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.

翻译：我们提出了一种高精度求解Gelfand-Levitan-Marchenko方程的算法。该算法基于Levinson型Toeplitz内边界算法的分块版本。为逼近积分，我们采用高精度的单侧和双侧Gregory求积公式。同时利用Woodbury公式构建计算算法，从而能够利用矩阵的近似Toeplitz结构实现快速计算。