 For instance, some methods compute formula_7 to avoid issues associated with clustering of the roots formula_8 near the ends of the interval formula_9 and formula_10 .Some methods utilize formulas to approximate the weights and then use a few iterations of Newton-Raphson to lower the error of the approximation to below machine precision .In 1969, Golub and Welsch published their method for computing Gaussian quadrature rules given the three term recurrence relation that the underlying orthogonal polynomials satisfy.