QR????????

1
QR????????
2008? 7?10? ???????? ????
?? ?
2
QR?????
?QR??
????

• ??????
• ??????????
• ?????(?????)???????????

???????

• ????????????
• ?????????

3
????????????QR??
??????????
?QR??????
4
???????QR??????
????Level-2 BLAS
?????????????,??????????
Level-3 BLAS??????????????
5
???????????????
?compact-WY representation
????? O(ml 2)
?Level-3 BLAS???????????????
2l ??Level-2 BLAS
3??Level-3 BLAS
6
???????QR????????
?Fixed-size Blocking (LAPACK??????????)
?????? l ????????
?
(?)??????QR??
??(??????)???????
Level-2 BLAS
(?)compact-WY representation???
Level-2 BLAS
(?)compact-WY representation???
Level-3 BLAS
7
QR????????????????
???????????????
Lv.2
Level-3 BLAS????????
????????????
???????????

• Variable-size Blocking
• Recursive Blocking
• ??????????????(???????)

????????????????????????
8
????????
?????

• ?????????(????????????)
• LAPACK?DGEQR2(??????)?DGEQRF(?????)

?????????????,QR?????????? (??????????????????????
??????)
???? ???????????? / ???????????
?????

• CPU Opteron(2.0GHz)
• BLAS AMD ACML

9
Level-2 BLAS?Level-3 BLAS?????

• DGEMV (Level-2 BLAS) n n ??? n ????????
• DGEMM (Level-3 BLAS) n n ??? n n ????

MFLOPS
67???????
n
10
???????????????
????????
????? ????(?) ????(?) ????(?) ????
????? ?????? ????? ????? ????
10001000 3.84 0.54 30 7.04
30001000 22.7 2.43 40 9.33
30003000 174 13.4 60 12.96
?LAPACK?????
????? ????(?) ????(?) ????(?) ????
????? ?????? (DGEQR2) ????? (DGEQRF) ????? ????
10001000 3.13 0.51 ? 6.09
30001000 13.3 1.99 ? 6.69
30003000 89.8 11.7 ? 7.67
11
???????????????
????(?)
(??????????)
(??? 30001000)
?????????,Level-3 BLAS???????
best
Level-3 BLAS????????? (?????QR?? WY
representation??)
?????
12
???????????????
????
BLAS????????
??????(????)
13
???

• ????????????QR????????????.
• compact-WY representation???????????????,Level-3
  BLAS????????????.
• ??????????????????,???13?????????????.
• LAPACK????????????????,BLAS???????????????????.

14
(??)QR?????????
?non-blocking
15
(??)QR?????????
?fixed-size blocking