Peter R. Turner (auth.)'s Guide to Numerical Analysis PDF

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Additional info for Guide to Numerical Analysis

Example text

GUIDE TO NUMERICAL ANALYSIS on the screen followed by a request for another set of initial values to be typed in. 581741E- 04 Note the sort of information which is included in the output. Obviously the intention is to provide the user with as much useful information as possible without burying him or her in reams of paper. ' and returns the program to line 40 if the answer is 'No'. The actual computation is seen therefore to be very simple in this case. 5). 2 1 Show that the equation 3x 3 - 5x 2 - 4x + 4 = 0 has a root in the interval [0, 1].

To prove this, let K > 0 be given. Then choosing N ~ K 1 '~ gives us, for n>N, >N~ ~K, 38 ITERATION AND CONVERGENCE which completes the proof. Since n- a = 1Ina, it now follows that, if IX < 0, then na-+ 0 as n-+ 00. In the last example, we used the fact that the terms of the sequence were growing continually so that we needed to find just one term x N ~ K to show that xn ~ K whenever n > N. This principle is very powerful. First, we need the following definition. 3 A set S of real numbers is said to be bounded above if there exists a number M such that x~M for all xES, and that it is said to be bounded below if there exists a number m such that x~m for all xES.

So, given K > 0, choosing N ~ KjJ implies that, for n > N, x" > 1 + nJ ~1+K >K, which completes the proof. For x = 1, x" = 1--+ 1 as n--+ oo. For x = -1, the sequence (x") oscillates finitely and, for x < - 1, it oscillates infinitely. For the sequence (na) we shall prove later that, if IX> 0, then na--+ oo as n--+oo, ifiX=O, then na= 1--+1 as n--+oo and, ifiX

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