Page 193 - Compact Numerical Methods For Computers

182 Compact numerical methods for computers

BUILD 3 4.1402114150E-02 4.6980937735E-14
LO-REDUCTION 5 1.2559406706E-02 4.6980937735E-14
HI -REDUCTION 7 3.4988133663E-03 4.6980937735E-14
HI-REDUCTION 9 7.8255935023E-04 4.6980937735E-14
. . .
SHRINK 59 3.1448995130E-14 1.0373099578E-16
SHRINK 63 2.4400978639E-14 1.0373099578E-16
HI-REDUCTION 65 1.7010223449E-14 1.0373099578E-16
. . .
HI--REDUCTION 117 6.0920713485E-24 3.9407472806E-25
Exiting from ALG19.pas Nelder Mead polytope minimiser
119 function evaluations used

Minimum function value found = 1.7118624554E-25
At parameters
B[l]= 1.1213869326E+00
B[2]= -4.0522273834E-01
alg20.pas -- axial search
Axis Stepsize function + function - rad. of curv. tilt
1 1.512415E-06 9.226758E-12 9.226726E-12 2.479099E-01 6.159003E-10
2 5.465253E-07 4.546206E-13 4.546053E-13 6.5702023-01 8.031723E-10

14.4. OTHER DIRECT SEARCH METHODS
The Nelder-Mead procedure presented here is by no means the only direct search
procedure, nor can it be taken to be the most economical of space or time. Dixon
(1972, chap 5) discusses a number of direct search methods. Some of these
perform various linear searches then form the resultant or sum of these over. say,
n directions. A new set of directions is produced with the first being the resultant
and the others orthogonal to this direction and to each other. This is the basis of
the method of Rosenbrock and that of Davies, Swann and Campey. These both
require storage of at least n vectors of n elements, equivalent to algorithm 19.
The major differences in the two methods mentioned occur in the linear search
and in the orthogonalisation procedure, which must be made resilient to the
occurrence of directions in which no progress can be made, since these will have
length zero and will cause a loss of dimensionality.
A method which is very simple to code and which uses only two vectors of working
storage is that of Hooke and Jeeves (1961). This is the only method 1 have tested
using the 77 test functions in Nash (1976). At that time, as reported in the first
edition, the performance of this method in BASIC on a Data General NOVA
computer was not satisfactory. However, for the types of functions encountered in
many teaching situations it seems quite reliable, and EASON and FENTON (1973)
showed a preference for this method. Furthermore. it is explicable to students whose
major interest is not mathematics, such as those in economics or commerce, who