For arbitrary , denote by the solution of the normal equations

Set The
The Moré-Hebden algorithm takes its cue from the simplest possible
case: *x* and *d* are one-dimensional, and *A* and *R* are scalars.
In that very special case,

The iteration proceeds as follows:

- initialize somehow
- until convergence do: replace by

- solve the normal equations for
*x*, compute the residual - compute the normal residual
*g*=*A*^{T}(*Ax*-*d*) - solve the auxiliary system for
*s* - compute
- replace by

A final detail: since must remain positive, I have replaced any large decrease implied by the above formula by a bisection strategy. Since , as soon as is too small (which forces the weight onto the regularization term and increases the residual), the algorithm produces regular increases in and converges very rapidly, usually in one or two steps, so long as the normal equations are solved successfully. This is not always the case, but failure to converge rapidly appears to signal large data components associated with very small eigenvalues and is a sure sign that the noise level estimate has been chosen too small.

4/20/1999