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An Algorithm for calculating the Gradient
of a function at an
-D point
.
The steepest descent method starts at a point
and, as many times as needed, moves from
to
by minimizing along the line extending from
in the direction of
,
the local downhill gradient. This method has the severe drawback of requiring a great many iterations for
functions which have long, narrow valley structures. In such cases, a
Conjugate Gradient Method is
preferable.
See also Conjugate Gradient Method, Gradient
References
Arfken, G. ``The Method of Steepest Descents.'' §7.4 in Mathematical Methods for Physicists, 3rd ed.
Orlando, FL: Academic Press, pp. 428-436, 1985.
Menzel, D. (Ed.). Fundamental Formulas of Physics, Vol. 2, 2nd ed. New York: Dover, p. 80, 1960.
Morse, P. M. and Feshbach, H. ``Asymptotic Series; Method of Steepest Descent.'' §4.6 in
Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 434-443, 1953.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T.
Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge
University Press, p. 414, 1992.