Ee1032bS10

Ee1032bS10 - Big "O" and Little "o"...

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Big “O” and Little “o” ,, ( ) , ,( ) , ( ) . Often we wish to compare a function say z h with the behavior of another function of h say h whose behavior is better understood than that of z h : Ex of functions of h (( ) () ) / ) ( ) ) / 2 ForwardD i f ference Approximation f x hf x h Central Difference Approximation f x h f x h h  : ' ( ) sin( ) ( ) , 0 Ex Let s compare z h h with h h as h  UCLA SEAS EE103 (SEJ) SLIDES 2B 1 Big “O” and Little “o” s i n lim lim We consider z hh 1 00  sin( ) ( ), "" We say h is O h pronounced big O of h , "intuitively", sin( ) 0 about as fast 0 and h as h 2 0 sin() : lim h h Ex h 2 0 2 (), , "intuitively", sin ( ) 0 faster 0 Wesay hiso h pronounced littleoo fh and h than h UCLA SEAS EE103 (SEJ) SLIDES 2B 2
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Big “O” and Little “o” 2 :/ s i n ( ) Ex h h 2 0 lim 0 sin( ) h h h 22 (sin( )), " sin( )" h is o h h is little o of h UCLA SEAS EE103 (SEJ) SLIDES 2B 3 Definition : " " little o 0 () lim 0 ( ) " ( ( ))" h zh means z h is o h h Definition : " " Big O 0 lim h B h or if || 0 B as h  () " ( () ) " h means z h is O h UCLA SEAS EE103 (SEJ) SLIDES 2B 4
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Ex: Forward Difference Approximation of the First Derivative 2 , 1 ( ) ( ) '( ) ''( ) / 2 () ( ) xh fx h fx f xh f h fx h fx  , 2 '( ) ''( ) f x fh h Move f to the left hand side take absolute valuestoobtain ', 2 fx h fx f h   , ( ) | '( ) || ''( ) / | h UCLA SEAS EE103 (SEJ) SLIDES 2B 5 Ex: Forward Difference Approximation of the First Derivative Divide by to obtain h and take the lim ( ) 00 22 hh f x h ff x h   , '( ) lim lim| ''( ) / | | ''( ) / | , . That is the forward difference approximation for the first derivative is O h The FDA converges to the value of the first derivative about . as fast as h converges to zero ,: Note in this example ( ) ' , z hf x a n d h h h  UCLA SEAS EE103 (SEJ) SLIDES 2B 6
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Ee1032bS10 - Big "O" and Little "o"...

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