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\begin{document}
\frontmatter
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\begin{center}
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{\bf\huge Summary of \\``Extrapolationsverfahren f{\"u}r\\ Zahlen-, Vektor- und
Matrizenfolgen und ihre Anwendung in der Theoretischen und Physikalischen
Chemie''\par}
\vfill
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{\bf\Large
Habilitationsschrift
\vskip 2cm
{\bf\Large Naturwissenschaftliche Fakult\"at IV\\
-- Chemie und Pharmazie --\\
Universit\"at Regensburg}
\vskip 2cm
vorgelegt von  \\[1cm]
{\bf\LARGE Herbert H. H. Homeier}
\vskip 1 cm
1996}
\end{center}
\end{titlepage}


\mainmatter
\chapter*{Summary}
In this contribution, extrapolation methods for number, vector and matrix sequences and their application in theoretical and 
physical chemistry are treated.

There are three parts of this work. In Part I, extrapolation methods are presented and characterized in their properties.
In Part II, applications of these methods to a series of different problems in the field of theoretical and 
physical chemistry are discussed. In Part III, supplementary material is made available in several appendices. 

After an introduction in Chapter 1, Part I starts. In Chapter 2, known methods for the extrapolation of number sequences
are discussed. Classication schemes of number sequences that are important for the successful choice
of extrapolation methods are discussed,  as are general criteria for  the construction of these methods, 
and also the r{\^o}le of recursive 
schemes for the representation and implemention of the methods. Important known methods are presented in the following that
are important for the present work.

In Chapter 3, iterative sequence transformations are treated. First, basic principles for, and problem of their construction
are sketched. A detailed discussion of the concept of hierarchical consistency follows that helps to solve these problems
and provides a new methodological basis for the construction of iterative methods. For this end, hierarchies of model
sequences are introduced and the iteration is interpreted as a mapping between the different levels of the hierarchie. The
next topic then is the  
$\mathcal{J}$ transformation. For the derivation and theoretical characterization of this transformation, the concept of
hierarchical consistency is exploited fully and proven to be very valuable. Since the 
$\mathcal{J}$ transformation is a very general extrapolation method, it is studied thoroughly. Several efficient
algorithms for its computation are provided, and basic mathematical properties of the transformation are proven. The 
kernel of the transformation is derived. This is insofar remarkable since for most
other iterative transformations, this has not been possible or the kernel is not fully known. For the theoretical
characterisation of the transformation, determinantal representations are provided. Various theorems 
concerning convergence properties of the $\mathcal{J}$ transformation   are proven. Extensively, the relation to other
sequence transformations is explained. It turns out that many known methods are special cases of the $\mathcal{J}$ transformation.
The latter proves to be a very flexible tool with many variants due to the various possible choices of the hierarchies
and remainder estimates. Hence, the $\mathcal{J}$ transformation can be easily adapted to a large number of different 
problems, and this can be done on the basis of simple heuristical principles. Consequently it is not surprising that
the numerical tests at the end of Chapter 3 show that suitable variants of the $\mathcal{J}$ transformation belong to
the best methods for well-studied model problems.

In Chapter 4, remainder estimates are investigated that are important in the context of Levin-type transformations.
Especially, the possibilities are studied that come into play if a Kummer transformation of the partial sums of infinite series 
is possible, i.e., if the asymptotical behavior of the terms of the series is known and if the corresponding asymptotically
related series can be evaluated in closed form. A new form of the remainder estimates
is proposed that is based on this related series. By numerical examples it is demonstrated
that this new choice of the remainder estimate can indeed be superior to known methods.

In Chapter 5, methods for the extrapolation of orthogonal expansions are considered. First,
methods for Fourier series are treated. The 
$\mathcal{H}$ transformation is derived on the basis of model sequence that generalizes that of Levin.
Recursive algorithms for the efficient computation of the transformation and basic mathematical
properties are derived. Theorems are proven  regarding convergence acceleration properties of the 
$\mathcal{H}$ transformation. It is shown that the transformation can be implemented by a very simple
program with very low storage requirements. This program is given in Appendix D as an example for the
short and efficient programs that are also possible for the other newly introduced extrapolation methods.
Numerical test result are given that show that the $\mathcal{H}$ transformation can be used very effectively 
for the convergence acceleration of Fourier series. A generalization of the $\mathcal{H}$ transformation for
several frequencies and an efficient recursive algorithm for its implementation are presented for the first
time. 
Further, the  $\mathcal{I}$ transformation is derived on the basis of the concept of hierarchical consistency. This iterative 
transformation can be implemented using various efficient algorithms. It can also be used for Fourier series as shown
by numerical test results. These data indicate that suitable variants of the $\mathcal{I}$ transformation
are approximately as efficient as the $\mathcal{H}$ transformation. It is discussed that close to point of
discontinuity or singularities special problem arise that can be tackled using the {\em method of
manyfold frequencies}. Alternatively to the use of special methods for the convergence acceleration of
Fourier series, it is proposed to rewrite the Fourier series in terms of other types of series for
which the methods introduced in Chapters 2 and 3 are applicable. One possibility is the use of 
alternating series. This turns out to be useful near singularities. Another possibility is to rewrite
the Fourier series as sums of complex power series. This approach is well-known for simple cases
({\em method of associated series}\footnote{Methode der  assozierten Reihen} for Fourier series with real, smooth, non-oscillating coefficients) but
could be generalized to the {\em method of adjunct series}\footnote{Methode der zugeordneten Reihen} and especially 
the newly introduced 
{\em generalized method of adjunct series} 
that are applicable to more complicated Fourier series. As a further topic, expansion in orthogonal polynomials are treated by 
methods that generalize those for Fourier series. As an iteration extrapolation method the $\mathcal{K}$ transformation
is derived on the basis of the concept of hierarchical consistency. This transformation is implemented by simple
algorithms. Alternatively it is shown that one can also use the {\em generalized method of adjunct series} upon
suitable modification. Both methods can be used in the vicinity of singularities if the {\em manyfold frequency} approach is
used that is easily adapted to the case of orthogonal expansions. The efficiency of the methods is demonstrated by
numerical examples.


In Chapter 6, extrapolation methods are presented that are taylored for problems in the field of perturbation theory. Here, the small value of
computable terms of the perturbation series hinders the application of other extrapolation methods. We discuss methods that are based on a simple
renormalization of the perturbation series, induced by a scaling of the unperturbed Hamiltonian by a constant factor that is determined variationally.
This leads to new approximations of the total energy in form of  Goldhammer-Feenberg and Feenberg series. Alternatively, one can use the method of
effective characteristic polynomials in which the coefficients of the polynom are not computed via linear variation but by comparison to perturbation theory.  


In Chapters 7 and 8, extrapolation methods for vector and matrix sequences are treated. In Chapter 7, known methods are reviewed starting from basic information 
about the concept of pseudo inverses that replace true inverses in the case of vectors, and about special features regarding iteration sequences and matrix functions
in general. Further, some known algorithms for vector and matrix sequences are presented.

In Chapter 8, generalizations of the $\mathcal{J}$ transformation to vector and matrix sequences are introduced. These depend on the pseudo inverses discussed in
Chapter 7 and represent transformations with a large potential for applications. This opinion is supported by numerical examples regarding the computation of the
matrix exponential function.
  
Part 2 starts with Chapter 9. The topic of that chapter is the application of extrapolation methods to the computation of the lineshape of spectral holes. In the
beginning, a simple model is sketched that involves certain convolution integrals. Subsequently, representations for these integrals are discussed that form the basis
for the computation of the convolution integrals. An example  is given where experimental data are fitted using programs based on extrapolation methods in order to 
determine parameters of spectroscopic interest.

In Chapter 10, the methods of Chapter 5 for orthogonal expansions are applied to multipole expansions of the electrostatic potential and to three-center nuclear
attraction integrals with exponential-type basis functions. For a simple example, it is shown that extrapolation methods, i.e. the $\mathcal{K}$ transformation, 
can lead to a pronounced convergence acceleration for multipole expansions and for expansions of three-center nuclear attraction integrals in terms of spherical
harmonics.


The calculation of quasi-particle corrections on the basis of the inverse Dyson equation is the topic of Chapter 11. With this method, one can describe the influence
of electron correlation in a one-particle context. The inverse Dyson equation in diagonal approximation is solved by direct iteration. It is shown that a suitable
extrapolation method, the Overholt process, can accelerate the convergence drastically.

In Chapter 12, the extrapolation methods of Chapter 6 are applied to the M{\o}ller-Plesset perturbation series. It is shown that these combined methods lead to 
{\em size-extensive} approximations. From a study of a large number of benchmark calculations of smaller molecules it is concluded that Feenberg,
Goldhammer-Feenberg and Pad{\'e} extrapolations and also the $\Pi_2$ method based on effective characteristic polynomials all normally lead to reliable and accurate
estimates for the correlation energy. It is proposed to assess the applicability of the  M{\o}ller-Plesset perturbation series in dependence of possible differences
of the various estimates from each other.

In Chapter 13, the  Ornstein-Zernike equation with various closure relations is treated. The solution of this equation yields the two-particle distribution function
$g$ of classical many-body theory and thus allows the calculation of the thermodynamical properties of the corresponding systems. The Ornstein-Zernike equation 
with a certain closure is a non-linear integral equation that can be solved by direct iteration. It is shown that the computational effort can be reduced by up to
50 \% by using a variant of the vector-${\mathcal J}$ transformation. Further, examples are presented that show that a divergent direct iteration sequence can be
transformed into a convergent sequence by extrapolation methods.

The present summary is the end of Part 2. 

In Part 3, supplementary material is included in appendices. In Appendix A, basic notations and definitions are given. In Appendices B and C some more technical
lemmas are proved that are used in Chapters 3 and 9. In Appendix D, a Fortran program for the $\mathcal{H}$ transformation is given as one example for the easy
coding of the various newly introduced extrapolation algorithms. Finally, in Appendix E it is shown how to extend the method of effective characteristic polynomials
of Chapter 6 to the simultaneous treatment of several perturbation series.  

\end{document}