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  • Articles  (42)
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  • Mathematics  (42)
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  • 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
    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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  • 3
    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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  • 4
    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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  • 5
    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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  • 6
    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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  • 7
    Electronic Resource
    Electronic Resource
    A direct method for sparse least squares problems with lower and upper bounds (1988)
    Springer
    Numerische Mathematik 54 (1988), S. 19-32 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65F20 ; CR: G1.3
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A direct method is developed for solving linear least squares problems $$\mathop {\min \left\| {Ax - b} \right\|_2 }\limits_x $$ , whereA is large and sparse and the solution is subject to lower and upper boundsl≦x≦u. The problem is initially transformed to upper triangular form by a sparseQR-factorization. An active set algorithm is then used. The key step is the stable updating of theR-factor associated with the columns ofA corresponding to the free variables, when theQ-factor is not available. For this a new method is developed, which uses the semi-normal equations and iterative refinement.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01403888
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  • 8
    Electronic Resource
    Electronic Resource
    A note on a one-sided Jacobi algorithm (1989)
    Springer
    Numerische Mathematik 56 (1989), S. 627-633 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65F15 ; CR: G1.3
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We propose a “one-sided” or “implicit” variant of the Jacobi diagonalization algorithm for positive definite matrices. The variant is based on a previous Cholesky decomposition and currently uses essentially one square array which, on output, contains the matrix of eigenvectors thus reaching the storage economy of the symmetric QL algorithm. The current array is accessed only columnwise which makes the algorithm attractive for various parallelized and/or vectorized implementations. Even on a serial computer our algorithm shows improved efficiency, in particular if the Cholesky step is made with diagonal pivoting. On matrices of ordern=25–200 our algorithm is about 2–3.5 times slower than QL thus being almost on the halfway between the standard Jacobi and QL algorithms. The previous Cholesky decomposition can be performed with higher precision without extra time or storage costs thus offering considerable gains in accuracy with highly conditioned input matrices.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396349
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  • 9
    Electronic Resource
    Electronic Resource
    Tchebychev acceleration technique for large scale nonsymmetric matrices (1989)
    Springer
    Numerische Mathematik 56 (1989), S. 721-734 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65F10 ; CR: G1.3
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The acceleration by Tchebychev iteration for solving nonsymmetric eigenvalue problems is dicussed. A simple algorithm is derived to obtain the optimal ellipse which passes through two eigenvalues in a complex plane relative to a reference complex eigenvalue. New criteria are established to identify the optimal ellipse of the eigenspectrum. The algorithm is fast, reliable and does not require a search for all possible ellipses which enclose the spectrum. The procedure is applicable to nonsymmetric linear systems as well.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01405199
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  • 10
    Electronic Resource
    Electronic Resource
    An a posteriori error estimation of the Rayleigh-Ritz eigenvalues (1989)
    Springer
    Numerische Mathematik 54 (1989), S. 639-654 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 35P15, 47A55, 65M15 ; CR: G1.3
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary LetA, B be essentially self-adjoint and positive definite differential operators defined inL 2(G). Using Svirskij's construction of the base operator and some results from the analytic perturbation theory of linear operators a formula providing eigenvalue lower bounds of the problemAu=λBu is derived. In this formula a rough lower bound of some higher eigenvalue and the residual convergence of the Rayleigh-Ritz eigenfunction approximations are needed. Some numerical results are presented.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396487
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