Note: We need to be careful of the possibility that is zero. For a practical algorithm, we would need to check if the second derivative is very small, and either terminate the algorithm, or threshold it to a larger positive value so as to allow the algorithm to continue.

In multiple dimensions this generalizes by using the gradient, $,$ and the Hessian matrix, :

$($Note that the second term in this expression can also be written as

when a column vector gradient is being used instead of a row vector gradient.$)$

Will the multidimensional Newton's method work for any convex loss function?