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# The Newton Method for Systems

The Newton method is useful for solving nonlinear systems of equations of the form
 (16.5)

There are independent variables ; in vector notation, these variables are the entries of the vector variable

The system is defined by functions (each is a function of the variables ). In vector notation,

Therefore, the system of equations ( ) can be written compactly as

We want to find a solution of the system, i.e. a set of values such that all the equations are simultaneously satisfied

Newton's method builds a sequence of points

which converge to the true solution of the problem

In one dimension, the sequence is built using Newton's formula

 (16.6)

For -dimensional problems, this formula generalizes to

 (16.7)

The place of the derivative is now taken by the Jacobian matrix defined as

Let

The Newton formula ( ) can be written as

 (16.8)

This is a system of linear equations; is a matrix, and and are -dimensional vectors.

One step of the Newton method (say, the step) proceeds as follows:

0
We have available, and want to compute the next iterate, ;
1
Evaluate the vector function (we evaluate each component function individually);
2
Evaluate the derivative matrix (each entry is a function of and need to be evaluated individually);
3
Solve the system ( ). For this, we need to
3.1
compute the LU decomposition (with pivoting) of the matrix ;
3.2
apply the back-substitution algorithm to the right hand side , to obtain the solution ;
This step cannot be performed if is singular; if this happens we need to terminate the computations.
4
Compute the next iterate as .

Note that one Newton step is very expensive. We have to evaluate functions (the entries of ) and we need to calculate the LU decomposition of each step.    Next: The Modified Newton Method Up: Nonlinear Equations Previous: Fixed point iterations   Contents