Numerical Analysis by John Todd

By John Todd

Numerical research.

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2. T h e s e examples suggest t h e following definition. Definition 3 . χ) 1004 1 Fig. 2 X 58 Chapter 5 to u(x) in X , if, given any ε > 0 , there is an no= ηο(ε), independent that if of x, such η>ηο(ε). It is clear that o u r first series is uniformly convergent to its sum in the interval X = [ 0 , 1 ] and that the second is not uniformly convergent in t h e interval X = [ 0 , 1 ] . It can be shown that the results (1) and (2) are true if the series is uniformly convergent in an interval including a or in the interval [a, b].

P = q= l , gives x_ = which has a relative error of t o avoid this pitfall. W e have % . It is easy x, = - i p + V{(p^/4)-l} and x_ = l/x+ A n o t h e r disadvantage of this formula is t h e following. F o r suitably chosen large values of - p , q we may find that although the d a t a and the solution are well inside machine range, there is overflow during t h e calcula­ tion and n o results. , if we t a k e P= , q = in o u r machine we fail since p ^ / 4 is o u t of range. W e can counter this by calculating the radical as psQRT[i-(q/p)/p].

In contrast, from the point of view of practical computation, an infinite descent is impossible: there are only a finite (but large) set of numbers available as outputs of a computation. In practical computation it is necessary to choose a starting value Xo and to choose a "stopping rule", to decide what Xn to accept as the square root of N. It is not appropriate here to discuss in detail how these choices should be made. , of Xo = 1 would be simpler than allowing Xo to depend on N, which would probably save time.

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