Accurate recovery of recursion coefficients from Gaussian quadrature formulas

摘要

We present an algorithm for the accurate recovery of recursion coefficients from quadrature formulas with positive weights, based on the differential form of the quotient-difference algorithm. The process is slightly faster than the Gragg–Harrod algorithm and is forward stable in the sense of componentwise relative error when the gaps between the nodes are available as data. This result shows constructively that the converse problem is well posed when the data are required to be accurate floating point numbers. This is not a contradiction of the examples given by Gragg and Harrod which are ill posed in a vector norm, because in that case very small numbers are not required to have any significant digits.