Next: IRLS algorithm Up: Lp MINIMIZATION WITH THE Previous: Lp MINIMIZATION WITH THE

## Normal equations

The iterative reweighted least-squares (IRLS) algorithm provides a mean by which linear systems can be solved by minimizing the Lp norm of the residuals (). Let us suppose that we try to solve the set of equations:

To find the Lp solution of this system we minimize the Lp 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)

Next: IRLS algorithm Up: Lp MINIMIZATION WITH THE Previous: Lp MINIMIZATION WITH THE
Stanford Exploration Project
1/13/1998