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3.4 Block-Iterative Methods 31
where ǫ is an arbitrarily small but fixed constant and M t(k) are given symmetric
and positive definite matrices with the control t(k), then any sequence generated
by (3.11) converges to a solution for (2.1). If in addition x 0 ∈ R(A T ), then x k
converges to the solution of minimum 2-norm.
The BICAV method has the property that for p = 1 the method becomes fully
simultaneous, i.e. it becomes the CAV method. For p = m we on the other
hand have that BICAV becomes the well-known Kaczmarz’s method.
3.4.4 Block-Iterative Diagonally Relaxed Orthogonal Projections
(BIDROP)
For the general SIRT methods we described a method called DROP and we
will now introduce its block-iterative generalization, which we will call Block-
Iterative Diagonally Relaxed Orthogonal Projections (BIDROP).
If we let Wt be positive definite diagonal matrices and Ut be symmetric positive
definite matrices for t = 1, 2, . . ., p, then the algorithm for the BIDROP method
looks as follows:
x k+1 = x k + λkU t(k)A T t(k) W t(k)(b t(k) − Atkx k ), (3.12)
where t(k) = (k mod p) + 1.
The following convergence theorem is derived for the BIDROP method:
Theorem 3.4 Let U be a given symmetric and positive definite matrix, and let
Wt be given positive definite diagonal matrices. If for all k ≥ 0,
0 < ǫ ≤ λk ≤ (2 − ǫ)/ρ(UA T t(k) W t(k)A t(k)),
where ǫ is an arbitrarily small but fixed constant, then any sequence generated
by (3.12) converges to a solution. If in addition x 0 ∈ R(UA T ), then the solution
has minimal U −1 -norm.
With only one block, i.e. p = 1, and U1 = S and W1 = W, then we have the
standard DROP method.
The BIDROP method is a general method since Ut and Wt is not specific given.
One of the variants of BIDROP is introduced in [5] and is called BIDROP1.