We develop the Akhiezer iteration, a generalization of the classical Chebyshev iteration, for the inner product-free, iterative solution of indefinite linear systems using orthogonal polynomials for measures supported on multiple, disjoint intervals. The iteration applies to shifted linear solves and can then be used for efficient matrix function approximation. Using the asymptotics of orthogonal polynomials, error bounds are provided. A key component in the efficiency of the method is the ability to compute the first $k$ orthogonal polynomial recurrence coefficients and the first $k$ weighted Stieltjes transforms of these orthogonal polynomials in $\mathrm{O}(k)$ complexity using a numerical Riemann--Hilbert approach. For a special class of orthogonal polynomials, the Akhiezer polynomials, the method can be sped up significantly, with the greatest speedup occurring in the two interval case where important formulae of Akhiezer are employed and the Riemann--Hilbert approach is bypassed.

翻译：我们发展了阿克西泽迭代，这是经典切比雪夫迭代的推广，用于无需内积的迭代求解不定线性系统，该方法利用支撑在多个不相交区间上的正交多项式。该迭代适用于平移线性求解，进而可用于高效的矩阵函数逼近。利用正交多项式的渐近性质，我们给出了误差界。该方法高效性的关键组成部分在于：能够利用数值黎曼-希尔伯特方法，以$\mathrm{O}(k)$复杂度计算前$k$个正交多项式递推系数及这些正交多项式的前$k$个加权斯蒂尔杰斯变换。对于一类特殊的正交多项式——阿克西泽多项式，该方法可显著加速，其中在双区间情形下加速效果最为显著，此时采用了阿克西泽的重要公式并绕过了黎曼-希尔伯特方法。