I expect this to be the first in a series of articles detailing geometric least squares. By geometric least squares I mean fitting measurements to equations, or coordinates to equations or a combination there of. Getting started we will assume that the user has a background in Calculus and Linear Algebra. A recent site that I have found is http://www.khanacademy.org/ , it has some excellent resources. In college, many years ago we studied this text http://www.amazon.com/Adjustment-Computations-Statistics-Surveying-Boundary/dp/0471168335/ref=sr_1_4?ie=UTF8&qid=1295317634&sr=8-4
Working the problems, the generic problems can be worked with MathCAD, Smath http://www.smathstudio.com/ , or another computer algebra system. If things go far enough along I will show how to use open source libraries to do the matrix math.
First step is to clearly define what you know about the condition, what you need to know and how they are related. Then you create a matrix, J. J is the constraint matrix as defined below. Basically each row represents a condition. Each Column represents an independent variable. Each cell in J represents the first derivative of the observation equation, in regard to the row and column of said cell… see the sketch.
After defining the observation conditions, make initial estimates of the variable values and solve the equations. Subtract the Calculated Values from the Measured Values and call it Column Vector, k. Make sure the rows of J, correspond to the correct row of k.
IMPORTANT : The number of observations, rows, MUST be greater than the number of variables, columns.
You MUST have more independent measurements that dependent variables.
Solve by ->>>>>>>
Where X will result in a column vector containing as many rows as Matrix J. The elements of X are correction values to the assumed values. Apply these corrections like : a= a + X, b = b + X, … etc. The process is iterative, recalculate Matrix J and k, solve for X again. With any luck the correction values in X should converge towards zero, if they don’t you either have something wrong or you have bad data. If you don’t have enough observations to give an extra degree of freedom then you will have a singular matrix.
I learn better by example where I can see the actual math, so here is points fitted to a parabola. Enjoy. Comments welcome and requested.