Chapter 8  Numerical Solutions of Nonlinear Systems of Equations

 

Steepest Descent:

   The method of Steepest Descent determines a local minimum for a multivariable function of the form . The connection between the minimization of a function from  to  and the solution of a system of nonlinear equations is due to the fact that a system of the form

,

,

,

has a solution at  precisely when the function  defined by

has the minimal value zero.

   The method of Steepest Descent for finding a local minimum for an arbitrary function  from  into  can be intuitively described as follows:

i.         Evaluate  at an initial approximation ;

ii.       Determine a direction from  that results in a decrease in the value of ;

iii.     Move an appropriate distance in this direction and call the new vector ;

iv.     Repeat steps i through iii with  by .

 An appropriate choice for  is

,             for some constant .

 To determine an appropriate choice of , we consider the single-variable function

        .

The value of  that minimize  is what we need.

 

References:

【1】         R. L. Burden and J. D. Faires, Numerical Analysis, PWS, Boston, 1993.