We recently found that the coordinate transformation method (the C method) equipped with well-established recursive algorithms for solving the system of linear equations is numerically instable when it is applied to thinly coated gratings. The origin of this new kind of numerical instability is not the exponential dependence of the field in the coated layers but the ill condition of the eigenvector matrix of the C method when the truncation number is high. Two simple and effective methods to circumvent the new instability are recommended. We also found that the popular recursive matrix algorithms have different (poor) immunities to the new instability, and they all perform inferiorly to the full matrix (nonrecursive) algorithm.

中文翻译：

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C方法应用于涂层光栅时的数值不稳定性及其避免方法。

我们最近发现，配备有完善的递归算法来求解线性方程组的坐标转换方法（C方法）在应用于薄涂层光栅时在数值上不稳定。这种新的数值不稳定性的根源不是涂层层中磁场的指数依赖性，而是当截断数高时C方法的特征向量矩阵的病态。建议使用两种简单有效的方法来规避新的不稳定性。我们还发现，流行的递归矩阵算法对新的不稳定性具有不同的（较差）免疫力，并且它们的性能均不如完整矩阵（非递归）算法。