We propose a high precision algorithm for solving the Gelfand–Levitan–Marchenko equation. The algorithm is based on the block version of the Toeplitz Inner-Bordering algorithm of Levinson’s type. To approximate integrals, we use the high-precision one-sided and two-sided Gregory quadrature formulas. Also we use the Woodbury formula to construct a computational algorithm. This makes it possible to use the almost Toeplitz structure of the matrices for the fast calculations. To the best of our knowledge, this is the first algorithm to solve this problem with an order of accuracy higher than the second.

中文翻译：

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求解 Gelfand-Levitan-Marchenko 方程的高阶分块 Toeplitz 内边界法


我们提出了一种高精度算法来求解 Gelfand-Levitan-Marchenko 方程。该算法基于 Levinson 类型的 Toeplitz Inner-Bordering 算法的块版本。为了近似积分，我们使用高精度单边和双边格里高利求积公式。我们还使用伍德伯里公式来构建计算算法。这使得使用矩阵的近似托普利茨结构进行快速计算成为可能。据我们所知，这是第一个解决该问题的算法，其精度比第二个高一个数量级。