## 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.

## References

- Walter Gautschi,
*The condition of orthogonal polynomials*, Math. Comp.**26**(1972), 923–924. MR**313558**, DOI 10.1090/S0025-5718-1972-0313558-9 - Walter Gautschi,
*Norm estimates for inverses of Vandermonde matrices*, Numer. Math.**23**(1975), 337–347. MR**378382**, DOI 10.1007/BF01438260 - John R. Rice,
*A theory of condition*, SIAM J. Numer. Anal.**3**(1966), 287–310. MR**211576**, DOI 10.1137/0703023 - Theodore J. Rivlin,
*The Chebyshev polynomials*, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. MR**0450850** - Arnold Schönhage,
*Approximationstheorie*, Walter de Gruyter & Co., Berlin-New York, 1971 (German). MR**0277960**

## 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