To construct a set of A-orthogonal search directions ${d}_{(i)}$.

High-level idea

Use $n$ linearly independent vectors ${u}_{0},{u}_{1},...,{u}_{n-1}$. To construct ${d}_{(i)}$, take ${u}_{i}$ and subtract out any components that are not A-orthogonal to the previous $d$ vectors. In other words, set ${d}_{(0)}={u}_{0}$, and for $i>0$, set

The difficulty with using Gram-Schmidt conjugation in the method of Conjugate Directions is that all the old search vectors must be kept in memory to construct each new one, and furthermore $O({n}^{3})$ operations are required to generate the full set.