Theorem (Broyden's Method for n-dimensional Systems).  To solve the non-linear system  ,  given one initial approximation  ,  and generating a sequence    which converges to the solution  ,  i.e.  .  Compute the initial Jacobian matrix  

        .  

Use it to compute the first approximation using Newton's method  

        .  

For  .  Suppose that    has been obtained, use the following steps to obtain  .  

Step 1.     Evaluate the function  .  

Step 2.  Update the approximate Jacobian using  ,  and  

        .   

Step 3.  Compute    using the Sherman-Morrison formula

          

Step 4.  Compute the next approximation   

        .  

Remark.

As the iteration proceeds,  ,  and then

        
    or
        

where, for  ,  

        ,  

        ,  

        ,  
then
        
    and
        .  

(c) John H. Mathews 2005