QR factorization. Let P 1, P 2, P n-1, be matrices such that Pn 1Pn 2... PPA

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1 QR facorzao Ay x real marx ca be wre as AQR, where Q s orhogoal ad R s upper ragular. To oba Q ad R, we use he Householder rasformao as follows: Le P, P, P -, be marces such ha P P... PPA ( R s upper ragular. These marces may be chose as orhogoal marces ad are kow as householder marces. The we wll have he requred facorzao wh Q ( P P... PP We frs fd P such ha P A wll have all elemes below he dagoal he frs colum as zero. We would he fd P such ha P P A wll have all zeroes below he dagoal boh he frs ad he secod colum. Ad so o ll we fd P - whch produces he upper ragular marx. The procedure s as follows: T (llusraed for a 3x3 marx [A] To fd ou he h (, - marx, P:. Take he h colum of he marx P - P - P P A (for, use s colum of A ad ormalze o have u L orm. Deoe hs vecor by {x}. For he gve marx, for he frs marx P : 4 {} x Compue he L orm of he subspace of {x} spag he dmesos hrough,.e., X x+ x x (use he egave roo f x >. Defe a ew vecor {y}, of u magude, whch has s frs - compoes as zero, he h compoe as x xk ad oher compoes (k+ o as. X Xy

2 X ; { y} Oba he marx P as P I yy T P Repeag he seps,, ad 3: PA {} x X.8783 ; { y} P R PPA

3 T Q ( PP Usg hs echque of facorzao ad afer abou 5 eraos of he QR mehod, we ge a dagoal marx showg he Egevalues as 5.496, -.74, ad.578 (a Excel fle showg he facorzao echque ad frs few QR eraos may be see a hp://home.k.ac./~raeshs/qr.xls. Some remarks (we assume ha all egevalues are real!:. For a x [A] marx, Q s always symmerc.. For 3x3 ad hgher, geeral Q wll o be symmerc eve whe A s symmerc. 3. If all he egevalues are dsc, he fal [A] marx would be dagoal for symmerc marces, ad upper ragular for o-symmerc marces. 4. Covergece of QR mehod s slow f wo egevalues are very close.

4 Fourer Seres Orhogaly: Couous: s xskxdx k k cos xcoskxdx k k s xcoskxdx s xdx cos xdx OR, erm of he expoeals x kx e e dx k k Dscree: α for x.e., M+ equally spaced pos bewee - (clusve ad (exclusve α M + M x kx k α α e e M + for eger α M + oherwse Fourer Expasos: Couous: ˆ( x where ( x f x ce c f xe dx OR: a fˆ( x + ( a cos x+ b s x where a f ( x cos xdx ad b f ( x s xdx

5 Dscree: (oe me perod s dvded o M+ ervals, he las po s o cosdered sce wll be same as he frs po due o perodcy α for xα.e., M+ equally spaced pos bewee - (clusve ad (exclusve M + If Ms eve: θ, km/ If M s odd: θ, k(m-/ fˆ( x ce where c f ( x e OR a fˆ( x k+ θ M x k M + α + k ( a k+ α x θ cos x+ b s x + a cos( k+ x M M where a f ( x cos x ad b f( x s x M + M + α α α α α α α Noe ha he form gve above resuls erpolao wh f ˆ( x equal o f(x a all he (M+ grd pos. Fewer erms could be used (.e., summao o carred up o k + θ o oba a leas squares f. The erm s he fudameal frequecy ad s he frs harmoc. Also oe ha ( he exbook uses a me perod of T, whle we have used he class. Therefore, whle he book has he fudameal frequecy as ω for he couous case, T we have ( he dscree approxmao gve he book requres complex compuaos whle ha gve above erms of se ad cose does o (alhough hey are equvale. Legedre Polyomals Defo: Orhogoaly: d P ( x for,,...: P ( x x! dx ( ( ( P, P PxPxdx ( ( + Recurso: P( x xp ( x P ( x,3,... P( x ; P( x x The equvale polyomals for dscree daa are Gram s polyomals. Followg s a plo of he Legedre Polyomal approxmao of f(x/(+x usg up o orders,, 4, ad 8. Noe he geeral mproveme f wh creasg order. However, for a specfc po he error may crease for a hgher order f (e.g., ear x.5, order f s beer ha d order f.

6 Acual P P P4 P8 f(x x

7 Tchebycheff Polyomals Defo: T ( x cos( cos x Orhogoaly: / (, ( ( ( T T T xt x x dx Recurso: T( x xt ( x T ( x,3,... T( x ; T( x x I he erval [-,], k Zeroes of T ( : cos x xk k,,...,( k (+ exrema T( x: x k cos k,,..., Values of T (x are (- k. Followg s a plo of he Tchebycheff Polyomal approxmao of f(x/(+x usg up o orders,, 4, ad 8. Noe he geeral mproveme f wh creasg order. However, for a specfc po he error may crease for a hgher order f (e.g., ear x.65, order f s beer ha d order f. Acual T T T4 T8 f(x x

8 Tchebycheff Polyomals For dscree case: If x k are he zeroes of T m+ (x ad he rage of ad s from o m: Cubc Sples m ( T, T T ( x T ( x k k k m+ m + Usg he local coordae sysem for he h x x segme, X ad usg he symbol o x x represe (x -x -, we ge (sarg from he fac ha he secod dervave s lear f ( X f + Xf x ˆ fx ( X f + Xf X( X( [ X f + (+ X f ] 6 f f X X f + X f f x Couy of he frs dervave gves us f + ( + f + + f + + f + f + f+ for o (m- + + whch alog wh he wo ed codos f ad f m could be solved usg he Thomas algorhm for rdagoal sysem. Sarg from he frs dervave, whch s a quadrac fuco of x, we ge (afer usg he wo ed codos for he frs dervave ad a ukow cosa C ( f f + CX + f f C X x Iegrag oce o oba he cubc polyomal ad applyg he wo ed codos for he fuco values, we ge ( ( ( X + + ( X + X fˆ 3X + X f + 3X X f + X x 6 6 f x ( X + X f + ( X X f + ( 4X + 3X f + ( X + 3X f 6 6 f + X f + X f + + X f + X ( ( ( 3 ( + 3 x X f f f

9 Couy of he secod dervave gves us + + f + + ( + f + + f + 3 f + 3 f + 3 f+ for o (m- + + The wo ed codos f ad f m gve he oher wo equaos as f f f + f 3 f f 3 m m f m + f m f f If we deoe he slope of he daa pos as s, we may wre hese equaos as x x fˆ x ( X f + Xf + X ( X ( X ( f s X ( f s f + + f + f 3 s + s for o (m- ( ( f + f 3s f + f 3s m m m m

10 Gram s Polyomal The geeral equao for geerag Gram s polyomals of order m for equdsa pos bewee - ad (x - + /m for o m s α G+ ( x αxg( x G ( x α m 4( + where α ad G ( x ; G( x + ( m+ ( + m+ x For order : G ; G x 3 For order : G ; G ; G x 3 3 The erpolao formula s he gve by m fˆ( x CG ( x ad he coeffces are obaed from he orhogoaly propery as m ( ( C f x G x For example, for m, he coeffces are gve by f + f + f f f 3 C C C ( f f f

11 Vecor ad Marx Norms L p orm of a -dmesoal vecor, x, s gve by: p p p p ( / x x + x + x x p p p p s he Eucldea orm ad p deoes he maxmum orm. The properes of Vecor ad marx orms are (x, y are vecors, A, B are marces ad α s a scalar x oly f x s a ull vecor; oherwse x > A oly f A s a ull marx; oherwse A > αx αa α α x A x+ y x + y A+ B A + B AB A B Also, for ay vecor orm here exss a cosse marx orm such ha Ax A x Smlar o he Eucldea orm of a vecor here s he Frobeus orm for a marx defed by A a. The orm whch s easy o compue ad s herefore commoly used s he, maxmum orm (also called uform orm whch, for a vecor, s he eleme wh larges magude ad, for a marx, s he larges row-sum of absolue values,.e., As show he class, for a lear sysem of equaos Axb, δx δ A κ( A x+ δ x A where κ(a s he codo umber of he marx A ad s equal o δx show ha κ ( A x or b wll produce large (relave chage x. A max a. A A. I ca also be δb. Thus for large codo umbers, small (relave chages A b

12 Covergece Properes of he Newo-Raphso Mehod + f ( x r As dscussed he class E E f ( x dcag quadrac covergece. If we assume ha for all pos x a ad x b ear he roo, + we ca wre M E ( M E ear he roo AND r f ( xa f ( x b has a upper boud of M,. Now f we assume ha he al guess x s suffcely M E <, ca be show ha he eraos wll coverge ad E ( M E M So, he N-R mehod wll always coverge f he al guess s suffcely close o he (smple roo AND magude of (M E s less ha. However, sce he roo s o kow beforehad s dffcul o use hs crero. A more usable crero for covergece s: If here s a erval [l,u] such ha f(l ad f(u have oppose sgs; f (x s o zero ad f (x does o chage sg he erval; ad f (x/f(x a boh l ad u s less ha (u-l; he N-R mehod wll coverge from ay al guess wh he erval. Error he Mehod of False Poso For solvg yf(x, ca be show ha afer a suffce umber of eraos, oe ed of he erval remas fxed. If we ake hs fxed po as x, we have x x x+ x + ( y y y whch s decal o Newo s dvded dfferece lear erpolao of fuco xg(y o oba x for y. The error of erpolao s gve by (x r s he roo + E xr x+ g ( y ( y( y where ỹ s he erval (y,y. Usg mea value heorem bewee x r ad x o oba y ad f ( x usg g ( y, we may wre f ( x [ ] 3 + f ( x E f ( x E f ( x E ( 3 [ f x ] where x s he erval (x,x, x s he erval (x,x r ad x s he erval (x,x r. Assumg ha he eraos coverge o he roo x r + f ( x f ( x f ( x r E E E 3 f ( x [ ] where x s he erval (x,x r. Ths shows lear covergece.

13 Error he Seca Mehod For hs mehod, we have x x x x + ( y + y y The error of erpolao s gve by (x r s he roo + E xr x+ g ( y ( y ( y where ỹ s he erval (y -,y. Aga, we may wre f ( x, f ( x f + ( x E E 3 E f ( x, where boh x, ad x are he erval (x -,x r, ad x s he erval (x,x r. Assumg ha he eraos coverge o he roo x r, we ge, for large, f ( xr E + αe E where α f ( xr If p deoes he order of he Seca mehod ad C s asympoc error cosa (such ha asympocally E + p CE we ge p p+ ad C / p α mplyg ha he order s.68 (beer ha Bseco ad False Poso bu o as good as Newo Raphso. As far as effcecy s cocered, f he compuaoal effor evaluag he dervave of a fuco s more ha.44 mes ha requred for a fuco evaluao, he Seca mehod s preferred ( Error he Muller s Mehod For hs mehod, sce we use quadrac erpolao o fd x for y, he error of erpolao s gve by (x r s he roo + E xr x+ g ( y ( y ( y ( y 6 where ỹ s he erval (y -,y -,y. Aga, assumg ha he eraos coverge o he roo x r, we may wre f ( xr E + αe E E where α 6 f ( xr If p deoes he order of he Muller mehod, we ge p 3 p +p+ mplyg ha he order s.839 (beer ha Seca bu o as good as Newo Raphso.