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  • Articles  (66)
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  • CR: G1.3  (42)
  • CR: G1.5  (24)
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  • Mathematics  (66)
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  • Articles  (66)
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  • Mathematics  (66)
  • 1
    Electronic Resource
    Electronic Resource
    Semi-iterative methods for the approximate solution of ill-posed problems (1986)
    Springer
    Numerische Mathematik 50 (1986), S. 263-271 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65F10 ; CR: G1.3
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We study semi-iterative (two step interative) methods of the form $$x_{n + 1} = T^* T(\alpha x_n + \gamma x_{n - 1} ) + \beta x_n + (1 - \beta )x_{n - 1} - (\alpha + \gamma )T^* y$$ for the approximate solution of ill-posed or ill-conditioned linear equationsTx=y in (infinite or finite dimensional) Hilbert spaces. We present results on convergence, convergence rates, the influence of perturbed data, and on the comparison of different methods.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01390704
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  • 2
    Electronic Resource
    Electronic Resource
    A convergence theorem for Newton-like methods in Banach spaces (1987)
    Springer
    Numerische Mathematik 51 (1987), S. 545-557 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS) ; 65G99 ; 65J15 ; CR: G1.5
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A convergence theorem for Newton-like methods in Banach spaces is given, which improves results of Rheinboldt [27], Dennis [4], Miel [15, 16] and Moret [18] and includes as a special case an updated (affine-invariant [6]) version of the Kantorovich theorem for the Newton method given in previous papers [35, 36]. Error bounds obtained in [34] are also improved. This paper unifies the study of finding sharp error bounds for Newton-like methods under Kantorovich type assumptions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01400355
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  • 3
    Electronic Resource
    Electronic Resource
    Lower bounds for the condition number of Vandermonde matrices (1987)
    Springer
    Numerische Mathematik 52 (1987), S. 241-250 
    Details
    ISSN: 0945-3245
    Keywords: AMS (MOS): 15A12 ; 49D15 ; 65F35 ; CR: G1.3 ; G1.6
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We derive lower bounds for the ∞-condition number of then×n-Vandermonde matrixV n(x) in the cases where the node vectorx T=[x1, x2,...,xn] has positive elements or real elements located symmetrically with respect to the origin. The bounds obtained grow exponentially inn. withO(2n) andO(2n/2), respectively. We also compute the optimal spectral condition numbers ofV n(x) for the two node configurations (including the optimal nodes) and compare them with the bounds obtained.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01398878
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  • 4
    Electronic Resource
    Electronic Resource
    Rehabilitation of the Gauss-Jordan algorithm (1989)
    Springer
    Numerische Mathematik 54 (1989), S. 591-599 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65F05, 65G05, 15A06 ; CR: G1.3
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In this paper a Gauss-Jordan algorithm with column interchanges is presented and analysed. We show that, in contrast with Gaussian elimination, the Gauss-Jordan algorithm has essentially differing properties when using column interchanges instead of row interchanges for improving the numerical stability. For solutions obtained by Gauss-Jordan with column interchanges, a more satisfactory bound for the residual norm can be given. The analysis gives theoretical evidence that the algorithm yields numerical solutions as good as those obtained by Gaussian elimination and that, in most practical situations, the residuals are equally small. This is confirmed by numerical experiments. Moreover, timing experiments on a Cyber 205 vector computer show that the algorithm presented has good vectorisation properties.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396364
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  • 5
    Electronic Resource
    Electronic Resource
    Homotopy algorithm for symmetric eigenvalue problems (1989)
    Springer
    Numerische Mathematik 55 (1989), S. 265-280 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS) ; 65H10 ; 58C99 ; 55M25 ; CR: G1.3
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The homotopy method can be used to solve eigenvalue-eigenvector problems. The purpose of this paper is to report the numerical experience of the homotopy method of computing eigenpairs for real symmetric tridiagonal matrices together with a couple of new theoretical results. In practice, it is rerely of any interest to compute all the eigenvalues. The homotopy method, having the order preserving property, can provide any specific eigenvalue without calculating any other eigenvalues. Besides this advantage, we note that the homotopy algorithm is to a large degree a parallel algorithm. Numerical experimentation shows that the homotopy method can be very efficient especially for graded matrices.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01390054
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  • 6
    Electronic Resource
    Electronic Resource
    Solving large nonlinear systems of equations by an adaptive condensation process (1986)
    Springer
    Numerische Mathematik 50 (1986), S. 633-653 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65H10 ; 65H15 ; 65K10 ; 65N20 ; 65N30 ; CR: G1.5
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We present an algorithm which efficiently solves large nonlinear systems of the form $$Au = F(u), u \in \mathbb{R}^n $$ whenever an (iterative) solver “A −1” for the symmetric positive definite matrixA is available andF'(u) is symmetric. Such problems arise from the discretization of nonlinear elliptic partial differential equations. By means of an adaptive decomposition process we split the original system into a low dimensional system — to be treated by any sophisticated solver — and a remaining high-dimensional system, which can easily be solved by fixed point iteration. Specifically we choose a Newton-type trust region algorithm for the treatment of the small system. We show global convergence under natural assumptions on the nonlinearity. The convergence results typical for trust-region algorithms carry over to the full iteration process. The only large systems to be solved are linear ones with the fixed matrixA. Thus existing software for positive definite sparse linear systems can be used.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01398377
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  • 7
    Electronic Resource
    Electronic Resource
    On efficient implementations of Kogbetliantz's algorithm for computing the singular value decomposition (1987)
    Springer
    Numerische Mathematik 52 (1987), S. 279-300 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65F15 ; CR: G1.3
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In this paper we compare several implementations of Kogbetliantz's algorithm for computing the SVD on sequential as well as on parallel machines. Comparisons are based on timings and on operation counts. The numerical accuracy of the different methods is also analyzed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01398880
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  • 8
    Electronic Resource
    Electronic Resource
    Comparison of Brown's and Newton's method in the monotone case (1987)
    Springer
    Numerische Mathematik 52 (1987), S. 511-521 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65H10 ; CR: G1.5
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In many cases when Newton's method, applied to a nonlinear sytemF(x)=0, produces a monotonically decreasing sequence of iterates, Brown's method converges monotonically, too. We compare the iterates of Brown's and Newton's method in these monotone cases with respect to the natural partial ordering. It turns out that in most of the cases arising in applications Brown's method then produces “better” iterates than Newton's method.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01400889
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  • 9
    Electronic Resource
    Electronic Resource
    A reliable argument principle algorithm to find the number of zeros of an analytic function in a bounded domain (1988)
    Springer
    Numerische Mathematik 53 (1988), S. 143-163 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65H05 ; CR: G1.5
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The argument principle is a natural and simple method to determine the number of zeros of an analytic functionf(z) in a bounded domainD. N, the number of zeros (counting multiplicities) off(z), is 1/2π times the change in Argf(z) asz moves along the closed contour σD. Since the range of Argf(z) is (−π, π] a critical point in the computational procedure is to assure that the discretization of σD, {z i ,i=1, ...,M}, is such that $$|\Delta _{{\text{[z}}_i {\text{,}} {\text{z}}_{i + 1} {\text{]}}} Arg f(z)| \leqq \pi $$ . Discretization control which may violate this inequality can lead to an unreliable algorithm. Mathematical theorems derived for the discretization of σD lead to a completely reliable algorithm to computeN. This algorithm also treats in an elementary way the case when a zero is on or near the contour σD. Numerical examples are given for the reliable algorithm formulated here and it is pointed out in these examples how inadequate discretization control can lead to failure of other algorithms.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01395882
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  • 10
    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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