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  • 1975-1979  (4)
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  • 1
    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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  • 2
    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)
    Fulltext
  • 3
    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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  • 4
    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)
    Fulltext
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