grammaticality - "need to do" vs "need do"

Consider:

1. I need to do this.


2. I need do this.

My English grammar knowledge tells me that "need" doesn't have the same status as the modal verbs "may", "can", "should" and what not. Hence the second usage where two verbs appear consecutively is incorrect.

But yesterday, my native English speaking friends (Americans) told me that "I need compute this." is a perfectly grammatical sentence, and one is simply omitting the "to". How can this be? Is it a colloquial usage but grammatically incorrect, or is it grammatically correct? If it is grammatically correct, is it because "need" is a semi-modal verb?

EDIT: in particular, is it okay to use "need compute" in a scientific paper?

EDIT2: the exact phrase that raised the question was "The advantage of this representation is that we need only compute sums and products"

Answer

Your friend is correct. "I need compute ..." is ungrammatical, but "I need only compute ..." is fine, if a little bit old-fashioned and formal.

Modal verbs do not use a "to". That is, you say

I can do this.

The verb "need" is a funny case; it is only modal in the negative. In the positive, we already have an equivalent modal verb; namely, "I must". However, there are two possible meanings for the opposite of "I must do this": "I am forbidden to do this" and "I am not required to do this". These two different meanings are conveyed by the modal verbs "I must not" and "I need not".

Searching with Google books, it appears to have been this way at least since 1600 (although back then, there was a positive construction "I must needs", which has since for the most part fallen out of use). Thus, you get various grammatical constructions.

In the negative, you have:

I need not do this.
I do not need to do this.

In the positive, you have:

I must do this.
I need to do this.
*I must needs do this. (obsolete)

"I need do this" is incorrect.

Over the last few centuries, "I don't need to" has slowly been replacing "I need not", but "I need not do this" is still used reasonably frequently, and is grammatical. However, if "I need do this" was ever grammatical, it was in the long distant past.

Finally, in the past you could say

it is not the case that I need compute this,

since that is a negative use (and this is why your friend might not be wrong). Today, I believe most people would use "need to" here. But if your friend was using "need compute" in the negative, there is a good case to make for it being grammatical.