**Chapter 8 Numerical
Solutions of Nonlinear Systems of Equations**

**Newton**** method:**

The iteration mapping function, , where

.

The solution of can be approximated by the iteration .

**Quasi-Newton method**:

The secant method uses the approximation

as a replacement for in

. (1)

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 .

**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,