  The complete elliptic integral of the first kind (CEI-1) plays in a
significant role in mathematics, physics and engineering. There is no simple
formula for its computation, thus numerical algorithms are essential for coping
with practical problems involved. The commercial implementations for the
numerical solutions, such as the functions ellipticK and EllipticK provided by
MATLAB and Mathematica respectively, are based on
$\mathcal{K}_{\mathrm{cs}}(m)$ instead of the usual form $K(k)$ such that $m =
k^2$ and $\mathcal{K}_{\mathrm{cs}}(k^2) = K(k)$. It is necessary to develop
open source implementations for the computation of the CEI-1 in order to avoid
potential risks of using commercial software and possible limitations due to
the unknown factors. In this paper, 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. Numerical
results show that our open source implementations based on $K(k)$ are
equivalent to the commercial implementation based on
$\mathcal{K}_{\mathrm{cs}}(m)$. The general algorithms for computing orthogonal
polynomials developed are significant byproducts in the sense of STEM education
and scientific computation.
