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We consider the theoretical and numerical aspects of the quadrature rulesassociated with a sequence of polynomials generated by a special $R_{II}$recurrence relation. We also look into some methods for generating the nodes(which lie on the real line) and the positive weights of these quadraturerules. With a simple transformation these quadrature rules on the real linealso lead to certain positive quadrature rules of highest algebraic degree ofprecision on the unit circle. This way, we also introduce new approaches toevaluate the nodes and weights of these specific quadrature rules on the unitcircle.

Quadrature rules from a $R_{II}$ type recurrence relation and associated quadrature rules on the unit circle