The results of the preceding sections define the method of conjugate
directions to consist of the following algorithmic steps:
Choose initial model and compute the residual
At -th iteration, choose the initial search direction .
If is greater than 1, optimize the search direction by
adding a linear combination of the previous directions, according to
equations (22) and (23), and compute the modified step
Find the step length according to equation
(6). The orthogonality principles (24) and (7) can
simplify this equation to the form
Update the model and the residual
according to equations (2) and (4).
Repeat iterations until the residual decreases to the required
accuracy or as long as it is practical.
At each of the subsequent steps, the residual is guaranteed not to
increase according to equation (8). Furthermore,
optimizing the search direction guarantees that the convergence rate
doesn't decrease in comparison with (9). The only assumption
we have to make to arrive at this conclusion is that the
operator is linear. However, without additional assumptions, we cannot
guarantee global convergence of the algorithm to the least-square
solution of equation (1) in a finite number of steps.
Least-square inversion with inexact adjoints.
Method of conjugate directions: A tutorial