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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) |

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Stanford Exploration Project

11/12/1997