Steepest Descent Method

An Algorithm for calculating the Gradient $\nabla f({\bf P})$ of a function at an $n$-D point ${\bf P}$. The steepest descent method starts at a point ${\bf P}_0$ and, as many times as needed, moves from ${\bf P}_i$ to ${\bf P}_{i+1}$ by minimizing along the line extending from ${\bf P}_i$ in the direction of $-\nabla f({\bf P}_i)$, 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.