WARNING: this is my opinion, but I consider myself educated on the topic. I was a straight-A Latin student for most of a decade, and geeking out over language and etymology is one of my hobbies.
The correct counterpart is tabular.
tabulatum, tabulati N N 2 2 N [XXXCX]
floor, story; layer, row; tier formed by the horizontal branches of a tree;
But the word you actually want, an adverb that means "operates on columns," is in fact column-wise, and its row-oriented counterpart is row-wise. The words "columnar" and "tabular," at least the way I see them used, more properly refer to data than to an algorithm.
This is because the suffix -ar (as in familiar) is derived from the Latin -aris, in turn derived from -alis (see, e.g. the "origin" section here) which is where we get the suffix -al (as in familial). Both suffixes mean the same thing as their Latin roots: "of or related to." Consider the difference in connotation between familiar, familial, and family-wise. Note also that, of these three, family-wise is the only one that makes sense as an adverb rather than an adjective.
- tabular: arranged in rows
- columnar: arranged in columns
- row-wise: operates on rows
- column-wise: operates on columns.
Also, my thanks to Glen_b for reminding me about the term "row-wise."