 Since formula_70 for j not equal to i, we have

Since both formula_72 and formula_67 are non-negative functions, it follows that formula_74.There are many algorithms for computing the nodes and weights of Gaussian quadrature rules.The most popular are the Golub-Welsch algorithm requiring operations, Newton's method for solving formula_75 using the three-term recurrence for evaluation requiring operations, and asymptotic formulas for large "n" requiring operations.