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CHAPTER 2. PRELIMINARIES 28
end
h k+1,k = ‖q k+1 ‖; q k+1 = q k+1 /h k+1,k ;
end
for k = 1 to m do
begin
√
cc = h 2 kk + h2 k+1,k ;
c = h kk /cc; s = h k+1,k /cc; h kk = cc;
for
(
i = k +
)
1 to
(
m do
) ( )
hk,i c s hk,i
=
;
(
h
) k+1,i (
s
)
−c
( )
h k+1,i
zk c s zk
=
;
z k+1 s −c 0
end
y m = z m /h mm ;
for i = m
(
down-to 1 do
y i = z i − ∑ )
m
j=i+1 h ijy j /h ii ;
x m = x 0 + ∑ m
i=1 y iq i ;
r m = ˆK −1 (b − Kx m );
x 0 = x m ; r 0 = r m ;
z 1 = ‖r 0 ‖; q 1 = (1/z 1 ) · r 0 ;
end
until |z 1 | < tolerance
2.5.1.2 BiCGstab
The stabilized bi-conjugate gradient method (BiCGstab) was introduced in [VdV92] (with
slight modifications in [SVdV94]). It is not optimal in each step, i.e. it solves the minimization
problem only approximately, but as it uses a short range recurrence for the
construction of the orthonormal basis of the Krylov space, it consumes considerably less
computer memory as GMRES.
Algorithm 2.12. BiCGstab. Iterative Solution of Kx = b, with preconditioner ˆK.
Choose starting solution x 0 ;
r 0 = ˆK −1 (b − Kx 0 );
Choose arbitrary ˆr 0 , such that ˆr 0 · r 0 ≠ 0, e.g. ˆr 0 = r 0 ;
ρ 0 = α = ω 0 = 1;
v 0 = p 0 = 0;
i ← 1;
repeat
begin
ρ i = ˆr 0 · r i−1 ; β = (ρ i /ρ i−1 )(α/ω i−1 );
p i = r i−1 + β(p i−1 − ω i−1 v i−1 );