关于Hahn渐近公式和稳定递推在快速、简单和稳定Chebyshev-Jacobi变换中的应用
On the use of Hahn's asymptotic formula and stabilized recurrence for a fast, simple, and stable Chebyshev--Jacobi transform
AI总结：
本文提出一种基于Jacobi展开在Chebyshev-Lobatto点数值评估的快速、简单且稳定的Chebyshev-Jacobi变换方法，利用Hahn渐近公式分解和稳定递推提高稳定性。
摘要：
我们描述了一种基于在Chebyshev-Lobatto点对Jacobi展开系数进行数值评估的快速、简单且稳定的Chebyshev展开系数到Jacobi展开系数及其逆变换方法。这通过将Hahn的内部渐近公式分解为小的对角缩放离散正弦和余弦变换之和，并利用稳定递推关系实现。已知Clenshaw-Smith算法在正交区间内不具有统一的稳定性，因此Reinsch的修改方法被扩展到Jacobi多项式，并在端点附近应用以提高数值稳定性。
英文摘要：
We describe a fast, simple, and stable transform of Chebyshev expansion coefficients to Jacobi expansion coefficients and its inverse based on the numerical evaluation of Jacobi expansions at the Chebyshev--Lobatto points. This is achieved via a decomposition of Hahn's interior asymptotic formula into a small sum of diagonally scaled discrete sine and cosine transforms and the use of stable recurrence relations. It is known that the Clenshaw--Smith algorithm is not uniformly stable on the entire interval of orthogonality. Therefore, Reinsch's modification is extended for Jacobi polynomials and employed near the endpoints to improve numerical stability.
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