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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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The condition of polynomials in power form
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by Walter Gautschi PDF
Math. Comp. 33 (1979), 343-352 Request permission

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 ${\left \| {{M_n}} \right \|_\infty }{\left \| {M_n^{ - 1}} \right \|_\infty }$ 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
  • © Copyright 1979 American Mathematical Society
  • Journal: Math. Comp. 33 (1979), 343-352
  • MSC: Primary 65D99; Secondary 41A10
  • DOI: https://doi.org/10.1090/S0025-5718-1979-0514830-6
  • MathSciNet review: 514830