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While Grover's tabular method to compute the mutual inductance of coaxial single-layer coils is convenient and fast when implemented on computer, its use has mostly been restricted to zeroth- and first-order calculations, since its accuracy is not well defined or quantified. This paper defines and quantifies the accuracy of Grover's tabular method for single-layer coaxial coils and shows that interpolation between the tabulated values is the primary factor limiting its accuracy. The results of this paper are used to develop an improved tabular-solution method that has accuracy approaching that of a finite-element field-code analysis. The analysis compares three mutual-inductance calculation methods for coaxial single-layer coils: a finite-element numerical method, an analytical method involving the numerical elliptic-integral evaluation, and Grover's tabular method. The finite-element numerical solution and the analytic solution are the most accurate and comparable to each other, while the tabular method is the least accurate. In terms of speed, however, the tabular method is the fastest while the finite-element method is the slowest. The interpolation techniques used in the analysis include cubic spline, cubic polynomial, polynomial, and linear. In comparison to finite-element numerical calculations, the improved tabular method developed in this paper decreases the computation time a factor of 4 times 103 while retaining an accuracy of at least four significant digits. The improved tabular method can be applied to similar-type problems that rely on the evaluation of elliptic integrals in their solution.