Least-square inversion with inexact adjoints. Method of conjugate directions: A tutorial

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## Second step of the improvement

Now let us assume and add some amount of the step from the -th iteration to the search direction, determining the new direction , as follows:
 (17)

We can deduce that after the second change, the value of numerator in equation (9) is still the same:
 (18)

This remarkable fact occurs as the result of transforming the dot product with the help of equation (4):
 (19)

The first term in (19) is equal to zero according to formula (7); the second term is equal to zero according to formula (15). Thus we have proved the new orthogonality equation
 (20)

which in turn leads to the numerator invariance (18). The value of the coefficient in (17) is defined analogously to (14) as
 (21)

where we have again used equation (15). If is not orthogonal to , the second step of the improvement leads to a further decrease of the denominator in (8) and, consequently, to a further decrease of the residual.

 Least-square inversion with inexact adjoints. Method of conjugate directions: A tutorial

Next: Induction Up: IN SEARCH OF THE Previous: First step of the

2013-03-03