1 0, x? x x. 1 Root finding. 1.1 Introduction. Solve[x^2-1 0,x] {{x -1},{x 1}} Plot[x^2-1,{x,-2,2}] 3

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1 Adrew Powuk - Math 49 (Numercal Aalyss) Root fdg. Itroducto f ( ),?,? Solve[^-,] {{-},{}} Plot[^-,{,-,}] Cubc equato Quartc equato Qutc equato Galos theory

2 Adrew Powuk - Math 49 (Numercal Aalyss) Solve[^-+S[]+Ep[]*Sqrt[],] Solve::smet: Ths system caot be solved wth the methods avalable to Solve.

3 Adrew Powuk - Math 49 (Numercal Aalyss) s l Plot[{Log[],-},{,,5}]

4 Adrew Powuk - Math 49 (Numercal Aalyss) Solve[S[]+(Cos[]) ((^-))==,] Plot[S[]+Cos[]/(^-),{,,}]

5 Adrew Powuk - Math 49 (Numercal Aalyss). (*) The Search Method a, b, b a a f f f ( ) #clude <ostream> double f(double ) { retur *-; } t ma(t argc, char** argv) { double a=-; double b= ; double eps=.; t =; double d=(b-a)/; double []; for(t =;<=;++) { []=a+d*; f((-eps<f[[])&&(f([])<eps)) { std::cout<<[]<<" "<<f([])<<"\"; } } retur ; } 5

6 Adrew Powuk - Math 49 (Numercal Aalyss) Results e For D N= For D N=^ For D N=^ 6

7 Adrew Powuk - Math 49 (Numercal Aalyss). Bsecto method.. Itroducto If f s a cotuous fucto. 7

8 Adrew Powuk - Math 49 (Numercal Aalyss) Eample , Iterato a, b f a f b f a f b f f ( ) error b a Iterato - f[_]=n[^-]; a=; c=(a+b)/; b=; Plot[f[],{,a,b}] f[a] f[c] f[b] c=/ f[a] = -. f[c]=.5 f[b]=. 8

9 Adrew Powuk - Math 49 (Numercal Aalyss) New terval [a=,c=/] Error=c-a=/-=/ Iterato f[_]=n[^-]; a=; c=(a+b)/;b=/; Plot[f[],{,a,b}] f[a] f[c] f[b] c=5/ f[a]=-. f[c]=-.475 f[b]=.5 New terval [c=5/4,b=/] Error=/-5/4=/4=/^ 9

10 Adrew Powuk - Math 49 (Numercal Aalyss) Iterato f[_]=n[^-]; a=5/4;b=/;c=(a+b)/ Plot[f[],{,a,b}] f[a] f[c] f[b] c=/ f[a]=-.475 f[c]=-.975 f[b]=.5 New terval [c=/8,b=/] Error=/-/8=/8=/^

11 Adrew Powuk - Math 49 (Numercal Aalyss) Iterato 4 f[_]=n[^-]; a=/8;b=/;c=(a+b)/ Plot[f[],{,a,b}] f[a] f[c] f[b] c=/ f[a]=-.975 f[c]=.6646 f[b]=.5 New terval [a=/8,b=/6] Error=/6-/8=/6=/^4

12 Adrew Powuk - Math 49 (Numercal Aalyss) Iterato 5 f[_]=n[^-]; a=/8;b=/6;c=(a+b)/ Plot[f[],{,a,b}] f[a] f[c] f[b] N[(b-a)] c=45/ f[a]=-.975 f[c]=-.469 f[b]=.6646 b-a=.65 New terval [a=45/,b=/6]

13 Adrew Powuk - Math 49 (Numercal Aalyss) Iterato 6 f[_]=n[^-]; a=45/;b=/6;c=(a+b)/ Plot[f[],{,a,b}] f[a] f[c] f[b] N[(b-a)] c=9/ f[a]=-.469 f[c]=.785 f[b]=.6646 b-a=.5 Results a=45/=.465 c=9/64=.488 error=c-a=9/64-45/=/64=/^6=.565 eect result =.44

14 Adrew Powuk - Math 49 (Numercal Aalyss).. Bsecto method - a program %%%%%%%%%%%%%%%%%%%%% fucto bsecto() a=; b=; N=6; dsp('start'); a b for =:N c=(a+b)/; dsp('terato'); f (f(a)*f(c) <=) a=a b=c else a=c b=b ed ed ed %bsecto %%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%% fucto y=f() y=^-; ed %%%%%%%%%%%%%%%%%%%% 4

15 Adrew Powuk - Math 49 (Numercal Aalyss) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%% Matlab/Octave %%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% fucto [ r ] = bsecto( f, a, b, N, eps_step, eps_abs ) % Check that that ether ed-pot s a root % ad f f(a) ad f(b) have the same sg, throw a ecepto. f ( f(a) == ) r = a; retur; elsef ( f(b) == ) r = b; retur; elsef ( f(a) * f(b) > ) error( 'f(a) ad f(b) do ot have opposte sgs' ); ed % We wll terate N tmes ad f a root was ot % foud after N teratos, a ecepto wll be throw. for k = :N % Fd the md-pot c = (a + b)/; % Check f we foud a root or whether or ot % we should cotue wth: % [a, c] f f(a) ad f(c) have opposte sgs, or % [c, b] f f(c) ad f(b) have opposte sgs. f ( f(c) == ) r = c; retur; elsef ( f(c)*f(a) < ) b = c; else a = c; ed % If b - a < eps_step, check whether or ot % f(a) < f(b) ad f(a) < eps_abs ad retur 'a', or % f(b) < eps_abs ad retur 'b'. f ( b - a < eps_step ) f ( abs( f(a) ) < abs( f(b) ) && abs( f(a) ) < eps_abs ) r = a; retur; elsef ( abs( f(b) ) < eps_abs ) r = b; retur; ed ed ed error( 'the method dd ot coverge' ); ed %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 5

16 Adrew Powuk - Math 49 (Numercal Aalyss).. Calculate three teratos of the bsecto method for f ad a, b. ( ) 5 ) a, b a b a, c.5, b 5 f ( a), f ( c), f ( b) 4 4 f ( a), f ( c), f ( b) ) a, b c old.5 a b 9 a, c, b f ( a), f ( c), f ( b) 6 4 f ( a), f ( c), f ( b) ) a, b c old a b 7 c f ( a), f ( c), f ( b) 64 6 f ( a), f ( c), f () b 9 a b 5 a c, b.5, c.875 old f ( a), f ( c), f ( b) ) et error b c ,

17 Adrew Powuk - Math 49 (Numercal Aalyss)..4 Calculate three teratos of the bsecto method for f ( ) ad a, b. ) a, b a b c 5 f a f f c f f b f 4 ( ) (), ( ), ( ) 8 a, b c old ) a b c f ( a) f (), f ( c) f, f ( b) f ) a, b c 4 old a b 4 9 c f ( a) f, f ( c) f, f ( b) f a= 4,b= 9 8 7

18 Adrew Powuk - Math 49 (Numercal Aalyss)..5 Calculate three teratos of the bsecto method for "bsecto method terato " "terval" a= b=4 "" c=(a+b)/ "accuracy" N[b-a] "values" N[f[a]] N[f[c]] N[f[b]] bsecto method terato terval 4 accuracy 4. values f ( ) ad a, b. "bsecto method terato " "terval" a=a b=c "" c=(a+b)/ "accuracy" N[b-a] "values" N[f[a]] N[f[c]] N[f[b]] bsecto method terato terval 8

19 Adrew Powuk - Math 49 (Numercal Aalyss) accuracy. values "bsecto method terato " "terval" a=c b=b "" c=(a+b)/ "accuracy" N[b-a] "values" N[f[a]] N[f[c]] N[f[b]] bsecto method terato terval / accuracy. values "bsecto method terato 4" "terval" a=c b=b "" c=(a+b)/ "accuracy" N[b-a] 9

20 Adrew Powuk - Math 49 (Numercal Aalyss) "values" N[f[a]] N[f[c]] N[f[b]] bsecto method terato 4 terval / 7/4 accuracy.5 values

21 Adrew Powuk - Math 49 (Numercal Aalyss)..6 Calculate three teratos of the bsecto method for f ( ) s ad a., b.5. Plot[S[]-/,{,,P}] Iterato a...5 c b.5 f a f c f b error b a.5 Iterato a c.75 b. 5 b f a f c f error b a.5

22 Adrew Powuk - Math 49 (Numercal Aalyss) Iterato a c.5 b.75 f a f c f b error b a.5

23 Adrew Powuk - Math 49 (Numercal Aalyss)..7 Calculate two teratos of the bsecto method for a, b. f( ) a, b a b c : fa ( ) 7 : fc ( ) 4 : fb ( ) a, b a b c 4 : fa ( ) : f ( c ) : f ( b ) 4 a, b 4 f[_]=-++^; a=;b=; Plot[f[],{,a,b}] f ( ) ad

24 Adrew Powuk - Math 49 (Numercal Aalyss) f[_]=-++^; a=;b=/; Plot[f[],{,a,b}]

25 Adrew Powuk - Math 49 (Numercal Aalyss)..8 Calculate two teratos of the bsecto method for a, b. 5 4 ad 5 4 a, b a b c : fa ( ) 5 4 * 5 : fc ( ) 5 4 * 4 : fb ( ) 5 4 * 6 c a, b a b 4 : fa ( ) 5 4 * 5 : fc ( ) 5 4 * : fb ( ) 5 4 * 4 a, b 4 5

26 Adrew Powuk - Math 49 (Numercal Aalyss)..9 Covergece aalyss, error b a b a, error b a b a b a, error b a b a b a, error..., error... error b a b a b a b a lm 6

27 Adrew Powuk - Math 49 (Numercal Aalyss).4 Newto's method.4. Itroducto f ' f Calculatos by had f ( ), f '( ) f( ) f '( ) f( ) f ().5 f '( ) f '() * f( ) f (.5) f '( ) f '(.5) * f '( ) f '(. 466). 466 f( ) f (.466) *

28 Adrew Powuk - Math 49 (Numercal Aalyss) f ' f f ' f f f ' f, f, f f ' f.5 f f '.4667 Plot[{^-,(^-)+**(-),((.5)^-)+*(.5)*(-.5)},{,,}]

29 Adrew Powuk - Math 49 (Numercal Aalyss) f( ) Step : set Step f( ) f ( ) f ( ) f '( )( ) f ( ) f ( ) f '( )( ) f '( )( ) f ( ) f( ) f '( ) f( ) f '( ) f( ) f '( ) f( ) f '( ) 9

30 Adrew Powuk - Math 49 (Numercal Aalyss).4. Eample f ( ), I[65]:= Clear[]; f[_] = ^ - ; df[_] = *; = ; = N[ - f[]/df[]] = N[ - f[]/df[]] = N[ - f[]/df[]] = N[ - f[]/df[]] = N[ - f[]/df[]] Calculatos by had f ( ), f '( ) f( ) f ( ), f '( ) f( ) f ().5 f '( ) f '() * f( ) f (.5) f '( ) f '(.5) * f '( ) f '(. 466). 466 f( ) f (.466) *.466

31 Adrew Powuk - Math 49 (Numercal Aalyss).4. Matlab code fucto ewto() eps=.; =; error=; terato=; whle(error>eps) New=-f()/df(); error=abs(new-); =New; dsp(sprtf('=%d =%f error=%f',terato,,error)); terato=terato+; ed ed fucto y=f() y=^-; ed fucto y=df() y=*; ed Output: octave:> ewto = =.5 error=.5 = = error=.8 = =.446 error=.45 =4 =.444 error=.

32 Adrew Powuk - Math 49 (Numercal Aalyss).4.5 (*) Covergece of Newto's method D y y (,) (-,-) D Fractal (*)

33 Adrew Powuk - Math 49 (Numercal Aalyss)

34 Adrew Powuk - Math 49 (Numercal Aalyss) 4

35 Adrew Powuk - Math 49 (Numercal Aalyss) 5

36 Adrew Powuk - Math 49 (Numercal Aalyss) 6

37 Adrew Powuk - Math 49 (Numercal Aalyss).4.6 (*) Problems wth multple roots f f( ) f '( ) ( ) ( ) f '( ) ( ) f ( ) ( ) f '( ) ( ) f( ) Sgularty the epresso for close to. f '( ), f ( ) f ( ) ( ) ( ) f '( ) ( ) ( )? 7

38 Adrew Powuk - Math 49 (Numercal Aalyss).4.7 (*) Startg pot eters a cycle.4.8 (*) Iftely may roots Plot[S[/],{,-,}]

39 Adrew Powuk - Math 49 (Numercal Aalyss).4.9 (*) Sample applcato Let us cosder the followg mechacal system P y L L 6EI P N 4. 4 I m 9 N E * m y y y y

40 Adrew Powuk - Math 49 (Numercal Aalyss) Clear[]; EE=*^9; L=; J=.^4/; P=^; Plot[P/(6*EE*J) ( +L L ),{,,L}] y=-.; F[_]=P/(6*EE*J) ( +L L )-y; df[_]=d[f[],]; =; =-F[]/dF[] =-F[]/dF[] =-F[]/dF[] =-F[]/dF[]

41 Adrew Powuk - Math 49 (Numercal Aalyss).4. Fd teratos of the Newto's method for f ( ) ad. f '( ) f( ) f '( ) f f '( ) f '() 4 ( ) f () f f( ) f '( ) f( ) f ' 4 f '( ) f f ' 4 4 4

42 Adrew Powuk - Math 49 (Numercal Aalyss).4. Fd teratos of the Newto's method for f ( ) ad. f( ) f '( ) f( ) f( ) f f ( ) 6. ( ) * f( ) f ( ) 5 * Fd teratos of the Newto's method for f ( ) 5 5 ad. f( ) 5 5 f '( ) 5 f ' f ' ' f 5 * f * 5 7 f f * * 5 7 4

43 Adrew Powuk - Math 49 (Numercal Aalyss).4. Fd teratos of the Newto's method for f ( ) 5 4 ad. f( ) 5 4 f( ) f '( ) f '( ) 4 f( ) 5 4 * 9 f '( ) 4 * 8 f( ) f '( ) * * 8 4

44 Adrew Powuk - Math 49 (Numercal Aalyss).4.4 Fd teratos of the Newto's method for f ( ) 7 ad. f ( ) 7, f '( )... ' ' * 7 f 7 7 * f 7 * f * f 7 *

45 Adrew Powuk - Math 49 (Numercal Aalyss).4.5 (*) Covergece aalyss f( ) f ( ) f ( ) f ( ) f '( )( ) f ''( )( ) (, ) f( ) f '( ) f ( ) f '( )( ) f ''( )( ) f ( ) f '( ) f ''( ) ( ) ( ) f '( ) f '( ) f '( ) f( ) f ''( ) ( ) f '( ) f '( ) f( ) f ''( ) ( ) f '( ) f '( ) f ''( ) ( ) f '( ) f ''( ) ( ) f '( ) f ''( ) M f '( ) M ( ) M ( ) M ( ) M ( ) M ( ).... Newto method has quadratc rate of covergece. 45

46 Adrew Powuk - Math 49 (Numercal Aalyss) M ( ) M ( ) M ( ) M ( ) M ( ) M ( ) M ( ) M ( ) M ( ) M ( ) M ( ) M ( ) M( ) f '( ) ( ) M f ''( ) f ''( ) f '( ) 46

47 Adrew Powuk - Math 49 (Numercal Aalyss).4.6 (**) Newto's method theorem (**) Newto's method theorem Let f : R R fd * R uder the followg assumptos. ) Ests at least oe soluto ). f * '( ) ) f ' Lp D * such that D where D R.e. f f * * ( ) s a ope terval. the * ) The Newto method coverge lm, ) f * ( ), * * ) k k p. Proof b We kow that f '( ) d f ( b) f ( a) b a b a f ( ) d f ( ) d b f z d f z b a ( ) ( ) a a f ( ) d f ( ) d b a b a k where D s a doma of the fucto f : R R. 47

48 Adrew Powuk - Math 49 (Numercal Aalyss) Lemma If f ' Lp D, the f ( y) f ( ) f '( ) y y Proof f y f f '( z) dz f '( ) y f '( ) dz the y y f y f f '( ) y f '( z) dz f '( ) dz '( ) '( ) '( ) f y f f y f z f dz Parameterzato z :, t z( t), y z t t ty y() y() y y y dz d t ty y dt dt y dz g z dz g z( t) dt g z( t) ( y ) dt g t ty ( y ) dt dt y f '( z) f '( ) dz f ' t ty f ' y dt f ' Lp D y f ' t ty f ' dt f '( y) f '( ) y the y y t ty dt y t y dt y tdt y f y f f y y '( ) Now t s possble to prove that p * * k k p * * k k 48

49 Adrew Powuk - Math 49 (Numercal Aalyss) f( ) k k k f '( ) k f( ) k k k k k * k * k f '( ) k * f '( ) f ( ) f '( ) * k k k k * * * k * * k k k k * * k k k f '( ) f '( ) f ( ) f ( ) f '( ) f '( ) f ( ) f ( ) f '( ) f ( ) f ( ) f '( ) * * k k k f '( ) '( ) y y * * ( ) ( ) '( ) f y f f k k f f f f '( ) f '( ) p * k k k * * k k k p m f '( ) k k p k * * k k k 49

50 Adrew Powuk - Math 49 (Numercal Aalyss).4.7 (*) Hgher order methods f( ) f( ) f ( ) f ( ) f '( )( ) f '( ) f ( ) f ( ) f '( )( ) f ''( )( )? 5

51 Adrew Powuk - Math 49 (Numercal Aalyss).5 Secat method f ( ) f ( ) f ( ) Fd teratos of the secat method for f ( ) f ( ) f ( ) f ( ), f ( ),, f ( ) f () f ( ) f ( ) f () f () 8 ( ).6 ( ) ( ) f ( ) f f ( ) f ( ) f f

52 Adrew Powuk - Math 49 (Numercal Aalyss).5. Relato wth the Newto s method f ' f f ' f f f f f f ' f f f ( ) f ( ) ( ) 5

53 Adrew Powuk - Math 49 (Numercal Aalyss).5. Dervato of the formula f ( ) f ( ) y f ( ) ( ) f ( ) f ( ) y f ( ) ( ) (, f ( )) (, f ( )) f( ) f ( ) f ( ) f f ( ) ( ) ( ) f ( ) ( ) f ( ) f ( ) ( ) f ( ) f ( ) f ( ) f( ) f ( ) f ( ) f f ( ) f ( ) or f f ( ) f ( ) f( ) f ( ) f ( ) ( ) ( ) 5

54 Adrew Powuk - Math 49 (Numercal Aalyss) f ( ) f ( ) y f ( ) ( ) f ( ) f ( ) f ( ) ( ) ( ) f ( ) ( ) f ( ) f f( ) f ( ) f ( ) f( ) f ( ) f ( ) f ( ) f ( ) f ( ) f ( f ( ) f ( ) ) 54

55 Adrew Powuk - Math 49 (Numercal Aalyss) Comparso to the Newto's method f( ) f '( ) f ( ) f ( ) f '( ) f ( ) f f ( ) f ( ) f ( ) f ( ) f ( ) f ( ) f ( ) ( ) f ( ),.. Clear[]; f[_]=^-; =; =; =N[-(-)*f[]/(f[]-f[])] =; =; =N[-(-)*f[]/(f[]-f[])] =; =; =N[-(-)*f[]/(f[]-f[])] =; =; =N[-(-)*f[]/(f[]-f[])]

56 Adrew Powuk - Math 49 (Numercal Aalyss).5. (*) Covergece aalyss Covergece aalyss f f f f f f f f f f f f f f f f f f f f f f f f f '' f '' f '' f '' f ( ) f ( ) f f f f ' f O f f f ' f '' O f ( ) f ( ) f f ' ' ' f f f f f f f 56

57 Adrew Powuk - Math 49 (Numercal Aalyss) 57 ( ) ( ) '' '' '' ' ' '' ' f f f f f f f f f f f f f 5.6 p p p p p p p p p A A A p A c A c p A Superler covergece 5 5 '' ' A f A f The method s faster tha the Newto s method because t requre oly oe fucto evaluato every terato.

58 Adrew Powuk - Math 49 (Numercal Aalyss) ma, ' ' , ' p p A p A A f A M f A A

59 Adrew Powuk - Math 49 (Numercal Aalyss).5.4 Fd teratos of the secat method for f ( ) f ( ) f ( ) f ( ), f ( ) f () f ( ) f ( ) f () f () 7 ( ) ( ) ( ) f ( ) f f ( ) f ( ) f f() f ( ),,.5.5 Fd teratos of the secat method for f ( ) f ( ) f ( ) f ( ), f ( ) f () f ( ) f ( ) f () f () 8 ( ).6 ( ) ( ) f ( ) f f ( ) f ( ) f f f ( ),, 59

60 Adrew Powuk - Math 49 (Numercal Aalyss).5.6 Fd teratos of secat method for f ( ) f ( ) f ( ) f ( ) , f f ( ) f () f ( ) f ( ) f () f () ( 5). ( 5) ( 5) 5 f ( ) 5 f f ( ) f ( ) 5 5 f f ( ) 5,,.5.7 Fd teratos of the secat method for f ( ),, f ( ),, f f ( ) f ( ) ( ) f ( ) f () f ( ) f ( ) f () f () 4 ( ). ( ) f ( ) f f( ) f ( ) 4 f f ()

61 Adrew Powuk - Math 49 (Numercal Aalyss).5.8 Fd teratos of the secat method for f (, ) 5 4, f ( ) 5 4,, f f ( ) f ( ) ( ) f ( ) f () f ( ) f ( ) f () f () ( 5 4 * ). 5 4 * ( 5 4 * ) 9 f ( ) 9 f f ( ) f ( ) 9 9 f f() * * 5 4 * 9 9 6

62 Adrew Powuk - Math 49 (Numercal Aalyss).5.9 Fd teratos of the secat method for f ( ),, f ( ),, f f ( ) f ( ) ( ) f ( ) f () f ( ) f ( ) f () f () f ( ) f ( ) f ( ) f ( ) f ( ) f ().5. Fd teratos of the secat method for f[_]=^-^--; Plot[f[],{,,5}] f ( ), 5,

63 Adrew Powuk - Math 49 (Numercal Aalyss) f ( ), 5, 4 f f ( ) f ( ) f ( ) f ( ) f ( ) f(4) f( 4) f(5) ( ) * * f ( ) f ( ) f ( ) * * * f(.8) f(.8) f(4).8.8 *

64 Adrew Powuk - Math 49 (Numercal Aalyss) f[_]=^-^--; =5; y=f[]; =4; y=f[]; Sl[_]=y+((y-y)/(-))*(-); Plot[{f[],Sl[]},{,.4,5}]

65 Adrew Powuk - Math 49 (Numercal Aalyss). (*) Regula fals c a f b f b b f a f a The method coverges faster tha the bsecto method. 65

66 Adrew Powuk - Math 49 (Numercal Aalyss) 66.6 Fed pot teratos Equato * * * * lm

67 Adrew Powuk - Math 49 (Numercal Aalyss) Eample 5 ( ), ( ), ( ) ( ) ( ) ( ) * lm * * * * 5 * true 67

68 Adrew Powuk - Math 49 (Numercal Aalyss) 68

69 Adrew Powuk - Math 49 (Numercal Aalyss).6. Fd fed pot teratos for f ( ) 5 ad 5. 5 ( ) 5 5 f( ) f 5 / : 5 / ( ) 5 ( ) Fed pot teratos 5 ( ), ( ), ( ) ( ) ( ) ( )

70 Adrew Powuk - Math 49 (Numercal Aalyss) Geeral formula for fed pot terato ( ) Covergece crtera a, b ma '( ) Eample 5, ( ), '( ).5. y '( ) Clear[]; f[_]=.5*(+5/); =7; =f[] =f[] =f[] =f[] =f[]

71 Adrew Powuk - Math 49 (Numercal Aalyss).6. Eample / 5 ( ) 5 ( ) 5 5 ( ) ( ) 5 ) ( Eample / 5 /*(-) 5 5 ( ) ( )

72 Adrew Powuk - Math 49 (Numercal Aalyss).6.4 Eample /:5 / ( ) ( ) ( ) ( ) ( ) ( )

73 Adrew Powuk - Math 49 (Numercal Aalyss).6.5 Eample / : 5 ( ) 5 ( ) ( ) ( ) ( ) 5... The lmt does t est. 5 5 ( ) ( ) ( ).666 7

74 Adrew Powuk - Math 49 (Numercal Aalyss).6.6 (*) Eample D y 4 4 y y 4 y 4 y 4 4 y X X y 4 y 4 X y 4 y X 4 X 4 y 4 X X y X X y X X

75 Adrew Powuk - Math 49 (Numercal Aalyss) 4 4 y y 4 y 4 y 4 y 4 y 4 y y 4 X AX B ( X ) AX B X ( X) X ( X ) AX B X ( X ) AX B A AX B B A X AB B X ( X ) AX B A AX B B A X AB B ( ) A AX B AB B A AX A B AB B A X A A B X A X A A B X A X A... A A B for A, lm AX S A... A A lm S (geometrcal seres) X AX B ( X ) AX B '( X) A A Matr orm 75

76 Adrew Powuk - Math 49 (Numercal Aalyss) A A ma a ma, j aj aj j j j ma a a, a a ma, ma, 4 y y 4 4 y 4 y A A ma a ma a, a j j j j j j ma, ma, a a a a ma, 76

77 Adrew Powuk - Math 49 (Numercal Aalyss).6.7 Geometrcal terpretato of the fed pot terato y y g( ) g( ) g( ) y y g( ).. 77

78 Adrew Powuk - Math 49 (Numercal Aalyss).6.8 Fed pot theorem (estece of the soluto).6.8. Itroducto Theorem (Estece of the soluto) Let be a cotuous fucto o[, ], ad suppose g satsfes the property a b a g( ) b g ( ) The the equato If D a, b g( ) the D D. ab ab has at least oe soluto the terval[, ]. b y a a b Remark Usg cotouty t s possble to prove that the soluto of the equato ests f g : a, b a, b or a, b g : ab,. g the terval ab, 78

79 Adrew Powuk - Math 49 (Numercal Aalyss) b y a a g : a, b a, b a, b g : ab, b 79

80 Adrew Powuk - Math 49 (Numercal Aalyss).6.9 Cotracto mappg theorem Theorem (Cotracto mappg theorem for Assume that g ( ) ad g '( ) the ) There s a uque soluto ) For ay tal estmate of ) 4) lm g '( ) for g ) are cotuous for a b ad assume that g satsfes the theorem, ad ma g'( ) a b of the equato g( ) (.e. g( ) ). ab, close to, the terates wll coverge to. we have g '( ). 8

81 Adrew Powuk - Math 49 (Numercal Aalyss) (*) Proof g g ' ' ' ' ma g ' g g g g g g ab, g g

82 Adrew Powuk - Math 49 (Numercal Aalyss) g g g ' ' ' ' ma g ' g g g g g g ab, By assumpto, the lm lm Because the by the sadwch theorem 8

83 Adrew Powuk - Math 49 (Numercal Aalyss) lm lm lm lm lm By cotuty of lm lm We kow the lm lm Uqueess Let s assume that there are two fed pots lm lm,,, lm lm lm D lm 8

84 Adrew Powuk - Math 49 (Numercal Aalyss) Theorem (***) (Baach fed pot theorem) Let (X, ) be a o-empty complete metrc space. Let : oegatve real umber * ad oly oe fed pot X T X X such that ( T ( ), T ( y)) (, y) for all Fed pot ca be foud as follows: start wth a arbtrary elemet or T ( ). The followg equalty descrbes the speed of covergece * (, ) (, ) * (, ) (, ) be a cotracto mappg o X,.e.: there s a, y X. The T admts oe X ad defe ad teratve sequece by.6.9. (*) Problem Is t possble to solve Aswer for,? I preseted eample. Rage of the fucto the terval, s (from the graph) :, :,, s cotuous (every polyomal s cotous) o the terval,. Fucto Because ad ), ), :, s cotuous (every polyomal s cotous) o the terval, The accordg to the theorem the soluto of the equato Plot[{,-},{,,}] for, ests. 84

85 Adrew Powuk - Math 49 (Numercal Aalyss) (*) Problem Is t possble to solve Aswer? for, I order to apply the theorem we eed to kow the rage of the fucto over the terval, ad the cotouty. Fucto s cotuous o the terval, because every polyomal s cotous. Rage of the fucto Plot[{,-^},{,,}]. ca be foud from the graph

86 Adrew Powuk - Math 49 (Numercal Aalyss) Because ) :, :,,, ad 4), :,, s cotuous (every polyomal s cotous) o the terval The accordg to the theorem the soluto of the equato, for, ests. 86

87 Adrew Powuk - Math 49 (Numercal Aalyss).6.9. Problem 5 Let us cosder ( ) ad,. estmate the error of the fed pot terato for Aswer From the graph whch s show below we ca see that ( ) : ab, a, b, or ( D) D, 5 the the soluto of the equato ests the terval Plot[{(/)*(+5/),},{,,}].,. ad ( D) D ,,,,,, m, ma, m ma, 87

88 Adrew Powuk - Math 49 (Numercal Aalyss) Let s calculate dervatve 5 ( ) 5 '( ) Let s D, the Crtcal umbers '( ) D 5 '( ) , 5.6 D ca be calculated by usg optmzato techques. [,] [,] ma ma ma, 5,.4 D m m m, 5, D D m, ma [,] [,] D D.5.5,.4, D 88

89 Adrew Powuk - Math 49 (Numercal Aalyss) Plot[(/)*(-5/^),{,,}] Plot[Abs[(/)*(-5/^)],{,,}] Defto of the rage [ ab, ], 89

90 Adrew Powuk - Math 49 (Numercal Aalyss) '' 5 ' 5 *5 5 Plot[5/^,{,-4,4}] I the terval [,] dervatve of the fucto '( ) s postve, the the fucto '( ) crease. The etreme values of the fucto '( ) are at the epots of the terval [,]. From the graph we see that 5 ma '( ) ma ma ', ' 5, ' ma,, a b the accordg to the theorem the sequece Let s estmate the error for the fed pot terato for s coverget to the fed pot.. 9

91 Adrew Powuk - Math 49 (Numercal Aalyss) 5,, ( ) Verfcato of the results We kow that the soluto of the equato After calculatos True error s much smaller tha the error estmator. s *

92 Adrew Powuk - Math 49 (Numercal Aalyss) Problem Let us cosder ( ) s ad,. Soluto Clear[]; f[_]=(/)*s[]; Plot[{f[],},{,-P,P}] =; =N[f[]] =N[f[]] =N[f[]] =N[f[]] =N[f[]] =N[f[]] =N[f[]] =N[f[]] =N[f[]] =N[f[]] estmate the error of the fed pot terato for ad

93 Adrew Powuk - Math 49 (Numercal Aalyss) Estece of the soluto for D [, ] D : D s :,, D D, true, Plot[{(/)*S[],/,-/},{,-P,P}]

94 Adrew Powuk - Math 49 (Numercal Aalyss) ' s ' cos ma ' ma cos D, Plot[{(/)*Cos[],/,-/},{,-P,P}] Plot[{(/)*Abs[Cos[]],/,-/},{,-P,P}]

95 Adrew Powuk - Math 49 (Numercal Aalyss) Error estmate s ( )

96 Adrew Powuk - Math 49 (Numercal Aalyss) Problem Let us cosder ( ) cos ad,. Aswer estmate the error of the fed pot terato for ad Clear[]; = ; = ; Do[{Prt[], = (/)*N[Cos[]]}, {,, }]

97 Adrew Powuk - Math 49 (Numercal Aalyss) D D,, m, ma,,,,, :, cos :,, m m cos,, ma ma cos,, D D,, true The f pot ests. ma ' ma s ma s D,, The f pot s uque. cos ( )

98 Adrew Powuk - Math 49 (Numercal Aalyss) Atke Error estmato ad Etrapolato

99 Adrew Powuk - Math 49 (Numercal Aalyss) after solvg that for or g( ) g( ) Because lm lm g '( ) the so t s possble to assume that Atke's etrapolato forma for error estmato ad g ' 99

100 Adrew Powuk - Math 49 (Numercal Aalyss) Eample, 5, ( ) ( )

101 Adrew Powuk - Math 49 (Numercal Aalyss).6. (*) Hgher order Error Estmato.6.. Frst order method g( ) g( ) g '( c ) g( ) g( ) g( ) g( ) g '( c ) g '( c ) g '( c ).6.. Secod order method If g '( ) g( ) g( ) g '( ) g ''( c ) g( ) g( ) g '( ) g( ) g( ) g '( ) g ''( c ) g ''( c ) g ''( c )

102 Adrew Powuk - Math 49 (Numercal Aalyss).7 (*) Order of covergece Defto (Rate of covergece) - The rate of covergece s at least ler. If c ad N such that * * C, for N - The rate of covergece s at least superler ff { } such that * ad N such that - The covergece s at least quadratc f C & N t * * C, I geeral for large p * *, C the p s the rate of covergece. p *, N

103 Adrew Powuk - Math 49 (Numercal Aalyss) l p * * C p * * C l p C p C C p C p l l l l l l l l l C p ( ) l l pl l l l p l l p l l p l l ~ p l l l l

104 Adrew Powuk - Math 49 (Numercal Aalyss) Sample code fucto ewto() eps=.; =; error=; terato=; whle(error>eps) em=log(abs(-sqrt())); New=-f()/df(); e =log(abs(new-sqrt())); error=abs(new-); New=New-f(New)/df(New); ep=log(abs(new-sqrt())); p=(ep-e)/(e-em); =New; dsp(sprtf('=%d =%f error=%f p=%f',terato,,error,p)); terato=terato+; ed ed fucto y=f() y=^-; ed fucto y=df() y=*; ed Output = =.5 error=.5 p=.855 = = error=.8 p=.9899 = =.446 error=.45 p= =4 =.444 error=. p=if 4

105 Adrew Powuk - Math 49 (Numercal Aalyss).8 Revew.8. Summer 7 Secto. Problem..7 Problem..8 Secto.4 Problem.4. Problem.4. Secto.5 Problem.5.7 Problem.5.8 Secto.6 Problem.6.9. Problem