1. A real number x is represented approximately by , and we are told that the relative error is 0.1 %. What is x? Note: There are two answers.

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1 PROBLEMS A real umber s represeted appromately by 63, ad we are told that the relatve error s % What s? Note: There are two aswers Ht : Recall that % relatve error s What s the relatve error volved roudg to? Ht : Roudg appromates to, thus, follows 3 How may decmals must be take alog π ad e order to compute (a) 334 π ad (b) 376 e to three correct decmals? Ht : Oe way s to geerate a sequece α d, α d d, wth creasg umber of correct decmals d, d, π (or e) where α f the fed umber ad s the teger part, utl the ew term of the covergg sequece agrees wth the prevous term to three correct decmals 4 Suppose that p appromates p to three sgfcat dgts Fd the largest terval whch p ca le f p s (a) 9, (b) 9, (c) 78, (d) 63 Ht : s sad to appromate to at least t-dgts whe < t Thus the terval follows Perform the followg computatos () eactly, () usg three-dgt choppg arthmetc, () usg three-dgt roudg arthmetc The determe ay loss sgfcat dgts, assumg that the gve umbers are eact (a) , (b) 8 79, (c) ( ) (43 + ), (d) ( 64 43) + (93 ) Ht : Recall that loss of sgfcat dgts occurs whe two early equal umbers are subtracted Further, t relatve error betwee two umbers whe compared agats s the way to determe the umber of dgts agreemet 6 Cosder the followg values of p ad p What s () the absolute error, () the relatve error appromatg p by p, ad to how may () decmal dgts, (v) sgfcat dgts does p appromate p? (a) p π, p 3 p π, p 3, (d) p 3, p 33 3, (b) 3 p, p 333, (c) Ht : Recall that relatve (absolute) error betwee two umbers whe compared agats yelds the umber of dgts (decmals) agreemet ( t ) t 7 Cout the umber of multplcatos ad addtos volved evaiuatg a polyomal usg ested multplcato Compare wth the work eeded whe the powers of are calculated by ad subsequetly multpled by a Note : I order to effcetly

2 evaluate a polyomal p () a + a + a + + a + a a ested multplcato ( a + ( a + + ( a ( a )))), we group the terms p() a + + Ths s llustrated by the followg MATLAB code, where a polyomal N p () a of degree 6 N for a certa value of s computed by usg the two methods: N +; a [:N]; ; tc % talze the tmer p sum(a*^[n-:-:]); p, toc % measure the tme tc, pa(); for :N p p* + a(); % ested multplcato ed p, toc 3 8 (a) Evaluate the polyomal p() at 73 Use 3-dgt arthmetc wth choppg Evalute the relatve error (b) Repeat (a) but epress (( ) + 6) p () + Evaluate the percet relatve error ad compare wth part (a) 9 Plot a seveth-degree polyomal by typg the followg statemets to the MATLAB commad wdow: X988::; y ^7-7*^6+*^-3*^4+3*^3-*^+7*-; plot(,y) The resultg plot does ot look aythg lke a polyomal It s ot smooth You are 4 seeg roudoff error acto The y-as scale factor s ty, The ty values of y 4 are beg computed by takg sums ad dffereces of umbers as large as 3 7 There s severe subtractve cacellato Note that y s the epaded form of y ( ) ad the rage for the -as s carefully chose to be ear If the values of y are computed stead by y (-)^7; the a smooth (but very flat) plot results (a) Show that the polyomal estg techque ca also be appled to the evaluato of 4 3 f () e 46e 3e + e 99 (b) Use 3-dgt roudg arthmetc, the assumpto that e 3 4 6, ad the fact that e (e ) to evaluate f (3) as gve part (a) (c) Redo the calculato part (b) by frst estg the calculatos (d) Compare the appromatos parts (b) ad (c) to the true three-dgt result f (3) 76

3 Recall that the dervatve of a fucto f at a pot s defed by the equato f ( + h) f () f () lm h h A computer has the capacty of mtatg the lmt operato by usg a sequece of 3 umbers h such as h 4,4,4,,4, for they certaly approach zero rapdly The followg MATLAB code s to compute f () at the pot wth f () s() clear all, clf ; h; ; % Try,,, ad for : hh/4; H()h; y(s(+h)-s())/h; % Forward dfferece formula % y(s(+h)-s(-h))//h; % Cetral dfferece formula ed error()abs(cos()-y); % Trucato error loglog(h,error), hold o N; % Try N ad loglog(h,h^(n),'--') Perform ths umercal epermet by rug the code ad terpret the results Also study the followg cetral dfferece formula: f ( + h) f ( h) f () as h h Ht : Is a possble loss of sgfcat dgt pheomea at play here? Where? The Taylor polyomal of degree for polyomal of degree e to fd a appromato to (a) 9 9 e (b) ( )! ( )! f () e s (!) Use the Taylor e by e (!) 9 e A appromate value of e 3 correct to three dgts s 674 Whch formula, (a) or (b), gves the most accuracy, ad why? Ht : The rule of thumb s that f you have the choce, always prefer the formula vod of possble cacellato (loss of sgfcat dgts) errors 3 I the computer, t ca happe that a + a whe Epla why Descrbe the set of for whch + your computer The followg MATLAB code llustrates the pheomeo a ; a; whle a + > a /; ed,

4 Ht : For a gve precso, ay umber eps satsfy + where eps s the so-called mache zero (type eps Matlab commad wdow) How s t related to the umber of dgts carred by the floatg pot represetato the mache? What f a? Ca you put a + a to form + y for some y? 4 Establsh that the recurso formula y + y represets the tegral y d for,, K + a) Compute the terms y, y, y3, y4, y, startg wth y l(6/) 8 a threedecmal computg evromet Do you observe aythg strage as we theoretcally epect that y > ad > y > y > y y y 3 4 > b) Now try the recurso formula the other drecto y y Appromate a startg value by settg y y9, thus gettg 9 7 to compute y 8, y7,, y y 6 Epla Ht : Forward, s a multpler ad backward, s a dvsor How does ths effect roudg error accumulato? It s kow that s k π 6 lm s, k Compute the respectve errors usg forward ad backward summato to compute s 3 for a four-dgt evromet usg the followg MATLAB code: dgts(4), s vpa(); for :e3 % Use e3:-: for backward summato ed s vpa(s+//); Iterpret the results Ht : The correct way s to allow small postve umbers to add up to somethg before they ecouter large umbers 6 Cosder the followg two formulas: ( + ) f () ad f () + + These are theoretcally equvalet, hece we epect them to gve eactly the same value How would you epla the result obtaed by rug the followg MATLAB program to compute the values of the two formulas? Ht : Is a possble loss of sgfcat dgt pheomea at play here? Where?

5 clear all f le('sqrt()*(sqrt( + ) - sqrt())',''); f le('sqrt()/(sqrt( + ) + sqrt())',''); ; format log e for k : fprtf('at %f, f%8f, f%8f \',,f(),f()); *; ed