T¬m tr»‹ ring b±ng pp qr

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  • 1. TM TR RING CA MA TRN BNG PHNG PHP QR Thc hin: Nhm 4

2. PHN TCH QR Dng pp Householder bin i ma trn cho v ma trn 3 ng cho, i xng. Dng ma trn quay P phn tch QR 3. 4. 5. 6. 7. 2 1 2 2 2 2 2 2 2 1 2 1 sin ; cos b a b a b a (1) (1) 2 2 A P A (1) 2 1 1 2 2 2 1 2 2 21 2 2 2 2 2 1 2 1 ( sin ) (cos ) 0 b a a b A a b b a b a 8. (1) 2 2 2 1 2 2 2 2 2 3 3 3 (1) 2 cos sin 0 0 sin cos 0 0 0 1 0 * * * 0 * * 0 * * P A a b b a b b a A 9. Tng qut, ta c: 1 1 1 1 1 1 ( ) 1 1 2 0 0 0 0 0 0 0 0 0 0 0 k k k i k k k k k k n n n z q r z q r A x y b a b b b a 10. Sau khi nhn vi ma trn quay: 1 1 1 ( ) ( ) 1 1 1 1 2 2 3 0 0 0 0 0 0 0 0 0 0 0 k k k i i k k k k k k k k n n n z q r z q r A P A x y b a b b b a 11. 1 1 1 1 1 1 1 1 1 1 2 2 2 2 1 1 ; k k k k k k n k k k k k k k k k I O O c s O OP s c O O I b x s c b x b x 12. 1 1 1 ( ) ( ) 2 1 0 0 0 0 0 0 i i n n n n n z q r R A r z q x 13. 14. TM TR RING BNG PP QR 15. QR tnh tin (shifted QR algorithm mthode QR avec translations) 16. ( ) ( ) 1 ( ) ( ) i i n n i i n n a b b a 17. 18. CC BC TM TR RING BNG PP QR 19. Xt ma trn vung 3x3 V D: S DNG PHNG PHP QR TM TR RING CA MA TRN 20. CC CNG THC CN LU ( ) ( ) ( ) ( 1) ( ) ( ) (1) (1) 2 2 (1) (1) 3 3 2 ( 2 ) (1) (1) (1) 3 2 2 3 i i i i i i t t A Q R A R Q A P A A P A A R Q P P A P P 21. 1 1 1 1 1 1 ( ) 1 1 2 0 0 0 0 0 0 0 0 0 0 0 k k k i k k k k k k n n n z q r z q r A x y b a b b b a Hng th k Ct th k 22. 1 1 1 1 1 1 1 1 1 1 2 2 2 2 1 1 ; k k k k k k n k k k k k k k k k I O O c s O OP s c O O I b x s c b x b x Ma trn quay 23. V D Cho ma trn l ma trn 3 ng cho, i xng 3 1 0 1 3 1 0 1 3 A 24. Ta tnh s1 bng cch tm tr ring ma trn vung 2x2 to bi dng th 2, 3 v ct th 2, 3 Ma trn trn c 2 tr ring l 1=2 v 2=4. Ta phi chn s1 l 1 trong 2 tr ring gn vi gi tr a3=A33=3 ( y chn 2 hay 4 u c). Chn s1=1=2. 3 1 1 3 25. (1) 1 1 3 1 0 1 0 0 1 1 0 1 3 1 2 0 1 0 1 1 1 0 1 3 0 0 1 0 1 1 A A s I 1 1 1 1 2 2 3 1 2 3 3 0 1; 1 0 x y A b a b x b b a 26. T cng thc sk v ck ta c Dng ca P2 1 1 1 2 2 2 2 1 1 ;k k k k k k k k b x s c b x b x 2 2 2 2 s c 2 2 2 2 2 2 2 0 2 20 2 20 0 2 2 0 0 1 0 0 1 c s P s c 27. Tm c: (1) (1) 2 2 1 1 1 2 2 3 3 2 3 2 2 0 2 2 1 1 0 2 2 0 1 1 1 2 2 0 1 10 0 1 22 2 2 0 0 2 0 0 1 1 0 0; 1 A P A z q r x y b a x b 28. Tnh c: Dng ca P Ta c th tnh nhng khng cn thit 3 3 3 3 3 1 0 0 1 0 0 0 0 0 1 0 0 1 0 P c s s c 3 3 0; 1s c (1) (1) 3 3 2 A P A 29. ( 2 ) (1) (1) (1) 3 2 1 2 3 22 0 2 2 21 2 2 20 0 2 t t A R Q P P A P P 30. Tip tc lp li cc bc nh trn u tin, ta tnh s2 bng cch tm tr ring ma trn vung 2x2 to bi dng th 2, 3 v ct th 2, 3 ca ma trn A(2) va thu c Tr ring ca ma trn trn l Vy nn ta chn gn vi a3=0 21 2 2 0 2 1 1 3 2 2 2 1 1 3 2 2 s 31. (3) 2, 6720277 0, 37597448 0 0, 37597448 1, 4736080 0, 030396964 0 0, 030396964 0, 047559530 A (3) 3 3 1 2 1, 5864151a s s 32. 2, 6720277 0, 37597448 0, 37597448 1, 4736080 1 1 1 2 4, 4141886s s 2 2 1 2 2, 9993964s s