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  • 1
    Electronic Resource
    Electronic Resource
    A Gautschi-type method for oscillatory second-order differential equations (1999)
    Springer
    Numerische Mathematik 83 (1999), S. 403-426 
    Details
    ISSN: 0945-3245
    Keywords: Mathematics Subject Classification (1991):65L05, 65L70, 65M12, 65M20
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary. We study a numerical method for second-order differential equations in which high-frequency oscillations are generated by a linear part. For example, semilinear wave equations are of this type. The numerical scheme is based on the requirement that it solves linear problems with constant inhomogeneity exactly. We prove that the method admits second-order error bounds which are independent of the product of the step size with the frequencies. Our analysis also provides new insight into the m ollified impulse method of García-Archilla, Sanz-Serna, and Skeel. We include results of numerical experiments with the sine-Gordon equation.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002110050456
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    Optimized look-ahead recurrences for adjacent rows in the Padé table (1996)
    Springer
    BIT 36 (1996), S. 264-286 
    Details
    ISSN: 1572-9125
    Keywords: Padé approximation ; row recurrence ; fast algorithm ; sawtooth recurrence ; look-ahead ; Toeplitz matrix ; Levinson algorithm ; Schur algorithm ; biorthogonal polynomials
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A new look-ahead algorithm for recursively computing Padé approximants is introduced. It generates a subsequence of the Padé approximants on two adjacent rows (defined by fixed numerator degree) of the Padé table. Its two basic versions reduce to the classical Levinson and Schur algorithms if no look-ahead is required. The new algorithm can be viewed as a combination of the look-ahead sawtooth and the look-ahead Levinson and Schur algorithms that we proposed before, but now the look-ahead step size is minimal (as in the sawtooth version) and the computational costs are as low as in the least expensive competing algorithms (including our look-ahead Levinson and Schur algorithms). The underlying recurrences link well-conditioned basic pairs,i.e., pairs of sufficiently different neighboring Padé forms. The algorithm can be used to solve Toeplitz systems of equationsTx = b. In this application it comes in several versions: anO(N 2) Levinson-type form, anO(N 2) Schur-type form, and a superfastO(N log2 N) Schur-type version. As an option of the first two versions, the corresponding block LDU decompositions ofT −1 orT, respectively, can be found.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01731983
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  • 3
    Electronic Resource
    Electronic Resource
    Exponential Integrators for Quantum-Classical Molecular Dynamics (1999)
    Springer
    BIT 39 (1999), S. 620-645 
    Details
    ISSN: 1572-9125
    Keywords: Numerical integrator ; oscillatory solutions ; Schrödinger equation ; quantum-classical coupling ; error bounds ; stability
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We study time integration methods for equations of mixed quantum-classical molecular dynamics in which Newtonian equations of motion and Schrödinger equations are nonlinearly coupled. Such systems exhibit different time scales in the classical and the quantum evolution, and the solutions are typically highly oscillatory. The numerical methods use the exponential of the quantum Hamiltonian whose product with a state vector is approximated using Lanczos' method. This allows time steps that are much larger than the inverse of the highest frequencies. We describe various integration schemes and analyze their error behaviour, without assuming smoothness of the solution. As preparation and as a problem of independent interest, we study also integration methods for Schrödinger equations with time-dependent Hamiltonian.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1022335122807
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  • 4
    Electronic Resource
    Electronic Resource
    A note on conjugate-gradient type methods for indefinite and/or inconsistent linear systems (1996)
    Springer
    Numerical algorithms 11 (1996), S. 181-187 
    Details
    ISSN: 1572-9265
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract To design a conjugate-gradient type method for a given linear system one has to choose an inner product space and to compute residual polynomials which minimize the induced norm. Here we propose a unified treatment of this approach for symmetric linear systems. We mainly focus on indefinite and/or inconsistent systems.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02142495
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    Paper (German National Licenses)
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  • 5
    Electronic Resource
    Electronic Resource
    On the use of two QMR algorithms for solving singular systems and applications in Markov chain modeling (1994)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 1 (1994), S. 403-420 
    Details
    ISSN: 1070-5325
    Keywords: Quasi-minimal residual iteration ; Non-Hermitian matrix ; Singular linear system ; Markov chain modeling ; Krylov-subspace method ; Convergence ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Recently, Freund and Nachtigal proposed the quasi-minimal residual algorithm (QMR) for solving general nonsingular non-Hermitian linear systems. The method is based on the Lanczos process, and thus it involves matrix - vector products with both the coefficient matrix of the linear system and its transpose. Freund developed a variant of QMR, the transpose-free QMR algorithm (TFQMR), that only requires products with the coefficient matrix. In this paper, the use of QMR and TFQMR for solving singular systems is explored. First, a convergence result for the general class of Krylov-subspace methods applied to singular systems is presented. Then, it is shown that QMR and TFQMR both converge for consistent singular linear systems with coefficient matrices of index 1. Singular systems of this type arise in Markov chain modeling. For this particular application, numerical experiments are reported.
    Additional Material: 2 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nla.1680010406
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  • 6
    Book
    Book
    Lanczos- und Krylov-Verfahren für nicht-Hermitesche lineare Systeme. Karlsruhe, Univ., Diss. 1992 (1992)
    Details
    Title: Lanczos- und Krylov-Verfahren für nicht-Hermitesche lineare Systeme. Karlsruhe, Univ., Diss. 1992
    Author: Hochbruck, Marlis
    Year of publication: 1992
    Pages: 108 S.
    Type of Medium: Book
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    ZIB Catalog
    Zuse Institute Berlin get availability status
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