Abstract. An algorithm for the construction of the polynomials associated with the weight function w(j)P{t) from those associated with w(z) is given for the case when P(t) is a polynomial which is nonnegative in the interval of orthogonality. The relation of the algorithm to the LR algorithm is also discussed.Introduction. In several problems of numerical analysis, particularly in the construction of Gaussian quadrature rules with preassigned nodes, the following problem arises. Given the orthogonal polynomials {/?,-(<)} associated with a weight function w(t) on the interval (a, b) and a polynomial P(t) of degree zzz which is nonnegative on the interval (a, b), construct the orthogonal polynomials {q¡(t)} associated with the weight function P(f)w(f) on the same interval.A theorem of Christoffel [1] gives an explicit expression for the polynomial qn(t) in the form

Abstract. Some algebraic methods are given to implement Uvarov's extended Christoffel theorem. The stability of the algorithms is discussed.

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Abstract. Gauss quadrature rules for evaluating integrals of the form 2tt-i'2 -/J exp ( -x*)f(x)dx have been calculated to 20S for one to twenty nodes. The coefficients for the threeterm recurrence relation of the first twenty orthogonal polynomials associated with the weight function exp ( -x2) on the interval [0, °o ) are also tabulated to 20S.

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