# High degree quadrature rules with pseudorandom rational nodes

## TL;DR

This paper introduces a method to construct high-degree quadrature rules with pseudorandom rational nodes by combining companion rules and mean rules, optimizing approximation accuracy.

## Contribution

It presents a novel framework for creating high-degree quadrature rules using linear combinations of lower-degree rules with rational pseudorandom nodes.

## Key findings

- Construction of higher-degree rules from lower-degree ones
- Mean rule as the best least-squares approximation
- Explicit examples of quadrature rule construction

## Abstract

After introducing the definitions of positive, negative and companion rules, from a given pair of companion rules we construct a new rule with higher degree of precision The scheme is generalized giving rise to a transformation which we call the mean rule. We show that the mean rule is the best approximation, in the sense of least-squares, obtained from a linear combination of two rules of the same degree of precision. Finally, we show that a rule of degree 2k+1 can be constructed as linear combination of k+1 rules of degree one and rational pseudorandom nodes. Several worked examples are presented.