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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

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 $ {M_n}$ which associates to each polynomial of degree $ \leqslant n - 1$ 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 $ {\left\Vert {{M_n}} \right\Vert _\infty }{\left\Vert {M_n^{ - 1}} \right\Vert _\infty }$ 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 $ 1 + \sqrt 2 $. In the more difficult case of an asymmetric interval around the origin we obtain upper bounds for the condition number which also grow exponentially.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1979-0514830-6
PII: S 0025-5718(1979)0514830-6
Keywords: Parametrization of polynomials, power basis, numerical condition
Article copyright: © Copyright 1979 American Mathematical Society