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## First step of the improvement

Assuming n>1, we can add some amount of the previous step to the chosen direction to produce a new search direction , as follows:
 (10)
where is an adjustable scalar coefficient. According to to the fundamental orthogonality principle (7),
 (11)
As follows from equation (11), the numerator on the right-hand side of equation (9) is not affected by the new choice of the search direction:
 (12)
However, we can use transformation (10) to decrease the denominator in (9), thus further decreasing the residual . We achieve the minimization of the denominator
 (13)
by choosing the coefficient to be
 (14)
Note the analogy between (14) and (6). Analogously to (7), equation (14) is equivalent to the orthogonality condition
 (15)
Analogously to (8), applying formula (14) is also equivalent to defining the minimized denominator as
 (16)

Next: Second step of the Up: IN SEARCH OF THE Previous: IN SEARCH OF THE
Stanford Exploration Project
11/12/1997