Dot Product (Dot)

Definition: Given a vector $x \in \Sn$ and vector $y \in \Rn$ , the dot product is the inner product of and , or .

How is it used?

Two vectors are Orthogonal if their dot product is equal to zero.

We will discuss algorithms for the closely related Apdot (alpha plus dot product) operation: $\alpha \becomes \alpha + x^T y$ .

$\begin{eqnarray*} \alpha &=& \FlaOneByTwo{x_{T}}{x_{B}} \FlaTwoByOne{y_{T}} {y_{B}} ~ + ~ \hat{\alpha} \end{eqnarray*}$

Details of how to derive the PME for this operation.

The dependencies in the above PME allow for two different loop-invariants to be identified:

Invariant 1	$\alpha = \tr{x}_T y_T + \hat{\alpha}$
Invariant 2	$\alpha = \tr{x}_B y_B + \hat{\alpha}$