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

翻译：在数学、物理和工程学方面,第一种完全的椭圆体(CEI-1)在数学、物理和工程学方面起着重要作用。没有简单的计算公式,因此数字算法和解决办法对实际问题至关重要。然而,我们发现,通过MATLAB和数学获得的数字解决办法是不可接受的,应当认真对待。为了获得CEI-1的正确和替代数字算法,无限系列方法、算术-地理平均数(AGM)方法、高斯-切比谢夫方法和高斯-Legendre方法与自上而下的战略详细讨论。计算CEI-1的四种关键算法是设计、核实、验证和测试的,可以在研发中加以利用并适当地再利用。在STEM教育、系统工程和计算思维的意义上,核查-估价-测试(VVT)阶段对于应用和教学大学生以避免不必要的损失至关重要。