The above method can be generalized to multi-variate case to solve n simultaneous
algebraic equations

where is an n-dimensional vector. This equation system can be more concisely represented in vector form as . The Newton-Raphson formula for multi-variate problem is

where is the Jacobian of function :

To derive this iteration, consider Taylor series

We ignore the terms of and higher and let be zero (i.e., is the zero-crossing of the tangent), and get

Solving this linear equation system for , we get

and the Newton-Raphson formula:

Ruye Wang 2018-03-26