Example: n n n R b R A , Introduction to Krylov Subspace Methods DEF: b Ax , , , 2 b A Ab b Krylov sequence Krylov sequence 10 -1 2 0 -1 11 -1 3 2 -1 10 -1 0 3 -1 8 A 1 11 118 1239 12717 1 12 141 1651 19446 1 10 100 989 9546 1 10 106 1171 13332 Ab b A 2 b A 3 b A 4 b
Example:
Jan 04, 2016
Introduction to Krylov Subspace Methods. DEF:. Krylov sequence. Example:. Krylov sequence. 1 11 118 1239 12717 1 12 141 1651 19446 1 10 100 989 9546 1 10 106 1171 13332. 10 -1 2 0 -1 11 -1 3 2 -1 10 -1 0 3 -1 8. - PowerPoint PPT Presentation
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Example:
nnn RbRA ,
Introduction to Krylov Subspace Methods
DEF:
bAx
,,, 2bAAbb Krylov sequence
Krylov sequence
10 -1 2 0 -1 11 -1 3 2 -1 10 -1 0 3 -1 8
A1 11 118 1239 127171 12 141 1651 194461 10 100 989 95461 10 106 1171 13332
Ab bA2 bA3 bA4b
Example:
Example:
Introduction to Krylov Subspace Methods
DEF:
} ,,, {),( 1bAAbbspanbA mm
Krylov subspace
10 -1 2 0 -1 11 -1 3 2 -1 10 -1 0 3 -1 8
A
Krylov subspace
),(
106
100
141
118
,
10
10
12
11
,
1
1
1
1
3 spanbA
DEF:
bAAbbbAK mm
1,,,),(
Krylov matrix
106
100
141
118
10
10
12
11
1
1
1
1
),(3 bA
Example:
Introduction to Krylov Subspace Methods
Remark: ))((3 AbAAbA
DEF:
bAAbbbAK mm
1,,,),(
Krylov matrix
106
100
141
118
10
10
12
11
1
1
1
1
),(3 bA
WHY: Krylov Subspace Methods
DEF:
bAAbbbAK mm
1,,,),(
Krylov matrix Example: Solve:
15
11
25
6
4
3
2
1
x
x
x
x 10 -1 2 0 -1 11 -1 3 2 -1 10 -1 0 3 -1 8
>> A=[10 -1 2 0; -1 11 -1 3;2 -1 10 -1; 0 3 -1 8]
>> poly(A)
1 -39 552 -3357 7395
7395335755239)( 234 xxxxxp
Characteristic poly
the Cayley-Hamilton theorem a matrix satisfies its characteristic polynomial, p(A) = 0. That is,
07395335755239 234 IAAAA
Multiplying with inv(A) & rearrange
IAAAA7395
3357
7395
55227395
3937395
11
Hence,
bAbbAbAbA7395
3357
7395
55227395
3937395
11
The key observation here is that the solution x to Ax = b is a linear combination of the vectors b and Ab,.. which make up the Krylov subspace
the solution to Ax = b has a natural representation as a member of a Krylov space,
Krylov subspace Methods
Krylov subspace
GMRES
Conjugate Gradient others
MINRESdefinite
symm
A
.
.symm
Ageneral
A
MATLAB commands
Conjugate Gradient definitesymmA & .
x = pcg(A, b, tol, maxit)
x = gmres(A,b,[],tol,maxit)
GMRESeneral gA
MINRES
x = minres(A, b, tol, maxit)
. symmA
Conjugate Gradient Method
nnn RbRA ,
Conjugate Gradient Method
We want to solve the following linear system
bAx
definite) positive (symmetric SPD A
0
0
x
AxxT
end
210for
111
111
1
1
00
00
kkkk
kT
kkT
kk
kkkk
kkkk
kTkk
Tkk
pβrp
r/rrrβ
Apαr r
pαxx
Ap/prrα
,..,, k
rp
Axbr
0 K=1 K=2 K=3 K=4x1
x2
x3
X4
)(kr
1
1
2
1
*x
Conjugate Gradient Method
Example: Solve:
15
11
25
6
4
3
2
1
x
x
x
x 10 -1 2 0 -1 11 -1 3 2 -1 10 -1 0 3 -1 8
Conjugate Gradient Method
end
210for
111
111
1
1
00
00
kkkk
kT
kkT
kk
kkkk
kkkk
kTkk
Tkk
pβrp
r/rrrβ
Apαr r
pαxx
Ap/prrα
,..,, k
rp
Axbr
0 0.4716 0.9964 1.0015 1.00000 1.9651 1.9766 1.9833 2.00000 -0.8646 -0.9098 -1.0099 -1.00000 1.1791 1.0976 1.0197 1.0000
31.7 5.1503 1.0433 0.1929 0.0000
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