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The complete elliptic integral of the first kind (CEI-1) plays in asignificant role in mathematics, physics and engineering. There is no simpleformulae for its computation, thus numerical algorithms and solution areessential in practical problems. However, we find that the numerical solutionsobtained via both MATLAB and Mathematica are not acceptable and should betreated seriously. For the purpose of obtaining correct and alternativenumerical algorithms for the CEI-1, the infinite series method,arithmetic-geometric mean (AGM) method, Gauss-Chebyshev method andGauss-Legendre methods are discussed in details with a top-down strategy. Thefour key algorithms for computing CEI-1 are designed, verified, validated andtested, which can be utilized in R & D and be reused properly. In the sense ofSTEM education, system engineering and computational thinking, theVerification-Validation-Testing (VVT) stage is crucial for applications andteaching college students in order to avoid unnecessary losses.

Correct and Alternative Numerical Algorithms for the Complete Elliptic Integral of the First Kind in MATLAB and Mathematica