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The iterative reweighted least-squares (IRLS) algorithm provides a mean by
which linear systems can be solved by minimizing the *L*^{p} norm of the residuals
(). Let us suppose that we try to solve the set of equations:
To find the *L*^{p} solution of this system we minimize the *L*^{p} norm
of the residuals:
is not a norm for *p*<1, so, for the moment, I will only consider
. I also omit the exponent 1/*p* .
To establish the normal equations, we set:
The partial derivatives can be expressed as follows:
where are the residuals.
Then:
| |
(1) |

To form the normal equations we can now replace *r*(*i*) by its expression:
or, after some reorganization:
| |
(2) |

There are *m* such equations (*k*=1 to *m*). They can be expressed shortly in a
matrix formulation:
| |
(3) |

and finally:
| |
(4) |

This formulation is implicit and non-linear since the residuals *r*(*i*), and
so the matrix *W*, depend on the unknown vector *x*. However, for the classical
least-squares inversion (*p*=2), *W* is the identity matrix, and
equation (4) is the usual least-squares equation:
| |
(5) |

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Stanford Exploration Project

1/13/1998