We develop a numerical method for computing with orthogonal polynomials that are orthogonal on multiple, disjoint intervals for which analytical formulae are currently unknown. Our approach exploits the Fokas--Its--Kitaev Riemann--Hilbert representation of the orthogonal polynomials to produce an $\OO(N)$ method to compute the first $N$ recurrence coefficients. The method can also be used for pointwise evaluation of the polynomials and their Cauchy transforms throughout the complex plane. The method encodes the singularity behavior of weight functions using weighted Cauchy integrals of Chebyshev polynomials. This greatly improves the efficiency of the method, outperforming other available techniques. We demonstrate the fast convergence of our method and present applications to integrable systems and approximation theory.

翻译：本文针对目前尚无解析公式的多区间、不相交区间上的正交多项式，发展了一套数值计算方法。该方法利用正交多项式的Fokas–Its–Kitaev Riemann-Hilbert表示，实现了计算前N个递推系数的$\OO(N)$算法。该方法还可用于复平面上多项式及其Cauchy变换的点态求值。通过引入Chebyshev多项式的带权Cauchy积分，该方法对权函数的奇异性行为进行编码，大幅提升了计算效率，并超越了现有其他技术。我们展示了该方法的快速收敛性，并将其应用于可积系统与逼近理论。