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** Up:** IN SEARCH OF THE
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Now let us assume *n* > 2 and add some amount of the step from
the (*n*-2)-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.

** Next:** Induction
** Up:** IN SEARCH OF THE
** Previous:** First step of the
Stanford Exploration Project

11/12/1997