A comparison of the average case numerical condition of the power and bernstein polynomial bases

Abstract

The relative numerical condition of a root x ₀, of arbitrary multiplicity, of a polynomial p(x) in the power and Bernstein bases is considered. The polynomial equation p(x)=0 and the linear algebraic equation that defines the transformation between the bases are used to show that the relative numerical condition of x ₀ in the bases is strongly dependent on the numerical condition of this equation. Furthermore, as the multiplicity of x _o increases for a given polynomial order, the relative numerical condition of x ₀ approaches unity. Computational examples that illustrate the theoretical results are presented.

Keywords:

• Numerical Condition
• Bernstein Basis
• Power Basis
• Polynomial Basis Conversion
• Polynomial Roots

C.R. Categories:

• G.1
• G1.5