AIR Tools - A MATLAB Package for Algebraic Iterative ...

3.1 Simultaneous Iterative Reconstructive Technique (SIRT) 19
of column j for each j = 1, 2, . . ., n:
sj = NNZ(aj), for j = 1, . . .,n.
We then define a i 2 S = n
j=1 a2 ij sj. Using this the CAV algorithm is as follows:
x k+1
j
= xk j
+ λk
m
i=1
wi
bi − a i , x k
a i 2 S
where wi > 0 are user-defined weights.
a i j for k = 0, 1, . . .,
We see that when A is dense we get the original Cimmino’s method, since sj = m
for all j = 1, . . .,n, and we have ai2 1
S = mai2 2.
To rewrite the CAV algorithm in matrix form we define S = diag(s1, s2, . . . , sn),
where the sj-values are defined as described above. We then let
wi
DS = diag for i = 1, . . .,m,
a i 2 S
where a i 2 S = (ai ) T Sa i and the CAV algorithm has the following matrix form
x k+1 = x k + λkA T DS(b − Ax k ),
which we recognize as (3.1) with M = DS and T = I.
3.1.5 Diagonally Relaxed Orthogonal Projections (DROP)
Another method in the SIRT class is the diagonally relaxed orthogonal projection
(DROP) method which is described in [5]. This method is another
extension of Cimmino’s method, which is inspired by the CAV method. In the
DROP method we also introduce a user-defined weighting of the equations. We
let wi > 0 denote this weighting.
The DROP method can then be written as:
m
x k+1 = x k + λk
i=1
wiS −1 (Pi(x k ) − x k ),
where Pi(x k ) is defined as in (3.3) and S is defined as above for CAV. Using
(3.3) we can rewrite the DROP algorithm into the following form:
x k+1
j
= xk j
1
+ λk
sj
m
i=1
wi
bi − a i , x
a i 2 2
a i j ,