Associated to a finite measure on the real line with finite moments are recurrence coefficients in a three-term formula for orthogonal polynomials with respect to this measure. These recurrence coefficients are frequently inputs to modern computational tools that facilitate evaluation and manipulation of polynomials with respect to the measure, and such tasks are foundational in numerical approximation and quadrature. Although the recurrence coefficients for classical measures are known explicitly, those for nonclassical measures must typically be numerically computed. We survey and review existing approaches for computing these recurrence coefficients for univariate orthogonal polynomial families and propose a novel "predictor-corrector" algorithm for a general class of continuous measures. We combine the predictor-corrector scheme with a stabilized Lanczos procedure for a new hybrid algorithm that computes recurrence coefficients for a fairly wide class of measures that can have both continuous and discrete parts. We evaluate the new algorithms against existing methods in terms of accuracy and efficiency.

翻译：这些复发系数经常是现代计算工具的投入,有助于评估和操纵该计量的多数值,而这种任务在数值近似值和二次曲线上具有基本意义。虽然古典计量的复发系数是明确已知的,但非古典计量的复发系数一般必须是数字计算的。我们调查并审查计算单数或多数值家庭复发系数的现有方法,并为一般连续计量类别提出新的“预变校正”算法。我们把预测-校正法与稳定的兰克佐斯程序结合起来,用于一种新的混合算法,将复发系数计算成相当广泛的、既具有连续性又具有离散部分的计量。我们根据现有方法评估新的算法,以准确性和效率为标准。