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.

11/11/1997