Efficient evaluation of infinite-series representations for overlap, two-center nuclear attraction, and Coulomb integrals using nonlinear convergence accelerators

Grotendorst, Johannes; Weniger, Ernst Joachim; Steinborn, E. Otto

doi:10.1103/physreva.33.3706

Cited by 85 publications

(40 citation statements)

References 21 publications

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“…However, experience indicates that these remainder estimates nevertheless give rise to very powerful sequence transformations [29,30,57,[60][61][62][63][64].…”

Section: Remainder Estimates For the Levin Transformationmentioning

confidence: 99%

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Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series

Weniger

1989

Computer Physics Reports

414

535

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Slowly convergent series and sequences as well as divergent series occur quite frequently in the mathematical treatment of scientific problems. In this report, a large number of mainly nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series are discussed. Some of the sequence transformations of this report as for instance Wynn's ǫ algorithm or Levin's sequence transformation are well established in the literature on convergence acceleration, but the majority of them is new. Efficient algorithms for the evaluation of these transformations are derived. The theoretical properties of the sequence transformations in convergence acceleration and summation processes are analyzed. Finally, the performance of the sequence transformations of this report are tested by applying them to certain slowly convergent and divergent series, which are hopefully realistic models for a large part of the slowly convergent or divergent series that can occur in scientific problems and in applied mathematics.

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“…However, experience indicates that these remainder estimates nevertheless give rise to very powerful sequence transformations [29,30,57,[60][61][62][63][64].…”

Section: Remainder Estimates For the Levin Transformationmentioning

confidence: 99%

“…(14.3-9) that the terms of the infinite series (14.3-10) behave like n −3/2 as n → ∞ (see p. 3709 of ref. [64]). In view of eqs.…”

Section: Some Numerical Test Seriesmentioning

confidence: 99%

Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series

Weniger

1989

Computer Physics Reports

414

535

View full text Add to dashboard Cite

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“…Putting S(a,p,t) = ((1 -t)aZ + tp2)1/2, one has (compare [21], [19], and [28]) (20) Alternatively, the Jacobi polynomial representation ( [13] …”

Section: Overlap Integrals With B Functionsmentioning

confidence: 98%

“…The convergence of the infinite series (23) can be further improved by the use of acceleration methods [20,28].…”

Section: Overlap Integrals With B Functionsmentioning

confidence: 99%

On the evaluation of overlap integrals with exponential‐type basis functions

Homeier

Steinborn

1992

Int J of Quantum Chemistry

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“…Moreover, numerous applications of sequence transformations have been reported in the literature. For example, the present author has applied sequence transformations successfully in such diverse fields as the evaluation of special functions [12][13][14][15][16][17], the evaluation of molecular multicenter integrals of exponentially decaying functions [18][19][20][21][22], the summation of strongly divergent quantum mechanical perturbation expansions [13,[23][24][25][26][27][28][29][30][31], and the extrapolation of crystal orbital and cluster calculations for oligomers to their infinite chain limits of stereoregular quasi-onedimensional organic polymers [32,33].…”

Section: Introductionmentioning

confidence: 99%

Irregular input data in convergence acceleration and summation processes: General considerations and some special Gaussian hypergeometric series as model problems

Weniger

2001

Computer Physics Communications

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Sequence transformations accomplish an acceleration of convergence or a summation in the case of divergence by detecting and utilizing regularities of the elements of the sequence to be transformed. For sufficiently large indices, certain asymptotic regularities normally do exist, but the leading elements of a sequence may behave quite irregularly. The Gaussian hypergeometric series 2F1(a, b; c; z) is well suited to illuminate problems of that kind. Sequence transformations perform quite well for most parameters and arguments. If, however, the third parameter c of a nonterminating hypergeometric series 2F1 is a negative real number, the terms initially grow in magnitude like the terms of a mildly divergent series. The use of the leading terms of such a series as input data leads to unreliable and even completely nonsensical results. In contrast, sequence transformations produce good results if the leading irregular terms are excluded from the transformation process. Similar problems occur also in perturbation expansions. For example, summation results for the infinite coupling limit k3 of the sextic anharmonic oscillator can be improved considerably by excluding the leading terms from the transformation process. Finally, numerous new recurrence formulas for the 2F1(a, b; c; z) are derived.

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Efficient evaluation of infinite-series representations for overlap, two-center nuclear attraction, and Coulomb integrals using nonlinear convergence accelerators

Cited by 85 publications

References 21 publications

Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series

Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series

On the evaluation of overlap integrals with exponential‐type basis functions

Irregular input data in convergence acceleration and summation processes: General considerations and some special Gaussian hypergeometric series as model problems

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