In this paper, we discuss the application of the author’s \(\tilde {d}^{(m)}\) transformation to accelerate the convergence of infinite series \({\sum }^{\infty }_{n=1}a_n\) when the terms a_n have asymptotic expansions that can be expressed in the form

We discuss the implementation of the \(\tilde {d}^{(m)}\) transformation via the recursive W-algorithm of the author. We show how to apply this transformation and how to assess in a reliable way the accuracies of the approximations it produces, whether the series converge or they diverge. We classify the different cases that exhibit unique numerical stability issues in floating-point arithmetic. We show that the \(\tilde {d}^{(m)}\) transformation can also be used efficiently to accelerate the convergence of infinite products \({\prod }^{\infty }_{n=1}(1+v_n)\), where \(v_n\sim {\sum }^{\infty }_{i=0}e_in^{-t/m-i/m}\) as \(n\to \infty \), t ≥ m + 1 an integer. Finally, we give several numerical examples that attest the high efficiency of the \(\tilde {d}^{(m)}\) transformation for the different cases.

在本文中，我们讨论了作者的\（\ tilde {d} ^ {（m）} \）变换在加速无限级数\（{\ sum} ^ {\ infty} _ {n = 1 } A_N \）时，术语一个_ñ具有可以在形式表示渐近展开

我们通过作者的递归W算法讨论\（\ tilde {d} ^ {（m）} \）转换的实现。我们展示了如何应用该变换，以及如何可靠地评估其产生的近似值的准确性，无论该序列是收敛的还是发散的。我们对在浮点算法中表现出独特的数值稳定性问题的不同情况进行分类。我们证明\（\ tilde {d} ^ {（m）} \）变换也可以有效地用于加速无限乘积\（{\ prod} ^ {\ infty} _ {n = 1}（ 1 + v_n）\），其中\（v_n \ sim {\ sum} ^ {\ infty} _ {i = 0} e_in ^ {-t / mi / m} \）为\（n \ to \ infty \），吨≥米+1个整数。最后，我们给出几个数值示例，以证明在不同情况下\（\ tilde {d} ^ {（m）} \）转换的高效性。