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 $\mathrm{O}(N)$ 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：我们发展了一种数值方法，用于计算在多个不相交区间上正交的多项式，这些多项式目前尚无解析公式。该方法利用正交多项式的Fokas–Its–Kitaev Riemann–Hilbert表示，提出了一个计算前N个递推系数的O(N)算法。该方法还可用于多项式及其Cauchy变换在整个复平面上的逐点求值。该方法通过Chebyshev多项式的加权Cauchy积分来编码权函数的奇异性行为，从而显著提升计算效率，优于现有其他技术。我们验证了该方法快速收敛的特性，并展示了其在可积系统与逼近理论中的应用。