The condition of polynomials in power form

Author:
Walter Gautschi

Journal:
Math. Comp. **33** (1979), 343-352

MSC:
Primary 65D99; Secondary 41A10

MathSciNet review:
514830

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Abstract: A study is made of the numerical condition of the coordinate map which associates to each polynomial of degree on the compact interval [*a, b*] the *n*-vector of its coefficients with respect to the power basis. It is shown that the condition number increases at an exponential rate if the interval [*a, b*] is symmetric or on one side of the origin, the rate of growth being at least equal to . In the more difficult case of an asymmetric interval around the origin we obtain upper bounds for the condition number which also grow exponentially.

**[1]**Walter Gautschi,*The condition of orthogonal polynomials*, Math. Comp.**26**(1972), 923–924. MR**0313558**, 10.1090/S0025-5718-1972-0313558-9**[2]**Walter Gautschi,*Norm estimates for inverses of Vandermonde matrices*, Numer. Math.**23**(1975), 337–347. MR**0378382****[3]**John R. Rice,*A theory of condition*, SIAM J. Numer. Anal.**3**(1966), 287–310. MR**0211576****[4]**Theodore J. Rivlin,*The Chebyshev polynomials*, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Pure and Applied Mathematics. MR**0450850****[5]**Arnold Schönhage,*Approximationstheorie*, Walter de Gruyter & Co., Berlin-New York, 1971 (German). MR**0277960**

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DOI:
https://doi.org/10.1090/S0025-5718-1979-0514830-6

Keywords:
Parametrization of polynomials,
power basis,
numerical condition

Article copyright:
© Copyright 1979
American Mathematical Society