To construct a set of directions which satisfy the conjugacy criterion (preconjugacy), we can start from an arbitrary set of model-space vectors and apply an orthogonalization process to their projections in the data space. An iterative orthogonalization is defined by recursion
p_n = c_n - _j=1^n-1_n^(j)p_j, where the following choice of the scalar coefficients assures condition (preconjugacy):
_n^(j) = (M c_n,Mp_j) ||Mp_j||^2. According to the fact that the residual vector is orthogonal to all the previous steps in the data space (equation (preresmpj)), the coefficient simplifies to
_n = (r_n-1,Mc_n-1) ||Mp_n-1||^2. Formulas (ortoprocess-cdalpha) define the method of conjugate directions Fomel (1996) also known as the preconditioned Krylov subspace method Kleinman and van den Berg (1991) and under several other names.
A particular choice of the initial directions transforms the method of conjugate directions into the method of conjugate gradients and introduces remarkable simplifications.