Acceleration of Linear and Logarithmic Convergence

• Published in: SIAM journal on numerical analysis 1979-04, Vol.16 (2), p.223-240
• Main Authors: Smith, David A., Ford, William F.
• Format: Article
• Language: English
• Subjects: Algorithms; Alternating series; Geometric series; Linear convergence; Mathematical theorems; Mathematical transformations; Methods; Neighborhoods; Partial sums; Perceptron convergence procedure; Series convergence
• ISSN: 0036-1429; 1095-7170
• DOI: 10.1137/0716017

Eleven different methods for accelerating convergence of sequences and series have been tested and compared on a wide range of test problems, including both linearly and logarithmically convergent series, monotone and alternating series. All but one of these methods are already in the literature, and they include both linear and nonlinear methods. The only methods found to accelerate convergence across the board were the u and v transforms of Levin and the θ algorithm of Brezinski. The paper gives detailed comparisons of all the tested methods on the basis of number of correct digits in the answer as a function of number of terms of the series used. A theorem of Germain-Bonne states that methods of a certain form which are exact on geometric series will accelerate linear convergence. The theorem applies to θ2, and we have extended it to apply to Levin's transforms. No corresponding theorem is known for logarithmic convergence, but u, v, and θ are exact on certain large classes of logarithmic series, and all tested methods lacking this property failed to accelerate some logarithmically convergent series.