Can a response paper agree with the initial paper, or must it disagree? A paper describes a computational method for solving a problem, and we made a significant improvement to the method and applied it to a new case study. Is it appropriate to publish a response paper?

Most response papers I've seen dispute the findings or methods of the paper they are responding to. Is it reasonable to write a response paper which agrees with the initial paper?

EDIT: It wasn't very clear when I first posted this, but the "response paper" I refer to is a specific mechanism that journals have for responding to an existing paper published in their journal. This is often called a comment or discussion paper. At Nature, they're called "brief communications arising". I might be the only one that calls it a "response paper," actually...
In general, 'comments' can be comments in support or comments in disagreement. Usually people tend to only submit comment when they are moved strongly to point out  a disagreement.

But, as question specifically referred to Nature's "brief communication arising" (BRA)  the answer is NO.  The BRA  notes that "Submissions should challenge the main conclusions of the Nature paper and contain new, unpublished data to support the arguments.". (i.e., pointing out errors, reinterpreting conclusions, coming to a different end result, comments to refute). Showing that there is an alternative method that is better than that presented in the paper doesn't seem to fall under that 'challenge' to the paper. (Unless perhaps you can write it as challenging some final conclusion that their method is the best possible ever? )

They offer the option to add extensive comments under the paper on the website. But then it will be difficult for others to be able to cite or even find your work. It sounds as if you should find a journal and submit your  your method as improvement in XX algorithm/method, citing the original Nature paper.

For comments and criticism there is a brilliant new service of pre- and post-publication reviews called Publons. For example, a recent and very harsh criticism of a sensational paper in Nature. Each review is assigned a DOI, so others may easily cite your comments. Also, it is well covered with Altmetric, which provides quite a visibility to your comment. Yet, no official impact factor for you reply publication.