principles and applications of microearthquake networks

128 5. Inversiori niid Optimization
steepest descent direction. If 0- 0, 6x is the Gauss-Newton adjustment
vector. The idea is to guarantee a decrease in the sum of squares of the
residuals via steepest descent when x is far from the minimum, and to
switch to the rapid convergence of Newton’s method as the minimum is
approached. Marquardt ( 1963) improved upon Levenberg‘s idea by devising
a simple scheme for choosing H at each iteration. The Levenberg-
Marquardt method is also known as the damped least squares method and
has been widely used.
In the last two decades, numerous papers have been written on nonlinear
least squares. Readers may consult, for example, Chambers (1977),
Gill and Murray (1978), Dennis c’t til. (1979), R. Fletcher (1980), and Gill et
trl. (1981) for more detailed information.