Horner's rule for interval evaluation revisited (Q1849567)

Horner's rule for polynomial interval evaluation is revisited. A new factorization scheme is introduced which can result in essentially tighter intervals than the interval evaluation of the original Horner representation. The first two sections present the subject and recall some basics of interval arithmetic. Section 3 is devoted to the Horner extension including overestimation problems with respect to the range of a polynomial considered over a real compact interval. Symbolic identities are used in the next section in order to modify Horner's rule. These identities combine pairs of monomials in an appropriate way and are of the form \[ ax^\alpha+ bx^{\alpha+\gamma}= bx^{\alpha-\gamma} \Biggl[\Biggl(x^\gamma+ {a\over 2b}\Biggr)^2- \Biggl({a\over 2b}\Biggr)^2\Biggr]. \] Here, \(\alpha\), \(\gamma\) play the role of parameters. They are positive integers of the same parity which satisfy \(\alpha> \gamma\). Strategies of combinations are given and tested in Section 5. Conclusions and some further topics of research are presented in the final section followed by an appendix which contains some proofs deferred from the preceding part of the paper.