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

3.2 Algebraic Reconstruction Techniques (ART) 25
relaxation parameters. We again let λ denote the constant relaxation parameter
of the orthogonal projection for the rows of A and we let α denote the constant
relaxation parameter for the orthogonal projection using the columns.
The extended Kaczmarz method has the following algorithm, where x 0 ∈ R n
and y 0 = b:
y k,0 = y k
y k,j = y k,j−1 − α
y k+1 = y k,n
b k+1 = b − y k+1
x k,0 = x k
aj, y k,j−1
aj 2 2
aj
j = 1, . . .,n
x k,i = x k,i−1 + λ bk+1 − ai , xk,i−1 a i , i = 1, . . . , m
x k+1 = x k,m .
a i 2 2
For the extended Kaczmarz method it is proven in [34] that for any x 0 ∈ R n
and for any λ, α ∈ (0, 2) the method converges to a least squares solution. This
method is not implemented in the package.
3.2.5 Multiplicative ART
Another method in the ART class is the multiplicative ART method. This
method was proposed by in [17]. For this method we assume that x 0 is a n
dimensional vector of all ones and that all the elements in A are between 0 and
1, 0 ≤ aij ≤ 1. The multiplicative ART method is given as:
x k+1
j
=
bi
〈ai , xk aij x
〉
k j,
where i = (k mod m) + 1. Originally when the method was presented is was
assumed that all the elements in A are either 0 or 1, but later is has been shown
that if
• all the entries of A are between 0 and 1
• A does not have zero rows
• the system (2.1) has a nonnegative solution