Analytical formula for numerical evaluations of the Wigner rotation matrices at high spins

The Wigner $d$ function, which is the essential part of an irreducible representation of SU(2) and SO(3) parameterized with Euler angles, has been know to suffer from a serious numerical errors at high spins, if it is calculated by means of the Wigner formula as a polynomial of cos and sin of half of the second Euler angle. This paper shows a way to avoid this problem by expressing the $d$ functions as the Fourier series of the half angle. A precise numerical table of the coefficients of the series is provided as Supplemental Material.

Figure 1

Maximum and root-mean-square errors in the values of $cos\nu \theta$ and $sin\nu \theta$, numerically calculated with 64-bit floating-point numbers, over $0\le \theta \le \pi$. The values of $\theta$ are sampled with a uniform spacing of ${10}^{-5}$ degree. The abscissa is $\nu =0,\frac{1}{2},1,\cdots ,100$. Recursion means the usage of Eq. (27) together with the initial values described in the following sentences.

Figure 2

The common logarithm of the error in the 64-bit floating-point numerical value of the $d$ function $d_{mk}^j\left(\theta \right)$ as a function of $m$ and $k$ for $j=40$ and $\theta ={30}^{\circ }$. The value of the $d$ function is calculated by means of the Wigner formula in part (a) and the Fourier series expression in part (b). Only a quarter of the possible combinations of $m$ and $k$ are plotted because the error is symmetric in the lines $m=\pm k$. Errors smaller than ${10}^{-20}$ are painted with the same color as that for the error of ${10}^{-20}$.

Figure 3

The same as in Fig. 2 but for $\theta ={60}^{\circ }$.

Figure 4

The same as in Fig. 2 but for $\theta ={90}^{\circ }$.

Figure 5

The common logarithm of the maximum error in the numerical value of the $d$ function $d_{mk}^j\left(\theta \right)$ vs $j$ calculated with the Wigner formula (blue [gray] solid line) and the Fourier series expression (red [gray] dashed line). The maximum is taken over the values of $m,k$, and $\theta$ for each $j$.