Effective and efficient algorithm for the Wigner rotation matrix at high angular momenta

The Wigner rotation matrix ($d$-function), which appears as a part of the angular-momentum-projection operator, plays crucial role in modern nuclear-structure models. However, it is long-standing problem that its numerical evaluation suffers from serious errors and instability, hinders precise calculations for nuclear high-spin states. Recently, Tajima [Phys. Rev. C 91, 014320 (2015)] has made significant step toward solving by suggesting high-precision Fourier method, however relies on formula-manipulation softwares. In this paper we propose an effective efficient algorithm $d$ function based Jacobi polynomials. We compare our method with conventional through some testing calculations, demonstrate can always give stable results similar cases (for special sets $j, m, k$ $\theta$) ours are even more accurate. Moreover, self-contained less memory consuming. A related code subroutines provided Supplemental Material present paper.