Chapter 8  Numerical Solutions of Nonlinear Systems of Equations


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 Newton’s method. For nonlinear systems,  is a vector, and the corresponding quotient is undefined. However, the method proceeds similarly in that we replace the matrix  in Newton’s method by a matrix  with the property that

.                                                     (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.



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