sparse image representation via combined transforms - Convex ...

5.3. MINRES 115
using the preconditioned Lanczos algorithm:
Set v k = Aw k − β k−1 q k−1 ,
α k = 〈v k ,w k 〉,
v k ← v k − α k q k ,
Solve M ˜w k+1 = v k ,
Set q k+1 = v k /β k and w k+1 = ˜w k+1 /β k , where β k = 〈v k , ˜w k+1 〉 1/2 .
(b) Work on the last column of T , that is:
If k>2, then
( ) ( )(
T (k − 2,k) ck−2 s k−2 0
←
T (k − 1,k) −¯s k−2 c k−2 T (k − 1,k)
)
.
If k>1, then
(
T (k − 1,k)
T (k, k)
) ( )(
ck−1 s k−1 T (k − 1,k)
←
c k−1 T (k, k)
−¯s k−1
)
.
Set
a = |T (k, k)|/ (|T (k, k)| + |T (k +1,k)|) ,
b = 1− a,
c k = a/ √ a 2 + b 2 ,
¯s k = c k T (k +1,k)/T (k, k).
Apply the k-th rotation to ξ and to the last column of T :
( ) ( )( )
ξ(k)
ck s k ξ(k)
←
,
ξ(k +1)
−¯s k c k 0
T (k, k) ← c k T (k, k)+s k T (k +1,k),
T (k +1,k) ← 0.