Chapter 8 Numerical Solutions of Nonlinear Systems of Equations
The iteration mapping function, , where
The solution of can be approximated by the iteration .
The secant method uses the approximation
as a replacement for in
This equation does not define a unique matrix, because it does not describe how operates on vectors orthogonal to . Since no information is available about the change in in a direction orthogonal to , we require additionally of that
, whenever . (2)
This condition specifies that any vector orthogonal to is unaffected by the update from , which was used to compute , to ,which is used in the determination of .
Conditions (1) and (2) uniquely define , as
It is the matrix that is used in place of to determine :
This method is then repeated to determine , using in place of and with and in place of and . In general, once has been determined, is computed by
where the notation and .
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.
R. L. Burden and J. D. Faires, Numerical Analysis, PWS,