SIMPLE FIXED-POINT ITERATION
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1 SIMPLE FIXED-POINT ITERATION The fed-pont teraton method s an open root fndng method. The method starts wth the equaton f ( The equaton s then rearranged so that one s one the left hand sde of the equaton For eample g( can be rearrange as f ( In some cases, separatng an out of the equaton f ( s not possble. In such cases, an can be added to both sdes of the equaton. For eample can be wrtten as f ( sn( sn( PROCEDURE Choose an ntal guess n the neghborhood of the root. stes.google.com/ste/zyadmasoud/numercal
2 Substtute n g ( to get a new appromaton of the root, ; g( Calculate the error and repeat untl tolerance s met. Eample Usng the fed-pont teraton, determne the non-trval root of the functon sn usng as an ntal guess, and.% tolerance. Rearrange the equaton as The relatve appromate error s Repeat wth. 97. Usng 4 sgnfcant fgures sn sn( sn(.97 a %.97 % 9.6%.97 stes.google.com/ste/zyadmasoud/numercal 4
3 Itr# a % ADVANTAGES As an open method, the fed-pont teraton has the followng advantages: Fast Fewer calculatons than the bracketng methods Requres one ntal guess Easer to program DISADVANTAGES Convergence s not guaranteed stes.google.com/ste/zyadmasoud/numercal 5
4 PSEUDO-CODE nput nput s old a s whle a g( s a old end whle output old CONVERGENCE Rearrange the functon f ( as g(. Plot both sdes of the equaton. The root of the equaton s at the ntersecton of the curves f ( and f ( g(. stes.google.com/ste/zyadmasoud/numercal 6
5 Now consder the case where the slope of the curve f ( g( s lower than the slope of the curve f (. Startng wth an ntal guess, the appromaton slowly approaches the root. On the other hand, when the slope of the curve f ( g( s hgher than the slope of the curve f (, and startng wth an ntal guess, the appromaton dverges. stes.google.com/ste/zyadmasoud/numercal 7
6 Concluson When the slope of g ( s less than 45 or g (, the convergence s guaranteed. Now consder the followng fgure where the slope of g ( s negatve but, more than 45. Startng wth an ntal guess, the appromaton slowly approaches the root. stes.google.com/ste/zyadmasoud/numercal 8
7 However, when the slope of g ( s negatve and less than 45 ntal guess, the appromaton dverges., then startng wth an Concluson When the slope of g ( s more than 45 or g (, the convergence s guaranteed. Therefore the convergence of the fed-pont teraton s guaranteed when or g ( g ( Ths condton s suffcent but not necessary. In some cases, the method converges for values of g (. stes.google.com/ste/zyadmasoud/numercal 9
8 Eample Use fed-pont teraton to determne a root of f (.8. 5 usng 5. Perform computatons untl a s less than s.5%. Frst we wll put the functon f ( n the fed-pont teraton form g(. st teraton the error s nd teraton the error s rd teraton a a.5.8 g( % % 6%.5 g( % % 85.7% stes.google.com/ste/zyadmasoud/numercal 4
9 the error s a g( % % 97.9% 454 It s clear that the teratons dverge away from the root. Now let us check the convergence condton on the chosen g (..5 g(.8 g(.8 (5 g( The above shows that the condton s not satsfed for the chosen g (. Another choce of g ( can be obtaned from f (.8. 5 as.8.5 Let us check the convergence condton on the chosen g (. stes.google.com/ste/zyadmasoud/numercal 4
10 g( g( g( (5.5 The above shows that the condton s satsfed for the new g ( whch must lead to a convergent soluton. st teraton the error s nd teraton the error s rd teraton a a g( %.8 g( % % 47.45% % 5.6%.9 stes.google.com/ste/zyadmasoud/numercal 4
11 g( the error s a % % 5.6%.789 On the 8 th teraton, the root appromaton becomes and the relatve appromate error s a 8.7.5% The eact soluton can be calculated drectly from the functon f (.8. 5 to be r.79 Now gong back to the f ( sn, the chosen g ( was g( sn( Now checkng the convergence condton wth the ntal guess cos( g( sn( cos( g(.945 sn( The convergence condton s satsfed whch had guaranteed convergence n the earler eample. stes.google.com/ste/zyadmasoud/numercal 4
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