diff --git a/content/introduction/derivation.md b/content/introduction/derivation.md index 03c7ecd..b3eedb6 100644 --- a/content/introduction/derivation.md +++ b/content/introduction/derivation.md @@ -33,7 +33,8 @@ Linear algebra operators exhibits many independent operations. For example, ther 4. dot product The operator {{< math >}}$z = alpha * x + y${{< /math >}} is frequently used, and although redundant, tends -to be added as the fifth operator, and referred to as the _saxpy_ operator for "Scalar Alpha X Plus Y". +to be added as the fifth operator, and referred to as the _axpy_ operator, short for "alpha X plus Y", +where _alpha_ is a scalar and _X_ and _Y_ are vectors. The _dot product_ is also referred to as the _inner product_. The _inner product_ is an operator that brings two vectors together into a scalar representing a measure how much the vectors point in the same direction. The _outer product_ is an operator that expands two vectors into a matrix: for vector @@ -52,12 +53,12 @@ arrangements is shown in Table 1: | Loop order | Inner Loop | Middle Loop | Inner Loop Data Access | |------------|------------|-------------------------------------------------|-----------------------------| -| ijk | dot | vector {{< math >}}$\times${{< /math >}} matrix | {{< math >}}$A${{< /math >}} by row, {{< math >}}$B${{< /math >}} by column | -| jik | dot | matrix {{< math >}}$\times${{< /math >}} vector | {{< math >}}$A${{< /math >}} by row, {{< math >}}$B${{< /math >}} by column | -| ikj | saxpy | row saxpy | {{< math >}}$B${{< /math >}} by row | -| jki | saxpy | column saxpy | {{< math >}}$A${{< /math >}} by column | -| kij | saxpy | row outer product | {{< math >}}$B${{< /math >}} by row | -| kji | saxpy | column outer product | {{< math >}}$A${{< /math >}} by column | +| ijk | dot | vector {{< math >}}$\times${{< /math >}} matrix | {{< math >}}$A${{< /math >}} by row, {{< math >}}$B${{< /math >}} by column | +| jik | dot | matrix {{< math >}}$\times${{< /math >}} vector | {{< math >}}$A${{< /math >}} by row, {{< math >}}$B${{< /math >}} by column | +| ikj | axpy | row axpy | {{< math >}}$B${{< /math >}} by row | +| jki | axpy | column axpy | {{< math >}}$A${{< /math >}} by column | +| kij | axpy | row outer product | {{< math >}}$B${{< /math >}} by row | +| kji | axpy | column outer product | {{< math >}}$A${{< /math >}} by column | *Table 1:* Matrix Multiplication: Orderings and Properties (see [2](#matrix-computations))