Multivariate Newton Rhapson Method

I could not find anything on this on the net though it must be done
somewhere - so I had a go myself. What I need is a multivariate form of
Newton Rhapson.

Suppose I have

f(X1)=f(X0) + f '(Xo)[X1-X0] (1)

where X1 and X0 are m square non-singular matrices. Now assume f(X1)=0
and re-write(1) above
as

0 = f(x0)+f '(x0) [ X1-X0]

and we eventually get the recursion

X(n)=X(n-1)-f ' (X(n-1) ^-1 f(X(n-1)

now my question is siimple - is this right? Or should (1) be

f(X1)=f(X0)+[X1-X0]f ' (X0)

since we are dealing with matrices which do not commute.

ie the X1-X0 before the f '(X0)

Also - how do I differentiate wrt a matrix to calculate f ' (X(n-1)?

Thanks

Naebad

.

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