1 CHAPTER 6 6 1 The function can be set up for fixed point iteration by solving it for x xi 1 sin x i Using an initial guess of x0 0 5 the first iteration yields x1 sin a 0 5 0 649637 0 649637 0 5 100 23 0 649637 Second iteration x2 sin a 0 649637 0 721524 0 721524 0 649637 100 9 96 0 721524 The process can be continued as tabulated below iteration 0 1 2 3 4 5 6 7 8 9 ratio xi a 0 500000 0 649637 23 034 0 721524 9 963 0 4325 0 750901 3 912 0 3927 0 762097 1 469 0 3755 0 766248 0 542 0 3688 0 767772 0 198 0 3663 0 768329 0 072 0 3653 0 768532 0 026 0 3650 0 768606 0 010 0 3649 Thus after nine iterations the root is estimated to be 0 768606 with an approximate error of 0 010 The table also includes a column showing the ratio of the relative errors between iterations ratio a i a i 1 As can be seen after the first few iterations this ratio is converging on a constant value of about 0 3649 Recall that the error of fixed point iteration is E t i 1 g E t i For our problem g x d sin dx x 0 5x cos x PROPRIETARY MATERIAL The McGraw Hill Companies Inc All rights reserved No part of this Manual may be displayed reproduced or distributed in any form or by any means without the prior written permission of the publisher or used beyond the limited distribution to teachers and educators permitted by McGraw Hill for their individual course preparation If you are a student using this Manual you are using it without permission 2 The value of this quantity in the vicinity of the true root 0 768648 agrees with the ratio obtained in the table confirming that the convergence is linear and conforms to the theory g 0 768648 0 5 0 768648 cos 0 768648 0 3648 6 2 a Graphical 8 4 0 4 0 1 2 3 4 8 Root 3 58 b Fixed point The equation can be solved in numerous ways A simple way that converges is to solve for the x that is not raised to a power to yield x 5 2 x 3 11 7 x 2 17 7 The resulting iterations are i xi a 0 1 2 3 3 3 180791 3 333959 3 442543 5 68 4 59 3 15 c Newton Raphson i xi 0 1 2 3 3 5 133333 4 26975 3 792934 f x 3 2 48 09007 12 95624 2 947603 f x a 1 5 55 68667 27 17244 15 26344 41 56 20 23 12 57 d Secant i xi 1 f xi 1 xi f xi a 0 1 2 3 3 4 3 326531 3 481273 3 2 6 6 1 96885 0 79592 4 3 326531 3 481273 3 586275 6 6 1 9688531 0 7959153 0 2478695 20 25 4 44 2 93 e Modified secant 0 01 PROPRIETARY MATERIAL The McGraw Hill Companies Inc All rights reserved No part of this Manual may be displayed reproduced or distributed in any form or by any means without the prior written permission of the publisher or used beyond the limited distribution to teachers and educators permitted by McGraw Hill for their individual course preparation If you are a student using this Manual you are using it without permission 3 i x 0 1 2 3 3 4 892595 4 142949 3 742316 f x 3 2 35 7632 9 73047 2 203063 dx x dx 0 03 0 048926 0 041429 0 037423 3 03 4 9415212 4 1843789 3 7797391 f x dx 3 14928 38 09731 10 7367 2 748117 a 38 68 18 09 10 71 6 3 a The function can be set up for fixed point iteration by solving it for x in two different ways First it can be solved for the linear x xi 1 0 9 xi2 2 5 1 7 Using an initial guess of 5 the first iteration yields x1 0 9 5 2 2 5 11 76 1 7 a 11 76 5 100 57 5 11 76 a 71 8 11 76 100 83 6 71 8 Second iteration x1 0 9 11 76 2 2 5 71 8 1 7 Clearly this solution is diverging An alternative is to solve for the second order x xi 1 1 7 xi 2 5 0 9 Using an initial guess of 5 the first iteration yields xi 1 1 7 5 2 5 3 496 0 9 a 3 496 5 100 43 0 3 496 a 3 0629 3 496 100 14 14 3 0629 Second iteration xi 1 1 7 3 496 2 5 3 0629 0 9 This version is converging All the iterations can be tabulated as iteration 0 1 2 3 4 5 6 7 8 9 xi a 5 000000 3 496029 3 062905 2 926306 2 881882 2 867287 2 862475 2 860887 2 860363 2 860190 43 0194 14 1410 4 6680 1 5415 0 5090 0 1681 0 0555 0 0183 0 0061 PROPRIETARY MATERIAL The McGraw Hill Companies Inc All rights reserved No part of this Manual may be displayed reproduced or distributed in any form or by any means without the prior written permission of the publisher or used beyond the limited distribution to teachers and educators permitted by McGraw Hill for their individual course preparation If you are a student using this Manual you are using it without permission 4 Thus after 9 iterations the root estimate is 2 860190 with an approximate error of 0 0061 The result can be checked by substituting it back into the original function f 2 860190 0 9 2 860190 2 1 7 2 860190 2 5 0 000294 b The formula for Newton Raphson is xi 1 xi 0 9 xi2 1 7 xi 2 5 1 8 xi 1 7 Using an initial guess of 5 the first iteration yields xi 1 5 a 0 9 5 2 1 7 5 2 5 3 424658 1 8 5 1 7 3 424658 5 100 46 0 3 424658 Second iteration xi 1 3 424658 a 0 9 3 424658 2 1 7 3 424658 2 5 2 924357 1 8 3 424658 1 7 2 924357 3 424658 100 17 1 2 924357 The process can be continued as tabulated below iteration 0 1 2 3 4 5 xi f xi f xi 11 5 2 23353 0 22527 0 00360 9 8E 07 7 2E 14 5 3 424658 2 924357 2 861147 2 860105 2 860104 7 3 4 46438 3 56384 3 45006 3 44819 3 44819 a 46 0000 17 1081 2 2093 0 0364 0 0000 After 5 iterations the root estimate is 2 860104 with an approximate error of 0 0000 The result can be checked by …
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