High-Order Block Toeplitz Inner-Bordering method for solving the Gelfand-Levitan-Marchenko equation
Sergey Medvedev, Irina Vaseva, Mikhail Fedoruk
TL;DR
This paper introduces a high-precision algorithm leveraging block Toeplitz inner-bordering and Gregory quadrature for efficiently solving the Gelfand-Levitan-Marchenko equation with improved computational speed.
Contribution
It presents a novel high-order algorithm combining Toeplitz inner-bordering, Gregory quadrature, and Woodbury formula for fast and accurate solutions.
Findings
Achieves high-precision solutions for the Gelfand-Levitan-Marchenko equation.
Utilizes the Toeplitz structure for computational efficiency.
Demonstrates improved speed and accuracy over existing methods.
Abstract
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.
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Taxonomy
TopicsDifferential Equations and Numerical Methods · Numerical methods for differential equations · Differential Equations and Boundary Problems