Herbert H. H. Homeier

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Sequence transformations are important tools for the convergence acceleration of slowly convergent scalar sequences or series and for the summation of divergent series. The basic idea is to construct from a given sequence { sn} a new sequence { s ′ n} = T ({ sn} ) where each s ′ n depends on a finite number of elements sn1 , . . . , snm . Often, the sn are(More)
We discuss Levin-type sequence transformations fsng ! fs 0 n g that depend linearly on the sequence elements sn, and nonlinearly on an auxiliary sequence of remainder estimates f!ng. If the remainder estimates also depend on the sequence elements, non-linear transformations are obtained. The application of such transformations very often yields new(More)
The acceleration of slowly convergent Fourier series is discussed. For instance, such series arise in spectral methods with irregular grids as are required to resolve shock waves or other narrow features, in the description of lattice-gas models, and in the Matsubara formalism. In extension of a method proposed by Sidi, it is shown that rather spectacular(More)
The solution of the Ornstein-Zernike equation with various closure approximations is studied. This problem is rewritten as an integral equation that can be solved iteratively on a grid. The convergence of the xed point iterations is relatively slow. We consider transformations of the sequence of solution vectors using non-linear sequence transformations,(More)
A new nonlinear sequence transformation, the iterative J transformation, was proposed recently For this transformation, a derivation based on the method of hierarchical consistency, alternative recursive representations, general properties, an explicit expression for the kernel, model sequences, and its relation to other sequence transformations have been(More)
A nonlinear sequence transformation is presented which is able to accelerate the convergence of Fourier series. It is tailored to be exact for a certain model sequence. As in the case of the Levin transformation and other transformations of Levin-type, in this model sequence the partial sum of the series is written as the sum of the limit (or antilimit) and(More)
Recently, the author proposed a new nonlinear sequence transformation, the iterative $$J$$ transformation, which was shown to provide excellent results in several applications (Homeier [15]). In the present contribution, this sequence transformation is derived by a hierarchically consistent iteration of some basic transformation. Hierarchical consistency is(More)
The stability of a large class of nonlinear sequence transformations is analyzed. Considered are variants of the J transformation [17]. Suitable variants of this transformation belong to the most successful extrapolation algorithms that are known [20]. Similar to recent results of Sidi, it is proved that the p {J} transformations, the Weniger S(More)