Page 246 - Compact Numerical Methods For Computers

Left-overs 233

To test the claimed magnetic-root properties of this system, 24 starting points
were generated, namely the eight points about each root formed by an axial step
of ±0·5 and ±0·1. In every case the starting point was still nearest to the root
used to generate it.
All the algorithms converged to the root expected when the starting point was
only 0·1 from the root. With the following exceptions they all converged to the
expected root when the distance was 0·5.

(i) The Marquardt algorithm 23 converged to R from (-0·5,0) = (b ,b ) instead
3 1 2
of to R .
1
(ii) The Nelder-Mead algorithm 19 found R from (0·5,0) instead of R .
2
1
(iii) The conjugate gradients algorithm 22 found R and the variable metric
3
algorithm 21 found R when started from (1·5,l), to which R is closest.
1 2
(iv) All algorithms found R instead of R when started from (-0·25, 0·5625).
1 3
(v) The conjugate gradients algorithm also found R instead of R from
1 3
(-1·25,0·5625).
Note that all the material in this chapter is from the first edition of the book.
However. I believe it is still relevant today. We have, as mentioned in chapter 17,
added bounds constraints capability to our minimisation codes included in Nash and
Walker-Smith (1987). Also the performance figures in this chapter relate to BASIC
implementations of the original algorithms. Thus some of the results will alter. In
particular, I believe the present conjugate gradients method would appear to perform
better than that used in the generation of table 18.5. Interested readers should refer
to Nash and Nash (1988) for a more modern investigation of the performance of
compact function minimisation algorithms.