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3.2 Algebraic Reconstruction Techniques (ART) 21
where V = diag(ςj) and W = diag 1
ςi
i , where ς and ςj denotes the row and
the column sums:
ς i =
ςj =
n
j=1
m
i=1
a i j
a i j
for i = 1, . . .,m
for j = 1, . . .,n.
For this method we assume that ai = 0 and aj = 0, such that A does not contain
any zero rows or columns.
Since the SART method has T = I, we cannot use theorem 3.1. The convergence
for SART was independently developed by Censor, Elfvind in [4] and Jiang,
Wang in [26]. Both showed that the convergence for SART is within the interval
(0, 2).
3.2 Algebraic Reconstruction Techniques (ART)
We now introduce a different class of methods which we will denote algebraic
reconstruction techniques (ART). All methods in the ART-class are fully sequential
method, i.e., each equation is treated at a time, since each equation is
dependent on the previous.
3.2.1 Kaczmarz’s Method
The classical and most known method of the ART class is called Kaczmarz’s
method, [27]. The method is a so-called row action method, since each iteration
consist of a ”sweep” through all the rows in the matrix A. Since the method uses
one equation in each step, an iteration consists of m steps. Figure 3.3 shows
an example of a sweep for the consistent case with the relaxation parameter
λk = 1.
The algorithm for Kaczmarz’s method updates x k in the following way:
x k,0 = x k ,
x k,i = x k,i−1 bi −
+ λk
ai , xk,i−1 ai2 2
x k+1 = x k,m .
a i , i = 1, 2, . . .,m,