Two of the main parts of the CG method, by looking at https://www.cs.cmu.edu/~quake-papers/painless-conjugate-gradient.pdf page 50 B2. Conjugate Gradients, are:
- find an appropriate $\alpha$ to follow the current descent direction $d$ to decrease the quadratic function as much as possible.
- find an appropriate $\beta$ to guarantee that the new descent direction is A-orthogonal to the previous one.
The usual initial $d$ for CG is the residual, but I'm wondering, if I use some other arbitrary descent direction for the initialization, $\alpha$ should still find an appropriate step size, and $\beta$ should update the descent direction to be A-orthogonal to the initial one. Is that correct? Are these invariants still satisfied? Would CG fail to converge if I use an arbitrary initial $d$?