Matrix Equations

The two sets of moments above form two vectors (each of length ten) for each (0)^(m). With these, and a matrix ${\bf H}$, whose elements are the coefficients from the earlier expression for R_ij, we can form a much simpler matrix form for the (0)^(m)s:
$$\begin{align}{}_{a'b'p'}(0)_{c'd'q'}^{(m)} = \frac{(2m-1)!!}{\vert DB\vert^{2m+1}}{\bf u}^{t}{\bf Hv}. \end{align}$$

Further efficiency is gained by forming ${\bf Hv}$ (which depends only on the www-theor and m) and then contracting the product with the necessary ${\bf u}$. The advantage of a matrix form is that the (0)^(m)s can now be generated using BLAS-2 and BLAS-3 constructs, which are linear algebra packages highly optimized for a variety of computer platforms.