifF is analytic in a ball of radiusR the radius of convergence increases to at leastR/2 withn.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02189351
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    Paper (German National Licenses)
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  • 8
    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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  • 9
    Electronic Resource
    Electronic Resource
    Integration and approximation in arbitrary dimensions (2000)
    Springer
    Advances in computational mathematics 12 (2000), S. 25-58 
    Details
    ISSN: 1572-9044
    Keywords: curse of dimension ; tractability ; multivariate integration ; multivariate approximation ; 41A05 ; 41A63 ; 65D05 ; 41A25
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We study multivariate integration and approximation for various classes of functions of d variables with arbitrary d. We consider algorithms that use function evaluations as the information about the function. We are mainly interested in verifying when integration and approximation are tractable and strongly tractable. Tractability means that the minimal number of function evaluations needed to reduce the initial error by a factor of ɛ is bounded by C(d)ɛ−p for some exponent p independent of d and some function C(d). Strong tractability means that C(d) can be made independent of d. The ‐exponents of tractability and strong tractability are defined as the smallest powers of ɛ{-1} in these bounds. We prove that integration is strongly tractable for some weighted Korobov and Sobolev spaces as well as for the Hilbert space whose reproducing kernel corresponds to the covariance function of the isotropic Wiener measure. We obtain bounds on the ‐exponents, and for some cases we find their exact values. For some weighted Korobov and Sobolev spaces, the strong ‐exponent is the same as the ‐exponent for d=1, whereas for the third space it is 2. For approximation we also consider algorithms that use general evaluations given by arbitrary continuous linear functionals as the information about the function. Our main result is that the ‐exponents are the same for general and function evaluations. This holds under the assumption that the orthonormal eigenfunctions of the covariance operator have uniformly bounded L∞ norms. This assumption holds for spaces with shift-invariant kernels. Examples of such spaces include weighted Korobov spaces. For a space with non‐shift‐invariant kernel, we construct the corresponding space with shift-invariant kernel and show that integration and approximation for the non-shift-invariant kernel are no harder than the corresponding problems with the shift-invariant kernel. If we apply this construction to a weighted Sobolev space, whose kernel is non-shift-invariant, then we obtain the corresponding Korobov space. This enables us to derive the results for weighted Sobolev spaces.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1018948631251
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  • 10
    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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