• *FormulaII*

Mysovskikh [23] derived the spherical integral formula according to the transformation group of the regular simplex. Lu and Darmofal [24] proposed a new fifth-degree cubature formula based on the integral formula of Mysovskikh, which is similar to the formula proposed by Stroud et al. It also decomposes the Gaussian weighted integral into the product of the spherical and radial integrals:

$$\begin{split} G(\mathfrak{u}) &\approx \frac{2(\pi)^{n/2}}{n+2} \mathfrak{u}(\mathfrak{d}) \\ &+ \frac{n^{2}(7-n)(\pi)^{n/2}}{2(n+1)^{2}(n+2)^{2}} \sum\_{k=1}^{n+1} \left( \mathfrak{u}\left(\sqrt{\frac{\mathfrak{u}}{2}+1}\mathfrak{a}\_{k}\right) + \mathfrak{u}\left(-\sqrt{\frac{\mathfrak{u}}{2}+1}\mathfrak{a}\_{k}\right) \right) \\ &+ \frac{2(n-1)(\pi)^{n/2}}{(n+1)^{2}(n+2)^{2}} \sum\_{k=1}^{n(n+1)/2} \left( \mathfrak{u}\left(\sqrt{\frac{\mathfrak{u}}{2}+1}\mathfrak{b}\_{k}\right) + \mathfrak{u}\left(-\sqrt{\frac{\mathfrak{u}}{2}+1}\mathfrak{b}\_{k}\right) \right) . \end{split} \tag{17}$$

In this formula, the values of the cubature points and parameters are given as follows:

$$\begin{aligned} a\_k &= \left( a\_{1,k}, a\_{2,k}, \dots, a\_{n,k} \right)^T, k = 1, 2, \dots, n+1 \\ a\_{i,k} &= \begin{cases} -\sqrt{\frac{n+1}{n(n-i+2)(n-i+1)}}, i < k \\ \sqrt{\frac{(n+1)(n-k+1)}{n(n-k+2)}}, i = k \end{cases}, \mathbf{b}\_k = \sqrt{\frac{n}{2(n-1)}} \mathbf{v}\_{k'} \\ \mathbf{V}\_{n \times \frac{n(n+1)}{2}} &= \underbrace{(\mathbf{a}\_1 + \mathbf{a}\_2, \dots, \mathbf{a}\_1 + \mathbf{a}\_{n+1}, \mathbf{a}\_2 + \mathbf{a}\_3, \dots, \mathbf{a}\_2 + \mathbf{a}\_{n+1}, \dots, \mathbf{a}\_n + \mathbf{a}\_{n+1})}\_{n(n+1)/2}, \end{aligned} \tag{18}$$

where *<sup>a</sup><sup>k</sup>* represents the *n* + 1 vertices of n-dimensional hypersphere *Sn*, and *<sup>b</sup><sup>k</sup>* represents the topological mapping of the midpoints of the vertices of the simplex on hypersphere *Sn*. The number of cubature points required by this formula is *n*<sup>2</sup> + 3*n* + 3. For low dimensional systems, this formula requires more cubature points than the cubature formula of HDCKF, and resulting in unnecessary costs. According to Theorem 2, the stability index of the formula can be calculated as stb = 3, indicating that the algorithm has good numerical stability. However, it is difficult to extend and improve the formula, because of the complex structure of the spherical simplex criterion.
