8. Computing Eigenvalues in Parallel

Transcription

1 8. Coptng Egenles n Prllel x egenector wth egenle λ, ff: x λx, x spd, then there exsts n orthogonl ss of egenectors λ,,,,n Λ or Λ,..., n ), Λ dg λ,..., λ n ) n generl y e coplex ntry nd Λ n pper trnglr coplex trx Schr decoposton) llowed opertons tht do not chnge the egenpr: QQ H wth ntry Q

2 rdgonlston Stndrd lgorth for coptng egenprs: QR-lgorth Prestep: rnsfor y Gens or Hoseholder trces to trdgonl for. 3 H G, G, 3 3 to elnte 3 nd 3 For nonsyetrc : rnsforton to pper Hessenerg for. For etter prllels se lock Hoseholder lke n the QR-decoposton.

3 QR-lgorth Frst step: By Hoseholder trces trnsfor y eqlence trnsfortons on trdgonl pper Hessenerg) for. HH Second step: Copte QR-decoposton of, QR nd replce old y new RQ new RQ Q ) Q Q Q herefore nd new he the se egenles Repet these QR-steps ntl conergence gnst dgonl pper trnglr) trx 3

4 4 Prllel Methods: Jco Method, ) : j n n j j n F off Descres the gntde of the nondgonl prt of shold ) We se Gens rottons to elnte pq for n ndex pr p,q O O O ) :,, c s s c q p J θ p q p q c cosθ), s snθ)

5 5 Jco Method c s s c c s s c qq qp pq pp qq qp pq pp qq pp!,θ, ) ) s c cs s c qq pp pq Effect of pplcton of J on on the nondgonl entres off). Consder p,q prt of J J B: qq pp pq qq pp J orthogonl Froensnor of nd B re the se: ) ) ) ) off off B B off pq pq qq pp q p F F <

6 Elnton Seqence Choose p nd q sch tht pq s ery lrge x). hen y J J the sze of the off-dgonl entres s redced y pq. Repet ths trnsforton for next choce of p nd q: dgonl. Dfferent strteges for choosng seqence of p,q: Mx pq optl, t seqentl nd expense! Cyclc y row: Frst se to elente frst row: p,q),),,3),,,n) hen for second row: p,q),3),.,,n) 33,, n-,n- Repet gn seqentl! 6

7 Jco Method n Prllel Choose seqence p,q) sch tht t llows strong prllels: Frst sweep: p,q),), 3,4), 5,6), 7,8) n prllel) Second sweep: p,q),4),,6), 3,8), 5,7),6), 4,8),,7), 3,5),8), 6,7), 4,5),,3) Fnd seqence of prttonngs of, n) n prs, sch tht ll ndces pper wth the se freqency. 7

8 8 Prllel rnsforton J : J ) J: Mltplctons wth J, resp. J cn e done n prllel.

9 9 Dde & Conqer pproch dde nd conqer pproch for coptng egenles of syetrc trdgonl trx. de: Splt n two trdgonl trces nd. Copte egenles of nd. Recoer the orgnl egenles of s pertrtons. Repet recrsely. n n n O

10 Splttng of ) : L L θ : Set : Generte zeros t the s/sperdgonl entres n the ddle of ) :, :!, θ θ θ θ θ θ

11 Relton etween nd, Λ Λ Λ?,, sse, tht we know the egenles nd egenectors of nd. How cn we get the egenprs of? Note, tht s rnk- pertrton of dg, ). Recoer the orgnl egenles s pertrtons y Newton s ethod. Λ Λ Λ Λ

12 Coptng the egenector Hence, we need to copte the egenles of trx of the for dgonl rnk- : D Let λ nd e n egenpr of D. hen t holds ) ) D λ D λ ) const D λ ) Hence, f we know λ, then we drectly get the egenector.

13 3 Egenles s Zeroes ) ) ) [ ] ) ) ) λ λ λ λ λ λ λ n n d d D f D D D L Frtherore, we get the eqton se Newton s ethod, to deterne the zeroes of fncton fλ) hese zeroes re the egenles of nd therefore lso of. Repet recrsely for nd. D

14 lgorths for coptng few egenprs: Vector terton: x k ) x k ) ) k ) ) x egenector to egenle wth x solte le Esy to prllelze only x), t slow conergence! Only λ x! Sspce terton: pply the se de to set of ectors ) x ),,x k) ) Consder egenles of )H ) nd then replce ) y ) nerse terton: pply ector terton on shfted prole σ ) - for coptng the egenector nerest to σ. Expense! ll-condtoned lner syste! 4

15 5 Rylegh Qotent terton ; ; ) ; y y y y y Rylegh Qotent terton: Strt wth ector y nd rel nd repet: ) y y y y y new Becse of: Fst conergence, t ncertn to whch egenle we wll conerge. Expense! nerse terton wth replcng the shft σ y the newest egenle estte.

16 rnold Lnczos): se the trnsforton on Hessenerg trdgonl) for descred for GMRES. Copte the egenles of the sll Hessenerg trx nd se the s pproxtons for the egenles of the orgnl trx. By rnold Orthogonlzton of the Krylo sspce,,, ) we get the relton j j j hk, j k j k k h k, j k... )... ) H, H, h, Egenles of H, s pproxtons for. Sll h, good pproxton). Good pproxton for extree egenles 6

17 Jco-Ddson de: - No Krylo sspce - choose sspce relte to egenle we re lookng for - nclde precondtonng; Strtng pont: Consder egenle pproxtons dered y V H V for sspce relte to V. he egenprs of V H V re sed s pproxtons to soe egenles of How to choose new sspce V wth ddtonl ector sch tht the new pproxton for specl egenles s strongly proed? For frst egenpr pproxton nd t H )/ H ), we try to proe these pproxtons y sll correctons nd t to get etter esttes nd t t! ) t t) ), 7

18 8 Jco-Ddson t t ), ) ) ) ) t t t t gnore correcton t of second order se orthogonl projecton wth fro the left. hs leds to H ) ) ) ) ) ) ) ) ) ) ) H H t t t t t H H H H H

19 Jco-Ddson For new pproxton we he to sole ) ) ) t t ) P H t ) P r or r H t gets ll-condtoned for t ner egenle, t P s projecton orthogonl to the ner snglr ector! s snglr, t lner syste s stll solle. Replce ll-condtoned y snglr syste. 9

20 Jco-Ddson V New egenector estte lso leds to new egenle estte t H )/ H ). Choose the new estte to enlrge the sspce V y the new ector to V. Copte egenprs of V H V. Repet ths step few tes. Restrt the whole process wth lst est pproxton s strtng ector, resp. -d sspce V. dntges: llows to copte lso nner egenles wthot solng ore nd ore ll-condtoned proles lke Rylegh Q.

21 Jco-Ddson V Mn step: Sole lner syste P pproxtely. t ) P r or r herefore, we se few steps of precondtoned cg or GMRES. Precondtoner: M - precondtoner for PM - P precondtoner for PP n ech terton step we he to ltply wth, wth P, nd sole n M. Sple precondtoner: M dg) Better precondtoner: SP or MSP

22 MRRR for trdgonl trces de: se nerse terton for coptng ll the egenles/ectors of trdgonl trx. Osertons: nerse terton chep, ecse of trdgonl for Prllel nd ndependent nerse terton for dfferent egenles Prole: Good strtng ector sch tht we need only one terton Need hgh ccrcy lso for sll or close together egenles

23 MRRR for egenectors Otlne of the lgorth: Copte egenle pproxton wth hgh relte ccrcy e.g. QR,..) Fnd the coln r of λ) - wth lrgest nor Perfor one step of nerse terton λ) z e r Bsc step: se fctorztons σ LDL for dfferent shfts llows the coptton of egenectors wth hgh ccrcy 3

24 wostep rdgonlzton n frst step redce trx to lock-nded for, nd n second step to trdgonl for. dntge: Frst step llows lock/bls3 opertons nd s good n prllel second step s chep. 4

25 Bothsded Hoseholder Copte Hoseholder ector n order to elnte strdgonl entres n the frst coln/row. pply - H )- H ) H ) ) H 4 H H ) H r H ) r) H y H -y H o redce BLS opertons work lockwse, Y H -Y H BLS3) t stll needed BLS). 5

26 Block-Bnd redcton n the frst step fnd y QR decoposton of : n, : n ) Where s the ndwdth nd n s lock sze. Copte QR decoposton of lck prt : pply, Q H ) fro the left leds to trnglr for of lck prt. Fro oth sdes: Bnd strctre. 6