The higher-order WKB Mathematica® code for computing quasinormal modes, whose accuracy was significantly enhanced through extensions to higher orders and, in particular, through the use of Pad´e resummation, has been widely employed in numerous studies over the past several years. In this work, we present an updated and optimized version of the code. The main improvement consists in expanding the effective potential in a Taylor series around its maximum, rather than evaluating the full analytic expression of the WKB formula for each specific potential. This modification leads to a substantial reduction in computation time. In cases where the effective potential is complicated and involves non-rational functions, the speed gain can reach several orders of magnitude, while preserving the accuracy of the method.
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An Efficient Higher-Order WKB Code for Quasinormal Modes and Greybody Factors
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Received: 08 Mar 2026 | Revised: 17 Mar 2026 | Accepted: 19 Mar 2026 | Published: 23 Mar 2026
Abstract
Keywords
WKB approximation | black holes | quasinormal modes | grey-body factors
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