3.4. DIRECT METHODS

Transcription

1 3.4. DIRECT METHODS 66 V = » flops %prit flop out s = 8» flops();h3_ivu(u)flops %iitilize flop out the ivert s = s = Gussi elimitio Gussi elimitio is oe of the most effiiet lgorithms for solvig lrge system of lier lgeri equtios. It is se o systemti geerliztio of rther ituitive elimitio proess tht we routiely pply to smll sy ( ) systems. e.g. x x = 4 x 4x = 3 From the first equtio we hve x = (4 x )/ whih is use to elimite the vrile x from the seo equtio viz. (4 x )/ 4x = 3 whih is solve to get x =.684. I the seo phse the

2 3.4. DIRECT METHODS 68 vlue of x is k sustitute ito the first equtio we get x =.63. We oul hve reverse the orer elimite x from the first equtio fter rerrgig the seo equtio s x = (3 4x ). Thus there re two phses to the lgorithm: () forwr elimitio of oe vrile t time util the lst equtio otis oly oe ukow; () k sustitutio of vriles. Also ote tht we hve use two rules urig the elimitio proess: (i) two equtios (or two rows) e iterhge s it is merely mtter of ook keepig it oes ot i y wy lter the prolem formultio (ii) we reple y equtio with lier omitio of itself with other equtio. A oeptul esriptio of ive Gussi elimitio lgorithm is show i figure 3.. All of the rithmeti opertios eee to elimite oe vrile t time re ietifie i the illustrtio. Stuy tht refully. We ll it ive sheme s we hve ssume tht oe of the igol elemets re zero lthough this is ot requiremet for existee of solutio. The reso for voiig zeros o the igols is to voi ivisio y zeros i step of the illustrtio 3.. If there re zeros o the igol we iterhge two equtios i suh wy the igols o ot oti zeros. This proess is lle pivotig. Eve if we orgize the equtios i suh wy tht there re o zeros o the igol we my e up with zero o the igol urig the elimitio proess (likely to our i step 3 of illustrtio 3.). If tht situtio rises we otiue to exhge tht prtiulr row with other oe to voi ivisio y zero. If the mtrix is sigulr we wil evetully e up with uvoile zero o the igol. This situtio will rise if the origil set of equtios is ot lierly iepeet; i other wors the rk of the mtrix A is less th. Due to the fiite preisio of omputers the flotig poit opertio i step 3 of illustrtio 3. will ot result usully i ext zero ut rther very smll umer. Loss of preisio ue to rou off errors is ommo prolem with iret methos ivolvig lrge system of equtios sie y error itroue t oe step orrupts ll susequet lultios. A MATLAB implemettio of the lgorithm is give i figure 3.3 through the futio guss.m. Exmple Let us otiue with the reyle prolem (equtio (.8) of Chpter ). First we oti solutio usig the uilt-i MATLAB lier equtio solver ( viz. x = A\ reor the flotig poit opertios (flops). The we solve with the Gussi elimitio futio guss ompre the flops. Note tht i orer to use the ive Gussi elimitio futio Note tht it is merely for illustrtig the oepts ivolve i the elimitio proess; MATLAB kslsh \ opertor provies muh more elegt solutio to solve Ax = i the form x = A\.

3 3.4. DIRECT METHODS 69 for j=i:; (ij)=(ij)/(ii); e for k=i:; (jk)=(jk)- (ji)*(ik); e STEP : Mke igol elemets (ii) ito. STEP : Arrge A ito ( x ) mtrix STEP 3 : Mke ll elemets i olum i elow igol ito for i=: for j=i: e E of forwr elimitio. Resultig mtrix struture is: e STEP 4: Bk sustitutio () = () for j=-:-:; (j) = (j) - (jj:)*(j:); e Solutio is reture i the lst olum (:) Figure 3.: Nive Gussi elimitio sheme

4 3.4. DIRECT METHODS 7 futio x=guss() % Nive Gussi elimitio. Cot hve zeros o igol % - (x) mtrix % - olum vetor of legth m=size(); % get umer of rows i mtrix =legth(); % get legth of if (m = ) error( o ot hve the sme umer of rows ) e %Step : form () ugmete mtrix (:)=; for i=: %Step : mke igol elemets ito. (ii:) = (ii:)/(ii); %Step 3: mke ll elemets elow igol ito for j=i: (ji:) = (ji:) - (ji)*(ii:); e e %Step 4: egi k sustitutio for j=-:-: (j) = (j) - (jj:)*(j:); e %retur solutio x=(:) Figure 3.3: MATLAB implemettio of ive Gussi elimitio

5 3.4. DIRECT METHODS 7 we ee to swith the 3r equtios to voi ivisio y zero.»a=[.36; % mtrix etry otiues o ext two lies»-;».7]; % Defiitio of mtrix A omplete»=[ ] ; % Defie right h sie olum vetor»flops() ; % iitilize flop out»a\ % solutio is : [ ]»flops % exmie flops for MATLAB iterl solver (s: 7)»flops() ; % iitilize flop out»guss(a) % solutio is of ourse : [ ]»flops % exmie flops for guss solver (s: 75)»flops() ; % iitilize flop out»iv(a)* % oti solutio usig mtrix iverse»flops % exmie flops for MATLAB iterl solver (s: ) Exmple - loss of ury & ee for pivotig The ee for pivotig e illustrte with the followig simple exmple. ɛx x = x x = where ɛ is smll umer. I mtrix form it will e [ ][ ] [ ] ɛ x = x I usig ive Gussi elimitio without rerrgig the equtios we first mke the igol ito uity whih results i x ɛ x = ɛ Next we elimite the vrile x from the equtio whih resutls i ( ) x = ɛ ɛ Rerrgig this usig k sustitutio we filly get x x s x = ɛ ɛ

6 3.4. DIRECT METHODS 7 Nive elimitio Built-i ɛ without pivotig MATLAB guss(a) A\ 5 [ ] [ ] 6 [ ] [ ] 7 [ ] [ ] Tle 3.: Loss of preisio ee for pivotig x = ɛ x ɛ The prolem i omputig x s ɛ shoul e ler ow. As ɛ rosses the threshol of fiite preisio of the omputtio (hrwre or softwre) tkig the ifferee of two lrge umers of omprle mgitue result i sigifit loss of preisio. Let us solve the prolem oe gi fter rerrgig the equtios s x x = ɛx x = pply Gussi elimitio oe gi. Sie the igol elemet i the first equtio is lrey uity we elimite x from the equtio to oti ( ɛ)x = ɛ or x = ɛ ɛ Bk sustitutio yiels x = x Both these omputtios re well ehve s ɛ. We tully emostrte this usig the MATLAB futio guss show i figure 3.3 ompre it with the MATLAB uilt-i futio A\ whih oes use pivotig to rerrge the equtios miimize the loss of preisio. The results re ompre i tle 3. for ɛ i the rge of 5 to 7. Sie MATLAB uses oule preisio this rge of ɛ is the threshol for loss of preisio. Oserve tht the ive Gussi elimitio proues iorret results for ɛ< Thoms lgorithm My prolems suh s the stgewise seprtio prolem we sw i setio.3. or the solutio of ifferetil equtios tht we will see i

7 3.4. DIRECT METHODS 73 (j) = (j) - {(j-)/(j-)}*(j-) (j) = (j) - {(j-)/(j-)}*(j-) STEP : Elimite lower igol elemets Give STEP : Bk sustitutio for i=-:-: (i) = {(i) - (i)*(i)}/(i); e () = ()/() Solutio is store i for j=: e Figure 3.4: Thoms lgorithm lter hpters ivolve solvig system of lier equtios Tx = with triigol mtrix struture. T =... x = x x. x x =. Sie we kow where the zero elemets re we o ot hve to rry out the elimitio steps o those etries of the mtrix T ; ut the essetil steps i the lgorithm remi the sme s i the Gussi elimitio sheme re illustrte i figure 3.4. MATLAB implemettio is show i figure 3.5.

8 3.4. DIRECT METHODS 74 futio x=thoms() % Thoms lgorithm for triigol systems % - igol elemets % - right h sie forig term % - lower igol elemets (-) % - upper igol elemets (-) =legth(); % get legth of =legth(); % get legth of =legth(); % get legth of =legth(); % get legth of if ( = = (-) = ) error( rry imesios ot osistet ) e =legth(); %Step : forwr elimitio for i=: ftr=(i-)/(i-); (i) = (i) - ftr*(i-); (i) = (i) - ftr*(i-); e %Step : k sustitutio () = ()/(); for j=-:-: (j) = ((j) - (j)*(j))/(j); e %retur solutio x=; Figure 3.5: MATLAB implemettio of Thoms lgorithm

9 3.4. DIRECT METHODS Gussi elimitio - Symoli represetio Give squre mtrix A of imesio it is possile to write it is s the prout of two mtries B C i.e. A = BC. This proess is lle ftoriztio is i ft ot t ll uique - i.e. there re iifitely my possililities for B C. This is ler with simple outig of the ukows - viz. there re ukow elemets i B C while oly equtios e otie y equtig eh elemet of A with the orrespoig elemet from the prout BC. The extr egrees of freeom e use to speify y speifi struture for B C. For exmple we require B = L e lower trigulr mtrix C = U e upper trigulr mtrix.this proess is lle LU ftoriztio or eompositio. Sie eh trigulr mtrix hs ( )/ ukows we still hve totl of ukows. The extr egrees of freeom is ofte use i oe of three wys: Doolitle metho ssigs the igol elemets of L to e uity. Crout metho ssigs the igol elemets of U to e uity. Cholesky metho ssigs the igol elemets of L to e equl to tht ofu - i.e. l ii = u ii. While simple egree of freeom lysis iites tht it is possile to ftorize mtrix ito prout of lower upper trigulr mtries it oes ot tell us how to fi out the ukow elemets. Revisitig the Gussi elimitio metho from ifferet perspetive will show the oetio etwee LU ftoriztio Gussi elimitio. Note tht the lgorithm outlie i setio is the most omputtiolly effiiet sheme for implimetig Gussi elimitio. The metho to e outlie elow is ot omputtiolly effiiet ut it is useful oeptul i i showig the oetio etwee Gussi elimitio LU ftoriztio. Steps 3 of figure 3. tht ivolve mkig the igol ito uity ll the elemets elow the igol ito zero is equivlet to pre-multuplyig A y L - i.e. L A = U or = () () () ().... () ()