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Also there is a method for controlling Adaptive filters with this name!
as Humungus said, it is a conventional method for linear regression and curve fitting, an important topic in statistic/math.
with this and other method of curve fitting it is possible to model systems from exprimental(measured) values of them. thus this method is widely used in device modeling(MOS,BJT,Caps,Inductors,...) in physics and other Advanced technologies.
I think the merit of this method in respect to other methods is simplicity and availability of analytical solution for it; also several functions which are not linear(such as log, exp) can be converted to linear function by some transforms.
To expand on the previous good points, in LMS the difference between the data points and a continuous math function that is going to be used to approximate the data points are squared. This is the squared error. Then the sum of the squared errors at all of the data points is made. The parameters of the math function are varied until this sum of errors is made the smallest.
Even though it is easy to compute and find it has some problems. One is that large errors count more than small ones and positive ones count the same as negative ones. This may not be what you need in your approximation.