 
Summary: COMPUTATION OF GAUSSKRONROD QUADRATURE RULES
WITH NONPOSITIVE WEIGHTS
G. S. AMMAR \Lambda , D. CALVETTI y , AND L. REICHEL z
Abstract. Recently Laurie presented a fast algorithm for the computation of (2n + 1)point
GaussKronrod quadrature rules with real nodes and positive weights. We describe modifications of
this algorithm that allow the computation of GaussKronrod quadrature rules with complex conjugate
nodes and weights or with real nodes and positive and negative weights.
Key words. orthogonal polynomials, indefinite measure, fast algorithm, inverse eigenvalue
problem
1. Introduction. Let dw be a nonnegative measure with support on the real
axis and an infinite number of points of increase. Assume that the moments ¯ k :=
R 1
\Gamma1 x k dw(x), k = 0; 1; 2; : : : , exist and are bounded. For notational convenience, we
assume that ¯ 0 = 1. An npoint Gauss quadrature rule for the integral
If :=
Z 1
\Gamma1
f(x)dw(x)
(1.1)
is a formula of the form
