If the ray-path matrix is very poorly conditioned, it may be advisable to modify the normal equations with a damping term. Then, (PSLSQR) changes to

& ^T & - = 0 , where is still the residual vector, but now the solution vector satisfies

= (^T+)^-1^T. The inverse matrix in (dampedvecs) exists for any value of , since the eigenvalues of are nonnegative and the addition of the diagonal term shifts all the eigenvalues up by .

Applying the Lanczos method to (LSQRdamped), I quickly discover that the terms proportional to always cancel out of the equations so the hard part of the analysis is the same as for the undamped problem. Thus, I find

_r_r_r^T and ^_r_r^-1_r^T, as before. The orthogonal matrices and are all identical to those for the undamped case.

Then, I must write the regularized normal matrix in terms of these orthogonal matrices and find its inverse. The resulting matrix is given by

^T+ = _r_r^T_r^T + = _r(_r^T_r + )_r^T + (-_r_r^T). The inverse matrix is easily shown to be

(^T+)^-1 = _r(_r^T_r + )^-1_r^T + 1(-_r_r^T). Substituting this result into (dampedvecs) gives

= _r(_r^T_r+)^-1_r^T_r^T = b_1_r(_r^T_r+)^-1_r^T 100, which should be compared to (solutionvecs). If the data vector and the ray-path matrix are inconsistent, then (dampedsfinal) must be modified in analogy to the modification made in transforming from (solutionvecs) to (ModLSQRinverse). Following the same analytical path in the Lanczos method for damped least-squares, the result is

= _r(_r+)^-1_r^T^T = D_1_r(_r+)^-1 100, which should be compared to (sfromlanczos).

The resulting resolution matrices are symmetric. The same results hold for
partial versions of these formulas, where the process is terminated early at
some *k* < *r* and *r*s are replaced by *k*s everywhere in the formulas
(dampedsfinal) and (dampedsfinallanczos).

11/17/1997