The Multiplicative ART (MART)

Related to the ART is the multiplicative ART (MART), also due to Gordon, Bender, and Herman [104]. While the ART applies to arbitrary systems of linear equations, the MART is restricted to a system of linear equations y = P x, in which the I by J matrix P has nonnegative entries, the entries of y are positive, and x has nonnegative entries; we shall also assume, for notational convenience, that the columns of P sum to one, although that is not necessary. The MART and its block-iterative versions, BI-MART, converge to nonnegative solutions of y = P x, whenever such solutions exist. The block-iterative version involving only a single block is the simultaneous MART (SMART), which also converges to an approximate solution when no nonnegative solution of y = P x exists.

The function minimized by the SMART is the Kullback-Leibler distance h( x) = KL( P x , y). With h _i( x) = KL(( P x) _i ,y _i), we see that h has the decomposition given by Equation (49.1).

The multiplicative algebraic reconstruction technique (MART) [104] begins with a strictly positive vector x ⁰ and has the iterative step

for j = 1 , 2 , ..., J and i = k(mod I) + 1.

Related Algorithms

The simultaneous MART...