Quaternion products with fewer real products

A couple days ago I wrote about Gauss’ trick to multiply two complex numbers using only 3 real multiplications. This post will do something similar with quaternions.

Just as the most direct approach to computing complex products requires 4 real multiplications, the most direct approach to quaternion products requires 16 real multiplications. (See the function mult here.)

We can represent a quaternion as a pair of complex numbers by observing

a + bi + cj + dk = (a + bi) + (c + di)j

Gauss’ trick can multiply two complex numbers using only three real multiplications. It would seem that we could use something analogous to multiply two quaternions using only 3 complex multiplications, then apply Gauss’ trick to do each of these complex multiplications with 3 real multiplications. The net result would be computing a quaternion product in only 9 real multiplications.

Except that doesn’t work. A direct application of Gauss’ trick doesn’t work because i and j don’t commute. Perhaps there’s some variation on Gauss’ trick that would work, but I haven’t found one.

So let’s back up and take a different approach.

You can represent the quaternions as 2 × 2 complex matrices via

a + bi + cj + dk \mapsto \begin{bmatrix}a - di & -c - bi \\ c - bi & \phantom{-}a + di\end{bmatrix}

So we could multiply two quaternions by multiplying two 2 × 2 complex matrices. That would require 8 complex multiplications [1], which could be carried out using 3 real multiplications each. This would let us “reduce” the number of multiplications from 16 to 24.

But by symmetry we could eliminate half the multiplications. This puts us at 12 real multiplications, which is less than the 16 required by the most direct approach.

To see this, let α = adi and β = c + bi. Then our matrix representation has the form

\begin{pmatrix} \alpha & -\beta \\ \bar{\beta} & \bar{\alpha} \end{pmatrix}

Then our quaternion product becomes the matrix product

\begin{pmatrix} \alpha & -\beta \\ \bar{\beta} & \bar{\alpha} \end{pmatrix} \begin{pmatrix} \gamma & -\delta \\ \bar{\delta} & \bar{\gamma} \end{pmatrix} = \begin{pmatrix} \alpha\gamma - \beta\bar{\delta} & -\alpha\delta - \beta\bar{\gamma}\\ \bar{\beta}\gamma + \bar{\alpha}\bar{\delta} & -\bar{\beta}\delta + \bar{\alpha}\bar{\gamma} \end{pmatrix}

We only need to compute the complex products in top row; we can recover the quaternion representation of the product from there. The products in the second row are negatives or conjugates of products in the first row.

Maybe there’s a Gauss-like trick to compute the top row of the matrix using only three complex products, but it can at least be done using four complex products.

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[1] You could use Strassen’s approach to multiply two 2 × 2 matrices in 7 multiplications. That would cut the number of real multiplications down to 21, but exploiting symmetry cuts the number of multiplications further.

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