11. (10/50)
•
A Rm m b Rm Rm
Kn(A; b) A b n
•
x(0) = 0 x(n) Kn(A; b) n 1
{x(n)}
# 0
K1(A; b) K2(A; b) Kn(A; b)
Kn(A; b) x(n)
Kn(A; b) = span{b, Ab, A2
b, …, An–1
b}
12. (11/50)
•
Kn(A; b) Kn+1(A; b)
Kn(A; b) = Kn+1(A; b)
•
A Kn(A; b) = Kn+1(A; b)
n Ax = b x* Kn(A; b)
•
Anb = c0b + c1Ab + + cn–1An–1b c1, c2, …, cn–1
c0 = 0 A n
c0 0 c0
A((1/c0)An–1b – (cn–1/c0)An–2b – – (c1/c0)b) = b x* Kn(A; b)
13. (12/50)
•
Kn(A; b) x(n) = d0b + d1Ab + + dn–1An–1b
r(n) = Ax(n) – b
Kn(A; b) x(n) 1 1
r(n)
= – b + d0Ab + d1A2
b + + dn–1An
b = n(A)b
n(z) = – 1 + d0z + d1z2
+ + dn–1zn
–1 n
22. (21/50)
II Ritz-Galerkin
•
r(n) = Ax(n) – b Kn(A; b) x(n)
Kn(A; b) 0
•
K1(A; b) K2(A; b) Kn(A; b)
r(0) = b, r(1), r(2), …, r(n–1) Kn(A; b)
# r(n–1) Kn(A; b) n
Ax(n)
– b Kn(A; b)
23. (22/50)
II Ritz-Galerkin
• A
(x) = (1/2) xTAx – xTb Ax = b (x)
x(n) Kn(A; b) x(n) = Qny(n) y(n) Rn
(x(n)) = (1/2) (Qny(n))T AQny(n) – (Qny(n))Tb
y(n) Qn
T(AQny(n) – b) = 0
Ax(n) – b Kn(A; b)
Ritz-Galerkin (x(n)) x(n)
25. (24/50)
III Petrov-Galerkin
•
Rm L1 L2 L3 Ln n
r(n) = Ax(n) – b Ln x(n)
Ln 0
Ln b* AT b*
Kn(AT; b*)
• Petrov-Galerkin
Bi-CG QMR
# Petrov-Galerkin
Petrov-Galerkin CGS Bi-CGSTAB
GPBi-CG
Ax(n)
– b Ln
26. (25/50)
•
–1 n Pn
n n(z)
• Ritz-Galerkin A
x(n) e(n) = x(n) – x* Ae(n) = Ax(n) – b = r(n)
Ritz-Galerkin
min P n(A)b 2n n
e(n)TAe(n) = r(n)TA–1r(n) = bT
n(A)A–1
n(A)b
= bTA–1
n(A)A n(A)A–1b
= e(0)T
n(A)A n(A)e(0) = n(A)e(0)
A
min P n(A)e(0)
An n
32. (31/50)
GMRES
• GMRES
n
#
1 n
# qn q1, q1, …, qn–1
#
• MINRES
Hn 3 1
n
x(n) ( (A))2 *
* H. A. van der Vorst “Iterative Krylov Methods for Large Linear Systems”, Cambridge Univ. Press, 2003.
33. (32/50)
CG
•
A
r(n) = Ax(n) – b Kn(A; b) x(n)
x(n) Kn(A; b) x(n) = Qny(n) y(n) Rn
Lanczos 3 Tn Qn = [q1| q2| |qn]
Tn = LnUn LU x(n)
•
q1, q2, …, qn
# n
Qn
T
(Ax(n)
– b) = Qn
T
(AQny(n)
– b) = Tn y(n)
– Qn
T
b = Tn y(n)
– e1 = 0.
y(n)
= Un
–1
Ln
–1
e1, x(n)
= Qny(n)
51. (50/50)
• L. N. Trefethen and D. Bau III: “Numerical Linear Algebra”, SIAM,
Philadelphia, 1997.
• H. A. van der Vorst: “Iterative Krylov Methods for Large Linear
Systems”, Cambridge University Press, Cambridge, 2003.
• Y. Saad: “Iterative Methods for Sparse Linear Systems”, PWS
Publishing Company, Boston, 1996.
• R. Barrett et al.: “Templates for the Solution of Linear Systems:
Building Blocks for Iterative Methods”, SIAM, Philadelphia, 1994.
• , : “ ”, , 1996.
• , : “ ”, , 2009.