Maxplus Function

semihoward - generalized maxplus eigenvalue eigenvector ( Howard algorithm )

Calling Sequence

[l,v,p,c,n]=semihoward(T,N)

Parameters

Description

Generalized right maxplus eigenvalues and eigenvectors of a timed event graph represented by a pair (T,N) of full or sparse maxplus matrices computed by a generalized Howard algorithm for delayed dynamic programming. That is the solutions (l,v) of max_j[(T_ij -N_ij x l_j)+v_j] = v_i . The matrices N and T must have the same nonzero entries. When T is irreducible l is constant.

The optimal policy p satisfies A_ip(i) v_p(i)= v_i where A denotes the matrix T-Nxdiag(l) in the standard algebra sense.

For performance evaluation of an event graph, N contains the numbers of tokens (initial marking) and T the minimal time that a token has to spend in a place. The eigenvalues l are interpreted as the average cost per time unit for the corresponding delayed dynamic programming problem and are computed by the Howard algorithm.

The values taken by the entries of l are the eigenvalues. If N is irreducible, l is constant: it is the eigenvalue and v is a corresponding eigenvector (in this case, there exits only one eigenvalue but there may exist more than one eigenvectors). If A can be decomposed into irreducible components (block diagonal with irreducible blocks), then, in each component, l is constant and this constant is the eigenvalue, the corresponding entries of v, completed by -inf, yields a corresponding eigenvector. For the block triangular case see howard command.

EXAMPLES

See Also