Complex Newton’s method

The general equation is

 Eqn 1

In this case

 

Therefore, the equation 1 is the set

 

In matrix notation

 

Set the left side to zero and multiply by the inverse matrix to find

 

The inverse matrix is given by

 

So that

 

This simplifies to

 

The denominator is given by the imaginary part of the product of the partials as shown below

 

The numerator of the expression for the change in parameters is given by the imaginary products shown below

 

Thus

 

Test Function

                for\cnewt.for is the test code in Fortan.