Evolution and Optimum Seeking

Multidimensional Strategies 81
Brown and Dennis (1972) and Gill and Murray (1972)have suggested other schemes
for obtaining the partial derivatives numerically from values of the objective function.
Stewart himself reports tests that show the usefulness of his rules insofar as the results are
completely comparable to others obtained with the help of analytically speci ed derivatives.
This may be simply because rounding errors are in any case more signi cant here,
due to the matrix operations, than for example in conjugate gradient methods. Kelley
and Myers (1971), therefore, recommend carrying out the matrix operations with double
precision.
3.2.3.3 Further Extensions
The ability of the quasi-Newton strategy of Davidon, Fletcher, and Powell (DFP) to
construct Newton directions without needing explicit second partial derivatives makes
it very attractive from a computational point of view. All e orts in the further rapid
and intensive development of the concept have been directed to modifying the correction
Equation (3.31) so as to reduce the tendency to instability because of rounding errors
and inexact line searches while retaining as far as possible the quadratic convergence.
There has been a spate of corresponding proposals and both theoretical and experimental
investigations on the subject up to about 1973, for example:
Adachi (1973a,b)
Bass (1972)
Broyden (1967, 1970a,b,c, 1972)
Broyden, Dennis, and More (1973)
Broyden and Johnson (1972)
Davidon (1968, 1969)
Dennis (1970)
Dixon (1972a,b,c, 1973)
Fiacco and McCormick (1968)
Fletcher (1969a,b, 1970b, 1972b,d)
Gill and Murray (1972)
Goldfarb (1969, 1970)
Goldstein and Price (1967)
Greenstadt (1970)
Hestenes (1969)
Himmelblau (1972a,b)
Hoshino (1971)
Huang (1970, 1974)
Huang and Chambliss (1973, 1974)
Huang and Levy (1970)
Jones (1973)
Lootsma (1972a,b)
Mamen and Mayne (1972)
Matthews and Davies (1971)
McCormick andPearson (1969)