sparse image representation via combined transforms - Convex ...

5.3. MINRES 113
2. For k =1, 2,...,
(a) Compute q k+1 , α k = T (k, k) andβ k = T (k +1,k)=T (k, k +1)using
the Lanczos algorithm:
˜q k+1 = Aq k − β k−1 q k−1 .
Set α k = 〈˜q k+1 ,q k 〉,
˜q k+1 ← ˜q k+1 − α k q k .
β k = ‖˜q k+1 ‖,
q k+1 = ˜q k+1 /β k .
(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.