Numerical Methods Contents - SAM

eplaced with κ(A) !
4.4.2 Iterations with short recursions
Iterative solver for Ax = b with symmetric system matrix A:
Iterative methods for general regular system matrix A:
MATLAB-functions:
• [x,flg,res,it,resv] = minres(A,b,tol,maxit,B,[],x0);
• [. ..] = minres(Afun,b,tol,maxit,Binvfun,[],x0);
Idea: Given x (0) ∈ R n determine (better) approximation x (l) through Petrov-
Galerkin condition
Computational costs : 1 A×vector, 1 B −1 ×vector per step, a few dot products & SAXPYs
Memory requirement: a few vectors ∈ R n
Extension to general regular A ∈ R n,n :
x (l) ∈ x (0) + K l (A,r 0 ): p H (b − Ax (l) ) = 0 ∀p ∈ W l ,
with suitable test space W l , dim W l = l, e.g. W l := K l (A H ,r 0 ) (→ biconjugate
gradients, BiCG)
Idea: Solver overdetermined linear system of equations
x (l) ∈ x (0) + K l (A,r 0 ): Ax (l) = b
in least squares sense, → Chapter 6.
x (l) = argmin{‖Ay − b‖ 2 : y ∈ x (0) + K l (A,r 0 )} .
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Zoo of methods with short recursions (i.e. constant effort per step)
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Memory requirements: 8 vectors ∈ R n
MATLAB-function: • [x,flag,r,it,rv] = bicgstab(A,b,tol,maxit,B,[],x0)
• [...] = bicgstab(Afun,b,tol,maxit,Binvfun,[],x0);
Computational costs :
2 A×vector, 2 B −1 ×vector, 4 dot products, 6 SAXPYs per step
➤
GMRES method for general matrices A ∈ R n,n
MATLAB-function: • [x,flag,r,it,rv] = qmr(A,b,tol,maxit,B,[],x0)
• [...] = qmr(Afun,b,tol,maxit,Binvfun,[],x0);
MATLAB-function: • [x,flag,relr,it,rv] = gmres(A,b,rs,tol,maxit,B,[],x0);
• [. ..] = gmres(Afun,b,rs,tol,maxit,Binvfun,[],x0);
Computational costs :
Memory requirements:
2 A×vector, 2 B −1 ×vector, 2 dot products, 12 SAXPYs per step
10 vectors ∈ R n
Computational costs : 1 A×vector, 1 B −1 ×vector per step,
: O(l) dot products & SAXPYs in l-th step
Memory requirements: O(l) vectors ∈ K n in l-th step
Remark 4.4.1 (Restarted GMRES).
little (useful) covergence theory available
stagnation & “breakdowns” commonly occur
Example 4.4.2 (Failure of Krylov iterative solvers).
After many steps of GMRES we face considerable computational costs and memory requirements for
every further step. Thus, the iteration may be restarted with the current iterate x (l) as initial guess →
rs-parameter triggers restart after every rs steps (Danger: failure to converge).
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⎛
0 1 0 · · · · · · 0
0 0 1 0 .
. . .. ... ...
A =
... .
⎜
. ... ... 0
⎝0 0 1
1 0 · · · · · · 0
⎞
⎟
⎠
⎛ ⎞
0.
, b =
⎜
.
⎟
⎝0⎠
1
x = e 1 .
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