link to my thesis

5.3. EFFICIENCY OF THE ND-SYSTEMS APPROACH 59
whereas the eigenvalues of the matrices A T x i
correspond to the i-th coordinate of the
stationary points (for which it is usually far less clear in advance in which range
the value at the global optimum will be located). Therefore, when dealing with the
matrix A T p λ
one may focus on the calculation of only a few eigenvalues, as one is
interested in the smallest real eigenvalue that corresponds to a real solution point.
Iterative methods allow the user to ‘zoom in’ on a few eigenvalues, whereas in case
of A T x i
all the real eigenvalues need to be computed, for each i =1, 2,...,n, and all
resulting real critical points have to be substituted into the criterion function to find
the global optimum.
In this thesis, approach (ii) of computing only the smallest real eigenvalue of the
matrix A T p λ
is investigated because only one eigenvalue is required when searching for
the global minimum.
5.3 Efficiency of the nD-systems approach
The following research questions are addressed in this section: (i) For a given multidimensional
time instant (t 1 ,...,t n ), what is the most efficient way to compute the
value of y t1,...,t n
from a given initial state w 0,...,0 using the n difference equations
(5.3)? And as a closely related question, what is the most efficient way to compute
the whole state vector w t1,...,t n
? (ii) Can we design a suboptimal heuristic procedure
that computes the state vector w t1,...,t n
at acceptable computational costs?
5.3.1 A linear complexity result for computing y t1 ,...,t n
The first research question raised above is especially important when the efficiency
of just one iteration of the iterative eigenvalue solver is under consideration. In this
single iteration the state vector w t1,...,t n
has to be computed in an efficient way. This
in contrast to the case where we study the efficiency of solving the eigenvalue problem
as a whole.
The following result addresses the computational complexity that can be achieved
by an optimal algorithm to compute y t1,...,t n
from w 0,...,0 .
Theorem 5.1. Consider a set of n multidimensional recursions of the form (5.3)
and let an initial state w 0,...,0 be given. Then every algorithm that computes the value
of y t1,...,t n
, using only the recursions (5.3), has a computational complexity which
increases at least linearly with the total time |t|.
Proof. Each recursion from the set (5.3) allows one to compute the value of
y t1,...,t n
from a set of values for which the total times are all within the range
|t| −m, |t| −m +1,...,|t| −1. The largest total time among the entries of the
initial state w 0,...,0 corresponds to y m−1,...,m−1 and is equal to n(m − 1). Therefore,
to express y t1,...,t n
in terms of the quantities contained in the initial state requires at
least ⌈(|t|−n(m − 1))/m⌉ applications of a recursion from the set (5.3). Hence, the
computational complexity of any algorithm along such lines increases at least linearly