2. Arnoldi Implicitly Restarted Arnoldi 3. Lanczos Thick Restart
Lanczos 4. A = AT A = AT AT A* AH Ax = x Ax = Bx 2 Ax + Bx + Cx = 0
A( )x = 0 5. 3 3 6. A = AT n A = A* 7. A n 1A n 1 {xi} 1 8. AT A AT
i A yi A i Schur A U T A Schur T A AT yi = i yi yi TA = i yi T UT
AU = T 9. Z C = Z1AZ C A i A yi C wi A 3 xi = Zwi 10. Ax = Bx A B B
= LLT L1ALTy = y x = LTy B B1Ax = x B B1A QZ 11. A 1 2 n 1 A x x =
i=1 ncivi i=1 n ci 2 = 1 xTAx = ( i=1 ncivi)T( i=1 nci ivi ) = i=1
nci 2 i c1 = 1, c2 = = cn = 0 1 n 12. Courant-Fischer A r r S Rn
xTAx / xTx x A Rayleigh 13. A A+ A x, y x y 14. A Jacobi 3 T 3
Householder 3 Lanczos 3 2QR T i ui A i vi A A 3 T = QTAQ Jacobi
Lanczos Householder 2 QR 15. A Hessenberg H Hessenberg Householder
Hessenberg Arnoldi Hessenberg QR U i ui A i vi A A Hessenberg H =
QTAQ Arnoldi Householder QR U = PTHP 16. x(0) A x(k) x(k) 17. A |
1| > | i| i 2 x(0) A x(0) = i=1 ncivi c1 0 x(0) v1 18. A | 1|/|
2| 2 2 1 19. A (A sI)1 s s Lanczos/Arnoldi x(0), x(1), , x(k) x(k)
QR n 20. Anoldi x(0), x(1), , x(k) x(0), x(1), , x(k) q(1), q(2), ,
q(k) q(1), q(2), , q(k) Kk(A; x(0)) Arnoldi Kk(A;
x(0))Ritz-Galerkin x r = Ax x Kk(A; x(0)) 21. Arnoldi Arnoldi 22.
Arnoldi Arnoldi Arnoldi Aqi q1, q2, , qi+1 A k Arnoldi Aqi = h1i q1
+ h2i q2 + + hi+1,i qi+1 AQk = Qk+1Hk nk (k+1)k , 23. Arnoldi
Arnoldi AQk = Qk+1Hk Qn T . Kk(A; x(0)) Qk Rk nHk Rk k 24. Arnoldi
x Kk(A; x(0)) x = Qy y Rk Ritz-Galerkin k k x = Qky Qk T(Ax x) = Qk
TAQky Qk TQky = Hky y = 0. Hky = y 25. Arnoldi Arnoldi Qk, Hk k k
Hky = y LAPACKQR x = Qky Arnoldi O(k2n) k k O(k3) O(kn) Arnoldi k n
Arnoldi 26. Arnoldi O(k2n) O(k2n) k qk q1 Arnoldi q1, q2, , qk1 27.
Implicitly Restarted Arnoldi (1) Kk(A; x(0)) l l < k l q1 +, q2
+, , ql + q1, q2, , qk l k Arnoldi AQl + = Ql+1 +Hl + [q1 +, q2 +,
, ql +] = [q1, q2, , qk]U U Rk l 28. Implicitly Restarted Arnoldi
(2) q1 +, q2 +, , ql + k-step Arnoldi AQk = QkHk + hk+1,k qk+1ek T
1, 2, , kl Hk kl QR Qk + = QkUHk + = UT HkU U = U1U2 UklQR l l-step
Arnoldi Arnoldi l AQk + = Qk + Hk + + hk+1, k qk+1ek TU AQl + = Ql
+ Hl + + hl+1, l ql+1 +el T 29. Implicitly Restarted Arnoldi (3) =
= = + + + l kl kl+1 A A A Qk + Qk + Ql + Qk + Qk + Ql + Hk + Hk +
Hl + ek T hk+1,k qk+1 U hk+1,k qk+1ek T U hl+1,l +ql+1 +el T QR
l-step Arnoldi k l 30. Implicitly Restarted Arnoldi (4) QR 1, 2, ,
kl R = Rkl R2 R1 Ri QR Qk + 1 q1 + implicit restart UR = (Hk + 1I)
(Hk + 2I) (Hk + kl I) q1 + = QkUe1 = Qk (Hk + 1I) (Hk + 2I) (Hk +
kl I) R1e1 = r11 1 (A 1I) (A 2I) (A kl I) q1 q1(A 1I) (A 2I) (A kl
I) q1 31. Implicitly Restarted Arnoldi (5) q1 32. Arnoldi AQk =
QkHk + hk+1,k qk+1ek T DO l = 1, 2, 3, 1, 2, , kl Hk k l QR U =
U1U2 Ukl Qk + = QkUHk + = UT HkU l = (Hk + )l+1,l l = Ukl ql+1 + =
ql+1 l + hk+1,k qk+1 l hl+1,l = ql+1 + 2 ql+1 + := ql+1 + / hl+1,l
Qk + l Ql + Hk + l l Hl + Arnoldi AQl + = Ql + Hl + + hl+1,l ql+1 +
el TArnoldi k l END DO 33. (1) SR8000 (F1) Arnoldi IRAM (ARPACK)
34. (2) Ax x / < 10-14 N = 13112, 51482 NE = 30, 100 IRAM
N=13112 N=51482 NE=30 930 50 1600 60 26.1G 7.69G Arnoldi IRAM 339G
40.6G Krylov N=13112 N=51482 NE=100 2000 140 160 242G 77.0G Arnoldi
IRAM 382G 35. (3) N=51482 1 , 30 IRAM 1.0E-14 1.0E-12 1.0E-10
1.0E-08 1.0E-06 1.0E-04 1.0E-02 1.0E+00 0.0 10.0 20.0 30.0 40.0
50.0 1 30 Arnoldi Step=100 IRAM Step=60 (Gflop) 36. (4) N=51482 100
IRAMArnoldi 1.0E-14 1.0E-12 1.0E-10 1.0E-08 1.0E-06 1.0E-04 1.0E-02
1.0E+00 0.0 100.0 200.0 300.0 400.0 500.0 100 Arnoldi Step=500 IRAM
Step=160 (Gflop) 37. Lanczos Arnoldi Kk(A; x(0)) Ritz-Galerkin A
Lanczos Lanczos 38. Lanczos Lanczos Qk Rk nTk Rk k . 3 Kk(A; x(0))
39. Lanczos Lanczos Qk, Tk k k Tky = y LAPACKQR x = Qky Ritzx Ritz
Lanczos O(kn) k k O(k3) O(kn) Lanczos k n Lanczos 40. Lanczos n =
20 1 1.2 1.4 1.6 1.8 2 2.2 2.4 0 2 4 6 8 10 12 14 Power method
Lanczos method 41. Lanczos Rayleigh xTAx / xTx Kk(A; x(0)) =
span{x(0), x(1), , x(k1)} Qk = [q1, q1, , qk] Tk y Tk A x = Qky
Lanczos Lanczos Rayleigh span{x(0), x(1), , x(k1)} v1 42. Lanczos r
r Courant-Fischer S Kk(A; x(0)) Tk r Lanczos Lanczos Kk(A; x(0))
Courant-Fischer 43. Lanczos q1, q2, , qk q1, q2, , qk O(k2n)
Lanczos 44. Lanczos k O(k2n) qk x(0) qk x(0) q1, q2, , qk Thick
Restart Lanczos TR-Lanczos* *
http://crd-legacy.lbl.gov/~kewu/ps/trlan.html 45. Thick Restart
Lanczos Lanczos k k k Ritz l l < k Lanczos l k Lanczos AVk =
VkTk + k vk+1 ek T Lanczos 46. (1) EP8000 /690Turbo Matrix Market
Lanczos TR-Lanczos Matrix n nz bcsstk16 4884 147631 bcsstk17 10974
219812 bcsstk18 11984 80519 bcsstk21 3600 15100 bcsstk23 3134 24156
bcsstk24 3562 81736 bcsstk25 15439 133840 bcsstk35 30237 740200
bcsstk36 23052 583096 bcsstk37 25503 583240 bcsstk39 46772 1068034
crystk02 13965 491274 Gap 0.58 0.88 0.73 0.13 0.98 1.00 1.00 0.99
0.82 0.60 0.34 0.99 Gap=( 1 100)/ 1 47. (2) Ax x / < 10-10 NE =
30, 100 TR NE=30 NE =100 0 50 100 150 200 250 300 350 400 bcsstk16
bcsstk17 bcsstk18 bcsstk21 bcsstk23 bcsstk24 bcsstk25 bcsstk35
bcsstk36 bcsstk37 bcsstk39 crystk02 0 100 200 300 400 500 600 700
bcsstk16 bcsstk17 bcsstk18 bcsstk21 bcsstk23 bcsstk24 bcsstk25
bcsstk35 bcsstk36 bcsstk37 bcsstk39 crystk02 simple TR 48. (3)
bcsstk39 28, 29, 30 1.0E-17 1.0E-15 1.0E-13 1.0E-11 1.0E-09 1.0E-07
1.0E-05 1.0E-03 1.0E-01 0.0 1.0 2.0 3.0 4.0 5.0 [Gflop] 28 30 29 TR
simple Conv. check Simple: 100 TR : 90 49. (4) bcsstk21 TR-Lanczos
1.0E-16 1.0E-14 1.0E-12 1.0E-10 1.0E-08 1.0E-06 1.0E-04 1.0E-02
1.0E+00 0.0 0.2 0.4 0.6 0.8 1.0 1.2 [Gflop] 28 30 29 TR simple
Conv. check Simple: 20 TR : 70 50. Arnoldi Lanczos Implicitly
Restarted Arnoldi Thick Restart Lanczos Jacobi-Davidson IRAMTRLANJD
51. L. N. Trefethen and D. Bau III: Numerical Linear Algebra, SIAM,
Philadelphia, 1997. Z. Bai, J. Demmel, J. Dongarra, A.Ruhe and H.
Van der Vorst (eds.): Templates for the Solution of Algebraic
Eigenvalue Problems: A Practical Guide, SIAM, Philadelphia, 2000.
G. W. Stewart: Matrix Algorithms, Vol. II: Eigensystems, SIAM,
Philadelphia, 2001. Y. Saad: Iterative Methods for Sparse Linear
Systems, SIAM, Philadelphia, 2011. , : , , 2009.
