论文标题
汇合范德尔蒙德与阿诺迪
Confluent Vandermonde with Arnoldi
论文作者
论文摘要
在本说明中，我们使用P. D. Brubeck，Y。Nakatsukasa和L. N. Trefethen的Arnoldi方法扩展了Vandermonde，以处理汇合的Vandermonde矩阵。要应用Arnoldi过程，至关重要的是找到一个生成汇合Vandermonde矩阵的列空间的Krylov子空间。为任何订单衍生物为此类Krylov子空间建立一个定理。这使我们能够将高度多项式的衍生物计算为高精度。如数值示例所示，它还使许多涉及导数的应用成为可能。我们注意到，其中一种方法仅对函数值进行正交，并且等同于P. D. Brubeck和L. N. Trefethen给出的公式。另一种方法正交化了Hermite数据。关于哪种方法比另一种方法更可取，我们进行了比较，结果取决于问题。
In this note, we extend the Vandermonde with Arnoldi method recently advocated by P. D. Brubeck, Y. Nakatsukasa and L. N. Trefethen to dealing with the confluent Vandermonde matrix. To apply the Arnoldi process, it is critical to find a Krylov subspace which generates the column space of the confluent Vandermonde matrix. A theorem is established for such Krylov subspaces for any order derivatives. This enables us to compute the derivatives of high degree polynomials to high precision. It also makes many applications involving derivatives possible, as illustrated by numerical examples. We note that one of the approaches orthogonalizes only the function values and is equivalent to the formula given by P. D. Brubeck and L. N. Trefethen. The other approach orthogonalizes the Hermite data. About which approach is preferable to another, we made the comparison, and the result is problem dependent.
