### Abstract

A method of numerically evaluating slowly convergent monotone series is described. First, we apply a condensation transformation due to Van Wijngaarden to the original series. This transforms the original monotone series into an alternating series. In the second step, the convergence of the transformed series is accelerated with the help of suitable nonlinear sequence transformations that are known to be particularly powerful for alternating series. Some theoretical aspects of our approach are discussed. The efficiency, numerical stability, and wide applicability of the combined nonlinear-condensation transformation is illustrated by a number of examples. We discuss the evaluation of special functions close to or on the boundary of the circle of convergence, even in the vicinity of singularities. We also consider a series of products of spherical Bessel functions, which serves as a model for partial wave expansions occurring in quantum electrodynamic bound state calculations.

Original language | English |
---|---|

Pages (from-to) | 28-54 |

Number of pages | 27 |

Journal | Computer Physics Communications |

Volume | 116 |

Issue number | 1 |

Publication status | Published - Jan 1999 |

### Fingerprint

### Keywords

- Calculations and mathematical techniques in atomic and molecular physics
- Computational techniques
- Numerical approximation and analysis
- Quantum electrodynamics (specific calculations)

### ASJC Scopus subject areas

- Computer Science Applications
- Physics and Astronomy(all)

### Cite this

*Computer Physics Communications*,

*116*(1), 28-54.

**Convergence acceleration via combined nonlinear-condensation transformations.** / Jentschura, U.; Mohr, Peter J.; Soff, Gerhard; Weniger, Ernst Joachim.

Research output: Contribution to journal › Article

*Computer Physics Communications*, vol. 116, no. 1, pp. 28-54.

}

TY - JOUR

T1 - Convergence acceleration via combined nonlinear-condensation transformations

AU - Jentschura, U.

AU - Mohr, Peter J.

AU - Soff, Gerhard

AU - Weniger, Ernst Joachim

PY - 1999/1

Y1 - 1999/1

N2 - A method of numerically evaluating slowly convergent monotone series is described. First, we apply a condensation transformation due to Van Wijngaarden to the original series. This transforms the original monotone series into an alternating series. In the second step, the convergence of the transformed series is accelerated with the help of suitable nonlinear sequence transformations that are known to be particularly powerful for alternating series. Some theoretical aspects of our approach are discussed. The efficiency, numerical stability, and wide applicability of the combined nonlinear-condensation transformation is illustrated by a number of examples. We discuss the evaluation of special functions close to or on the boundary of the circle of convergence, even in the vicinity of singularities. We also consider a series of products of spherical Bessel functions, which serves as a model for partial wave expansions occurring in quantum electrodynamic bound state calculations.

AB - A method of numerically evaluating slowly convergent monotone series is described. First, we apply a condensation transformation due to Van Wijngaarden to the original series. This transforms the original monotone series into an alternating series. In the second step, the convergence of the transformed series is accelerated with the help of suitable nonlinear sequence transformations that are known to be particularly powerful for alternating series. Some theoretical aspects of our approach are discussed. The efficiency, numerical stability, and wide applicability of the combined nonlinear-condensation transformation is illustrated by a number of examples. We discuss the evaluation of special functions close to or on the boundary of the circle of convergence, even in the vicinity of singularities. We also consider a series of products of spherical Bessel functions, which serves as a model for partial wave expansions occurring in quantum electrodynamic bound state calculations.

KW - Calculations and mathematical techniques in atomic and molecular physics

KW - Computational techniques

KW - Numerical approximation and analysis

KW - Quantum electrodynamics (specific calculations)

UR - http://www.scopus.com/inward/record.url?scp=0032760685&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0032760685&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0032760685

VL - 116

SP - 28

EP - 54

JO - Computer Physics Communications

JF - Computer Physics Communications

SN - 0010-4655

IS - 1

ER -