By Pier, Jean-Paul

ISBN-10: 0470210664

ISBN-13: 9780470210666

ISBN-10: 0582014808

ISBN-13: 9780582014800

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12) n of a sequence of (finite) interval respect to a non-negative measure da(x). [a,b] We consider with the condition of this basis on the interval of orthogonality, [a,b]. 1) of any polynomial are now representable as Fourier coefficients of n it is easy to estimate the condition of Schwarz inequality. 1 One finds 1/2 cond M < max L

ISBN 0-12-208360-1 46 W A L T E R GAUTSCHI space. The sensitivity of the m a p small relative changes in totic) condition number cond(M;x) = lim 6+0 x M at x , with respect to , will be measured by the (cf. asymp­ (1966)) sup • ||h|| = 6 provided the limit exists. 1) measures the o maximum amount by which a relative perturbation of by 6 / | | x || ) is magnified under the map infinitesimal perturbations. bers are called If M M x (given , in the limit of Maps with large condition num­ ill-conditioned.

9 x l 0 " , t, = 2 . , at t,. 01*10 , -12 1 5 1 5 1 4 0 t ? 67x10 ). 2 may be viewed as an elementary example of a bifurcation phenomenon (catastrophe theory, if you w i l l ) , the special feature, here, being the al­ most infinitesimal time scale on which the phenomenon takes place. 2 (Wilkinson The roots £ v (1963, p. ,n. V v rapidly near the origin, which at first might suggest that they become more and more illconditioned. ditioned. 9) 136. 32. . n. v=l 2 The condition is thus bounded by a relatively small number condition numbers go) , uniformly in Since the bound in n .

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Amenable Banach algebras by Pier, Jean-Paul


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