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15
Accelerating Filtering Techniques for Numeric CSPs
, 2002
"... Search algorithms for solving Numeric CSPs (Constraint Satisfaction Problems) make an extensive use of filtering techniques. In this paper we show how those filtering techniques can be accelerated by discovering and exploiting some regularities during the filtering process. Two kinds of regularit ..."
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Cited by 11 (3 self)
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Search algorithms for solving Numeric CSPs (Constraint Satisfaction Problems) make an extensive use of filtering techniques. In this paper we show how those filtering techniques can be accelerated by discovering and exploiting some regularities during the filtering process. Two kinds of regularities are discussed, cyclic phenomena in the propagation queue and numeric regularities of the domains of the variables. We also present in this paper an attempt to unify numeric CSPs solving methods from two distinct communities, that of CSP in artificial intelligence, and that of interval analysis. 2002 Elsevier Science B.V. All rights reserved.
Extrapolation algorithms and Padé approximations: a historical survey
, 1994
"... This paper will give a short historical overview of these two subjects. Of course, we do not pretend to be exhaustive nor even to quote every important contribution. We refer the interested reader to the literature and, in particular to the recent books [5, 22, 29, 24, 38, 46, 48, 68, 78, 131]. For ..."
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Cited by 10 (2 self)
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This paper will give a short historical overview of these two subjects. Of course, we do not pretend to be exhaustive nor even to quote every important contribution. We refer the interested reader to the literature and, in particular to the recent books [5, 22, 29, 24, 38, 46, 48, 68, 78, 131]. For an extensive bibliography, see [23]. 1 Extrapolation methods Let (S n ) be the sequence to be accelerated. It is assumed to converge to a limit S. An extrapolation method consists in transforming this sequence into a new one, (T n ), by a sequence transformation T : (S n ) \Gamma! (T n ). The transformation T is said to accelerate the convergence of the sequence (S n ) if and only if lim n!1 T n \Gamma S S n \Gamma S =<F13.
Matrix and Vector Sequence Transformations Revisited
 Proc. Edinburgh Math. Soc
, 1995
"... . Sequence transformations are extrapolation methods. They are used for the purpose of convergence acceleration. In the scalar case, such algorithms can be obtained by two different approaches which are equivalent. The first one is an elimination approach based on the solution of a system of linear ..."
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Cited by 9 (7 self)
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. Sequence transformations are extrapolation methods. They are used for the purpose of convergence acceleration. In the scalar case, such algorithms can be obtained by two different approaches which are equivalent. The first one is an elimination approach based on the solution of a system of linear equations and it makes use of determinants. The second approach is based on the notion of annihilation difference operators. In this paper, these two approaches are generalized to the matrix and the vector cases. Key words. Convergence acceleration, extrapolation. AMS(MOS) subject classifications. 65B05. There exist many algorithms for transforming a sequence of numbers, or a sequence of vectors, or a sequence of matrices into a new sequence of objects of the same type. Such sequence transformations are used for accelerating the convergence of the initial sequence. They are often much useful, and even essential, since many sequences and many iterative processes used in numerical analysis ...
A Hierarchically Consistent, Iterative Sequence Transformation
 NUMER. ALGO
, 1994
"... Recently, the author proposed a new nonlinear sequence transformation, the iterative J transformation, which was shown to provide excellent results in several applications [H. H. H. Homeier, Some applications of nonlinear convergence accelerators, Int. J. Quantum Chem. 45, 545  562 (1993)]. In the ..."
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Cited by 8 (7 self)
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Recently, the author proposed a new nonlinear sequence transformation, the iterative J transformation, which was shown to provide excellent results in several applications [H. H. H. Homeier, Some applications of nonlinear convergence accelerators, Int. J. Quantum Chem. 45, 545  562 (1993)]. In the present contribution, this sequence transformation is derived by a hierarchically consistent iteration of some basic transformation. Hierarchical consistency is proposed as an approach to control the wellknown problem that the basic transformation can be generalized in many ways. Properties of the J transformation are studied. It is of similar generality as the wellknown E algorithm [C. Brezinski, A general extrapolation algorithm, Numer. Math. 35, 175  180 (1980). T. Havie, Generalized Neville type extrapolation schemes, BIT 19, 204  213 (1979)]. It is shown that the J transformation can be implemented quite easily. In addition to the defining representation there are alternative algorithms for its computation based on generalized differences. The kernel of the J transformation is derived. The expression for the kernel is relatively compact and does not depend on any lowerorder transforms. It is shown that several important other sequence transformations can be computed in an economical way using the the J transformation.
A Convergent Renormalized Strong Coupling Perturbation Expansion for the Ground State Energy of the Quartic, Sextic, and Octic Anharmonic Oscillator
 NY) 246, 133
, 1989
"... The RayleighSchrodinger perturbation series for the energy eigenvalue of an anharmonic oscillator defined by the Hamiltonian H (m) (fi) = p 2 + x 2 + fi x 2m with m = 2; 3; 4; . . . diverges quite strongly for every fi 6= 0 and has to summed to produce numerically useful results. How ..."
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Cited by 8 (6 self)
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The RayleighSchrodinger perturbation series for the energy eigenvalue of an anharmonic oscillator defined by the Hamiltonian H (m) (fi) = p 2 + x 2 + fi x 2m with m = 2; 3; 4; . . . diverges quite strongly for every fi 6= 0 and has to summed to produce numerically useful results. However, a divergent weak coupling expansion of that kind cannot be summed effectively if the coupling constant fi is large. A renormalized strong coupling expansion for the ground state energy of the quartic, sextic, and octic anharmonic oscillator is constructed on the basis of a renormalization scheme introduced by F. Vinette and J. C'izek [J. Math. Phys. 32 (1991), 3392]. This expansion, which is a power series in a new effective coupling constant with a bounded domain, permits a convenient computation of the ground state energy in the troublesome strong coupling regime. It can be proven rigorously that the new expansion converges if the coupling constant is sufficiently large. Mo...
Convergence Acceleration During the 20th Century
 J. Comput. Appl. Math
, 2000
"... This paper, which is based on [31], but includes new developments obtained since 1995, presents my personal views on the historical development of this subject during the 20th century. I do not pretend to be exhaustive nor even to quote every important contribution (if a reference does not appear be ..."
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Cited by 6 (2 self)
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This paper, which is based on [31], but includes new developments obtained since 1995, presents my personal views on the historical development of this subject during the 20th century. I do not pretend to be exhaustive nor even to quote every important contribution (if a reference does not appear below, it does not mean that it is less valuable). I refer the interested reader to the literature and, in particular to the recent books [55, 146, 33, 144]. For an extensive bibliography, see [28]
Mathematical properties of a new Levintype sequence transformation introduced by Číˇzek
 Journal of Mathematical Physics
, 2003
"... Číˇzek, Zamastil, and Skála [J. Math. Phys. 44, 962 – 968 (2003)] introduced in connection with the summation of the divergent perturbation expansion of the hydrogen atom in an external magnetic field a new sequence transformation which uses as input data not only the elements of a sequence {sn} ∞ ..."
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Cited by 6 (6 self)
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Číˇzek, Zamastil, and Skála [J. Math. Phys. 44, 962 – 968 (2003)] introduced in connection with the summation of the divergent perturbation expansion of the hydrogen atom in an external magnetic field a new sequence transformation which uses as input data not only the elements of a sequence {sn} ∞ n=0 of partial sums, but also explicit estimates {ωn} ∞ n=0 for the truncation errors. The explicit incorporation of the information contained in the truncation error estimates makes this and related transformations potentially much more powerful than for instance Padé approximants. Special cases of the new transformation are sequence transformations introduced by Levin [Int. J.
A General Extrapolation Procedure Revisited
, 1993
"... The Ealgorithm is the most general extrapolation algorithm actually known. The aim of this paper is to provide a new approach to this algorithm. This approach gives a deeper insight into the Ealgorithm, allows to obtain new properties and to relate it to other algorithms. Some extensions of the ..."
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Cited by 5 (1 self)
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The Ealgorithm is the most general extrapolation algorithm actually known. The aim of this paper is to provide a new approach to this algorithm. This approach gives a deeper insight into the Ealgorithm, allows to obtain new properties and to relate it to other algorithms. Some extensions of the procedure are discussed.
Linear difference operators and acceleration methods
, 1997
"... The aim of this paper is the study of the kernel and acceleration properties of sequence transformations of the form T n = L(S n =D n)=L(1=D n) , where (S n) is the sequence for which we want to compute the limit, (D n) is an error estimate and L is a linear difference operator. We will obtain thos ..."
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Cited by 3 (1 self)
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The aim of this paper is the study of the kernel and acceleration properties of sequence transformations of the form T n = L(S n =D n)=L(1=D n) , where (S n) is the sequence for which we want to compute the limit, (D n) is an error estimate and L is a linear difference operator. We will obtain those properties for different classes of operators L and we will give a procedure for constructing, for a given class of sequences, an operator for which the corresponding transformation accelerates that class.
Vector and Matrix Sequence Transformations Based on Biorthogonality
 Appl. Numer. Math
, 1996
"... Sequence transformations are used for the purpose of convergence acceleration. An important algebraic property connected with a sequence transformation is its kernel, that is the set of sequences transformed into a constant sequence (usually the limit of the sequence). In this paper, we show how to ..."
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Cited by 2 (1 self)
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Sequence transformations are used for the purpose of convergence acceleration. An important algebraic property connected with a sequence transformation is its kernel, that is the set of sequences transformed into a constant sequence (usually the limit of the sequence). In this paper, we show how to construct transformations whose kernels are the sets of vector or matrix sequences of the forms x n = x + Z n ff, x n = x + Z n ff n and x n = x+Z n ff n +Y n fi where Z n and Y n are known matrices, ff; ff n and fi unknown vectors or matrices. Recursive algorithms for their implementation are given. Applications to the solution of systems of linear and nonlinear equations are also discussed.