2. Elementary Linear Algebra Problems
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1 . Eleety e lge Pole. BS: B e lge Suoute (Pog pge wth PCK) Su of veto opoet:. Coputto y f- poe: () () () (3) N Full y tee Depth te tep log()n
2 Veto updte the f- poe wth N : ) ( ) ( ) ( ) ( ) ( ) ( ) ( N N N N ( ) ( ) [ ] ( ) ( ) [ ] ed ed fo N fo N N ; ) ; ; ( ) ; ; ( log()n te tep pllel fo veto of legth
3 evel- BS BS oute wth () pole (veto oly y ε IR ). Ft eple: DT-podut y f-: y T y y y 3 y 3 4 y 4 5 y 5 6 y y 7 8 y 8 3
4 Te tep pllel of the DT-podut ot ette th log(). Evey oputto volvg f- wll te log() te tep pllel! ddedu: Coputg the DT-podut o pel htetue: Dtute dt o -deol y of poeo wth / poeo 3 (-) P P P 3 P / P - P Ptto veto d y lo of veto wth legth. Eh poeo h to opute ptl DT-podut: ( ) y( ) y... 4
5 Te tep pllel oputto ed o th odel: ( dd ult ) te fo ddto d ultplto pllel poeo p fo ptl DT-podut fte oputg the ptl DT-podut eh poeo pllel teedte eult hve to e olleted d dded up to the fl eult. To th the poeo P (P ) ed t eult to the eghog poeo P (P - ) tht dd th ue to t ow ptl DT-podut. The the eult ed to the et egho tht dd two ptl eult g d o o utl the fl eult oputed the poeo P /. ( ) 3 (-) P P P 3 P / P - P 5
6 Totl te depedg o d : f ( ) ( dd ( dd ult) ( ed dd) dd ed ult) ze the totl te f() :! f '( ) ( dd ult) dd ed Ο ( ) ptu: ( ) te tep dt dtuted o ( ) poeo. f ( ) / Ο ( ) Cope f-: poeo log() te tep! 6
7 Futhe evel- BS Pole BS-Notto: defe SXPY: S --- gle peo (D fo doule C fo ople) --- α l X --- veto P --- plu opeto Y --- veto y α y Y α X Y Vetozto of SXPY y ppelg: α α y : α ultplto ddto α y y 7
8 8 SXPY Pllelzto y Pttog { } R I I I > < R R Y Y y y y y X X wth hot veto X d Y of legth /R. Eh poeo P get ptl veto X d Y d opute Y α X Y R. Reult: SXPY vey good vetozle d pllelzle X X X X X
9 Futhe evel-bs Route SCPY: y o y Cope SXPY No: T Cope DT-podut 9
10 evel- BS t-veto opeto wth ( ) opeto (equetlly) BS-Notto: S --- gle peo G E geel t V --- veto defe SGEV t-veto podut: y α y the evel- BS: Solvg tgul yte wth tgul t
11 evel-3 BS t-t opeto wth ( 3 ) opeto (equetlly) BS-Notto: S --- gle peo G E geel t --- t defe SGE t-t podut: C α B C PCK uoute fo olvg le equto let que pole QR-deopoto egevlue gul vlue ed o BS
12 . ly of t-veto- Podut ( ) IR IR IR Vetozto DT-podut of legth SXPY of legth (GXPY)
13 Peudoode: ()-fo: ; fo fo ed ed DT podut Dot podut of -th ow of wth veto 3
14 Peudoode: ()-fo: ; fo fo ed ed SXPY GXPY SXPY updtg veto wth -th olu of GXPY: Sequee of SXPY elted to the e veto dvtge: veto tht updted e ept ft eoy! No ddtol dt tfe. 4
15 .. Pllelzto y uldg lo Ide: Redue t-veto podut o lle t-veto podut. < > < > {... } I I I {... } J J J S R I dut: J I J fo fo Ue -deol y of poeo P. P get t lo :(I J ) :(J ) :(I ). S S ( ) I J I J 5
16 Peudoode fo R fo S () ; ed ed fo R ; fo S () ; ed ed Sll depedet t-veto podut. No outo eey! Blowe olleto d ddto of veto. Rowwe outo! F-. 6
17 7 Spel e: S P P.. No outo eey etwee poeo P P R The oputto of vetozle y GXPY. R : ( ) e depedet. The olleto of ptl eult fo poeo P P. F-. Fl u oe poeo: vetozle y GXPY.
18 Rule () Ie loop of pog hould e ple vetozle () ute loop of pog hould e uttl depedet pllelzle fo uttl d pllelzle fo. ed ple vetozle ed (3) Reue of dt (Che l dt tfe log) 8
19 9..3 fo Bded t Bdwdth (yet) dgol: dg. udg. upedg. : tdgol
20 N N N Stog ete dgolwe () t ted of. ow fo... d d ow dgol d d [ ] { } { } [ ] l [ ] { } { } [ ] l
21 Coputto of the t-veto podut ed o th toge hee o veto poeo: Fo : l l lgoth: Fo - : Fo {-} : {-} ed ed Geel tde No SXPY o Fo : Fo {--} : {-} ed ed Ptl Dot podut Spty le opeto ut lo lo of effey thee opeto
22 Pllel Pttog: R < > U fo ed I I dut l ; Poeo P get ow to de et I :[ ] ode to opute t pt of the fl veto. Wht pt of veto doe poeo P eed ode to opute t pt of?
23 Neey fo I : : l { } { } { } { } Poeo P wth de et I eed fo the de [ { } { } ] 3
24 4.3 ly of t-t Podut ( ) ( ) ( ) q q B C B ed ed q Fo Fo : :
25 ()-Fo: lgoth Fo : Fo : q Fo : ; ed ed ed Dot podut of legth B fo ll ll ete e fully oputed oe fte othe. e to d C owwe to B oluwe. 5
26 6 Dffeet vew o the t-t podut: ( ) ( ) ( ) T T e e e e t odeed oto of olu o ow ( ) T T e e e e B u of full te y oute podut of the -th olu of d the -th ow of B Full q - te
27 ()-Fo lgoth Fo : q Fo : Fo : ; ed ed ed SXPY Veto updte. GXPY Sequee of SXPY fo the e veto. C oputed oluwe; e to oluwe. e to B oluwe ut delyed. 7
28 ()-Fo lgoth 3 Fo : Fo : q Fo : ; ed ed ed SXPY No GXPY Sequee of SXPY fo dffeet veto. Veto updte. e to oluwe. e to B delyed. C oputed wth teedte vlue () whh e oputed oluwe. 8
29 vevew ove dffeet Fo e to y e to B y Coput -to of C ow ow olu olu olu ow ow olu ow ow ow olu olu olu Coput det delyed delyed det delyed delyed -to Veto opeto Veto leght DT GXPY SXPY DT GXPY SXPY q q Bette: GXPY loge veto legth. e to te odg to toge hee (owwe o oluwe) 9
30 t-t Pllel R < > U I < > U K < q > U J S T t t. Dtute the lo eltve to de et I K d J t y P t : to poeo K J t J t K. I () t I B t () Poeo P t opute ll t-t podut ll Poeo pllel. ( ) t B t () Copute u y f- : S ( t t ) 3
31 Spel Ce S J t J t. I t I B t I th e eh poeo P t opute t pt of t depedetly wthout y outo. Eh poeo eed full lo of ow of eltve to de et I d full lo of olu of B eltve to de et J t ode to opute t eltve to ow I d olu J t. Epelly wth q poeo eh poeo h to opute oe DT podut t t wth () pllel te tep. F- y q ddtol poeo fo ll thee Dot podut edue the ue of pllel te tep to (log()). 3
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