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Recall the fitting goals (10)
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(10) |

Without preconditioning we have the search direction
| |
(11) |

and with preconditioning we have the search direction
| |
(12) |

The essential feature of preconditioning is not that we perform
the iterative optimization in terms of the variable .The essential feature is that we use a search direction
that is a gradient with respect to not .Using we have
.This enables us to define a good search direction in model space.

| |
(13) |

Define the gradient by and
notice that .
| |
(14) |

The search direction (14)
shows a positive-definite operator scaling the gradient.
Each component of any gradient vector is independent of each other.
All independently point a direction for descent.
Obviously, each can be scaled by any positive number.
Now we have found that we can also scale a gradient vector by
a positive definite matrix and we can still expect
the conjugate-direction algorithm to descend, as always,
to the ``exact'' answer in a finite number of steps.
This is because modifying the search direction with
is equivalent to solving
a conjugate-gradient problem in .

** Next:** NULL SPACE AND INTERVAL
** Up:** Preconditioning
** Previous:** The preconditioned solver
Stanford Exploration Project

4/27/2004