Reading BLAS names

BLAS (Basic Linear Algebra Subprograms) routine names look cryptic — gemv, ger, syr2, gemm — but they are systematic: each name is a compressed description of what the routine does. Once you can read the pieces, the names are self-documenting and easy to remember. This page is the key.

The three levels

BLAS is organized into levels by how much arithmetic a routine does relative to the data it touches — which is exactly what decides whether it is memory-bound or compute-bound, and how it tiles:

level	shape	work / data	examples
Level 1	vector–vector	$O(n)$ / $O(n)$	axpy, dot, nrm2, scal
Level 2	matrix–vector	$O(n^2)$ / $O(n^2)$	gemv, ger, symv, syr
Level 3	matrix–matrix	$O(n^3)$ / $O(n^2)$	gemm

Level 3 is the prize: $O(n^3)$ work on only $O(n^2)$ data means each element is reused $O(n)$ times, so it tiles with high arithmetic intensity — the reason gemm dominates DL compute.

Anatomy of a name

A Level-2/3 name is three fields glued together: [type][structure][operation].

Field 1 — data type (dropped throughout this catalog, where everything is real):

s	d	c	z
single	double	complex	double complex

Field 2 — matrix structure — which part of the matrix actually carries data:

ge	sy	he	tr	gb / sb / tb
general	symmetric	Hermitian	triangular	banded

Field 3 — operation:

mv	mm	r	r2	sv
matrix–vector product	matrix–matrix product	rank-1 update	rank-2 update	triangular solve

Read left to right and the name expands itself:

• gemv = ge + mv = GEneral Matrix–Vector product
• ger = ge + r = GEneral Rank-1 update
• syr2 = sy + r2 = SYmmetric Rank-2 update
• gemm = ge + mm = GEneral Matrix–Matrix product

The catalog, decoded

Level 1 — the names predate the field scheme, so they are mnemonics rather than [type][structure][op]:

name	reads as	operation
axpy	a·x plus y	$y := \alpha x + y$
dot	dot product	$s := x^{\mathsf T} y$
nrm2	norm, 2-norm	$s := \lVert x\rVert_2$
asum	absolute sum	$s := \sum_i \lvert x_i\rvert$
scal	scale	$x := \alpha x$
swap	swap	$x \leftrightarrow y$
copy	copy	$y := x$
rot	(Givens) rotation	apply a plane rotation to $(x,y)$
iamax	index of abs max	$\arg\max_i \lvert x_i\rvert$

Level 2 — [structure][operation]:

name	expands to	operation
gemv	general matrix–vector	$y := \alpha A x + \beta y$
ger	general rank-1	$A := A + \alpha x y^{\mathsf T}$
trmv	triangular matrix–vector	$x := T x$
symv	symmetric matrix–vector	$y := \alpha A x + \beta y$, $A = A^{\mathsf T}$
syr	symmetric rank-1	$A := A + \alpha x x^{\mathsf T}$
syr2	symmetric rank-2	$A := A + \alpha(x y^{\mathsf T} + y x^{\mathsf T})$

Level 3:

name	expands to	operation
gemm	general matrix–matrix	$C := \alpha A B + \beta C$

Why the shorthand helps here

In the domain-flow view the name’s fields often predict the recurrence shape:

• the operation field tells you the index-space geometry — mv is a reduction (a matvec contracts an axis), r/r2 are fully-parallel outer products (a rank update has no reduction), and mm is the full 3-D reduction cube;
• the structure field tells you the matrix is referenced only in a triangular half — sy/he exploit symmetry to read half the elements, tr is triangular. For the rank updates syr/syr2 this restricts the writes too, so their domain of computation is a triangular prism rather than a box; the matrix–vector cases symv/trmv still produce a full vector, they just read half the matrix.

So gemv and ger being adjacent in the catalog is not a coincidence: they are the inner-product and outer-product halves of the same matrix–vector interaction, and their SUREs are almost mirror images (a reduction vs. a broadcast-only map).