A GENERAL CONDITION NUMBER FOR POLYNOMIALS

作者：Barrio, Roberto^*; Jiang, Hao; Serrano, Sergio

来源：SIAM Journal on Numerical Analysis, 2013, 51(2): 1280-1294.

DOI：10.1137/120864581

• polynomial condition number
• polynomial basis
• orthogonal polynomials
• Clenshaw algorithm
• floating-point arithmetic
• running-error bound

This paper presents a generic condition number for polynomials that is useful for polynomial evaluation of a finite series of polynomial basis defined by means of a linear recurrence. This expression extends the classical one for the power and Bernstein bases, but it also provides us a general framework for all the families of orthogonal polynomials like Chebyshev, Legendre, Gegenbauer, Jacobi, and Sobolev orthogonal polynomial bases. The standard algorithm for the evaluation of finite series in any of these polynomial bases is the extended Clenshaw algorithm. The use of this new condition number permits us to give a general theorem about the forward error for that evaluation algorithm. A running-error bound of the extended algorithm is also presented and all the bounds are compared in several numerical examples.

• 出版日期2013
• 单位中国人民解放军国防科学技术大学