This is a great method! It avoids arbitrary floating point arithmetics[1] and thus makes it more comparable with the method presented in the article.
As far as I can see, this algorithm is very similar to the matrix operations of the article, at least in terms of complexity.
[1] Only the very last step might involve fp arithmetics, because you'll have to convert the result pair (x,y) back to x+y*sqrt(5). However, we already know that the result has to be an integer, so the its second part will always be 0, no fp arithmetics required.
As far as I can see, this algorithm is very similar to the matrix operations of the article, at least in terms of complexity.
[1] Only the very last step might involve fp arithmetics, because you'll have to convert the result pair (x,y) back to x+y*sqrt(5). However, we already know that the result has to be an integer, so the its second part will always be 0, no fp arithmetics required.