ast QR Iteration or Companion Matrices Da vid Bindel Ming Gu Da vid Gar mire ames Demmel Shivkumar Chandr asekar an ast QR Iter ation or Companion Matr ices PDF document - DocSlides

Page 1
ast QR Iteration or Companion Matrices Da vid Bindel Ming Gu Da vid Gar mire ames Demmel Shivkumar Chandr asekar an ast QR Iter ation or Companion Matr ices .1/18
Page 2
Moti ation: olynomial oot finding standard algor ithm: QR iter ation on companion matr ix Rob ust softw are xists It nor mwise bac kw ard stab le It used in Matlab But it tak es time and stor age Use str ucture in QR iter ates to get time space ast QR Iter ation or Companion Matr ices .2/18
Page 3
Companion matrices Giv en Define companion matr ix or matr ix polynomials replace and ast QR Iter ation or Companion Matr ices .3/18
Page 4
Companion matrix structur Wr ite as or thonor mal lo ank: This str ucture is preser ed under unitar similar ity ast QR Iter ation or Companion Matr ices .4/18
Page 5
Reminder: CS decomposition An loc 2-b y-2 unitar has CS decomposition: 11 12 21 22 where and are also unitar Theref ore ank 21 ank 12 ast QR Iter ation or Companion Matr ices .5/18
Page 6
Off-diagonal rank Compute Schur or 11 12 22 Then ank 12 Proof: Wr ite where := and := AQ ank 21 ank 21 ank 12 ank 12 12 ank 12 ank 12 ast QR Iter ation or Companion Matr ices .6/18
Page 7
Off-diagonal rank Similar argument sho ws: Hessenberg or ms similar to ha ank(off-diagonal) Real Schur or ms similar to ha ank(off-diagonal) Matr ix polynomials with coefficient siz ha ank 12 ast QR Iter ation or Companion Matr ices .7/18
Page 8
SSS structur Wr ite Hessenberg matr ices with lo off-diagonal ank as narro band matr ix SSS matr ix. Wr ite ij +1 or where and otal stor age cost: ast QR Iter ation or Companion Matr ices .8/18
Page 9
SSS structur transf ormed 11 21 22 32 33 43 44 ansf or second ro and column reflection. Only aff ects the band, and What about tr ansf or mations that cross loc edges? Merge loc ks so it ne er happens then split them. Result: ulge-chasing pass tak es time vs ast QR Iter ation or Companion Matr ices .9/18
Page 10
Splitting blocks +1 +2 +1 +1 ast QR Iter ation or Companion Matr ices .10/18
Page 11
Splitting blocks Do piv oted QR decomposition to split Choose with or thonor mal columns ast QR Iter ation or Companion Matr ices .11/18
Page 12
Balancing Cannot diagonally scale and maintain lo w-r ank str ucture But can use geometr ic scaling. If is companion matr ix, so is The erron ector of theref ore has the or c Let diag diag Then is optimally balanced in the -nor m. ast QR Iter ation or Companion Matr ices .12/18
Page 13
Implementation Modified DLAHQR (basic doub le-shift QR) to use the ne data str ucture Con er to ight proper or m. Choose shifts (logic from LAP CK). Bulge chase top to bottom con er to left proper or m. Chec or con ergence deflate (logic from LAP CK). ast QR Iter ation or Companion Matr ices .13/18
Page 14
erf ormance Compared against LAP CK DLAHQR (not DHSEQR). Compiled using g77 without optimizations LAP CK with standard optimizations TLAS BLAS entium laptop at 700 MHz. olynomials of deg ree 50-2000. or deg ree 10000, current code tak es 41 min utes ast QR Iter ation or Companion Matr ices .14/18
Page 15
erf ormance 200 400 600 800 1000 1200 1400 1600 1800 2000 100 200 300 400 500 600 700 800 900 1000 ast QR Iter ation or Companion Matr ices .15/18
Page 16
erf ormance 200 400 600 800 1000 1200 1400 1600 1800 2000 10 −2 10 −1 10 10 10 10 ast QR Iter ation or Companion Matr ices .16/18
Page 17
Generalizations and Futur ork Profiling and basic tuning. Modified ulti-shift DHSEQR code Balancing. olynomial eigen alue prob lems Sequences of symmetr ic lo ank prob lems ast QR Iter ation or Companion Matr ices .17/18
Page 18
Conclusions Schur({ symmetr ic sk unitar lo ank) has lo off-diagonal ank. Can use lo w-r ank str ucture or ast ulge-chasing. Only or thonor mal tr ansf or mations so bac kw ard stab le Can re-use most xisting QR lore and code Can still do -nor balancing or companion matr ices ast QR Iter ation or Companion Matr ices .18/18