Iterative Optimization in Inverse Problems

152 Iterative Optimization in Inverse Problems

for m =1, ..., N . With D the invertible diagonal matrix with entries D

mm

=

S

mm

we can write one cycle of Jacobi’s method as

z

new

= z

old

+ D

−1

(h − Sz

old

). (10.20)

The Jacobi overrelaxation (JOR) method has the following full-cycle iter-

ative step:

z

new

= z

old

+ ωD

−1

(h − Sz

old

); (10.21)

choosing ω = 1 we get the Jacobi method. Convergence of the JOR iteration

will depend, of course, on properties of S and on the choice of ω.WhenS =

Q,whereQ is Hermitian and nonnegative-definite, for example, S = A

†

A

or S = AA

†

, we can say more. Note that such Q can always be written in

the form Q = AA

†

or Q = A

†

A, for appropriately ...