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Precise Numerical Differentiation of Thermodynamic Functions with Multicomplex Variables



Ian Bell, Ulrich Deiters


The multicomplex finite-step method for numerical differentiation is an extension of the popular Squire--Trapp method, which uses complex arithmetics to compute first-order derivatives with almost machine precision. In contrast to this, the multicomplex method can be applied to higher-order derivatives. Furthermore, it can be applied to functions of more than one variable and obtain mixed derivatives. It is possible to compute various derivatives at the same time. This work demonstrates the numerical differentiation with multicomplex variables for some thermodynamic problems. The method can be easily implemented into existing computer programs, applied to equations of state of arbitrary complexity, and achieves almost machine precision for the derivatives. Alternative methods based on complex integration are discussed, too.
Journal of Research (NIST JRES) -


multicomplex arithmetics, numerical differentiation


Bell, I. and Deiters, U. (2021), Precise Numerical Differentiation of Thermodynamic Functions with Multicomplex Variables, Journal of Research (NIST JRES), National Institute of Standards and Technology, Gaithersburg, MD, [online],, (Accessed May 26, 2024)


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Created November 29, 2021, Updated November 29, 2022