Abstract In this paper and in the framework of the Asymptotic Numerical Method (ANM), we investigate numerically improved vectorial Pade Approximants. The ANM is a branch-by-branch continuation algorithm, each branch is represented by a vectorial Taylor series with respect to a path parameter. In the ANM, the vectorial Pade approximants have been introduced to increase the validity range of vectorial Taylor series. In a recent work, we have introduced a new matrix generalized definition of vectorial Pade approximants of vectorial Taylor series truncated at order N. By using this definition, the vectorial Pade approximants of type [L, M] (N = L + M) have been investigated. We show also numerically, in this paper, that the vectorial Pade approximants [1, N − 1] when the coefficients bi are computed from the Gram-schmidt orthogonalization technique make it possible to reduce the number of ANM steps compared to the classical vectorial Pade appoximant. The effectiveness of these vectorial Pade approximants will be demonstrated on numerical examples of post-buckling of shells.

中文翻译：

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ANM算法中改进的Padé近似：应用于壳的后屈曲

摘要 在本文中，在渐近数值方法 (ANM) 的框架内，我们研究了数值改进的向量 Pade 逼近。ANM 是一种逐个分支的延续算法，每个分支由一个关于路径参数的向量泰勒级数表示。在 ANM 中，引入了向量 Pade 逼近，以增加向量泰勒级数的有效性范围。在最近的一项工作中，我们引入了在 N 阶截断的向量泰勒级数的向量 Pade 逼近的新矩阵广义定义。 通过使用这个定义，类型 [L, M] (N = L + M) 的向量 Pade 逼近有被调查。在本文中，我们还通过数值证明了矢量 Pade 逼近 [1, N - 1] 当系数 bi 从 Gram-schmidt 正交化技术计算时，与经典矢量 Pade 近似法相比，可以减少 ANM 步骤的数量。这些矢量 Pade 近似的有效性将在壳后屈曲的数值示例中得到证明。