Everyone,
I am taking an Intro to Least Squares class, working towards a BAS in Land Surveying at Great Basin College. It's been over 20 years since I had a college level math class, and linearizing equations is really giving me trouble.
Problem 2 for this week's home work is to linearize the equation A=LW. The variable to be used are: Lo=100'; Li=105'; Wo=150'; Wi=158'
for this example, Lo is (L zero), Li is (L at some point from Lo), same for W. (d) stands for the derivative of the variable following the (d).
I have arrived at this equation: A=LW+WdL(Li-Lo)+(Wi-Wo)
My answers are not coming out to be close to the original equation, which they should be somewhat close.
I am looking at several online tutoring services, as well as the tutors at the college. I've got to get this process, as the class is just going to get harder.
The next one that I think will give me trouble is_P=2L+2W; where Lo=105'; Li=105'; Wo=150'; Wi=158'
I have not worked through the problems to get to that one yet.
Any advice you can help with is greatly appreciated.
Jimmy
Hey I do not have an answer to your question, however I have a question for you. ?ÿI am looking into signing up for the BAS program there and wanted to ask you how you like it!? ?ÿI just applied online yesterday actually and I feel like it??s fate that you made this post and I came across it haha. ?ÿAnything you don??t like about it or anything you can suggest as far as taking it? ?ÿI am a registered LSI in Colorado and will be sitting for my PLS in October. ?ÿJust want to really further my career and knowledge in geomatics and geodesy. ?ÿ
Wolf & Ghilani text has some examples of linearizing equations, with enough of the steps given that you should be able to follow if you are acquainted with partial derivatives (derivatives with respect to one variable at a time).
I think there is an old edition somewhere on the web you can freely download.
In your example, I think you forgot the multiplier on the last part.
A ~ LW+?ÿ (Li-Lo) dA/dL + (Wi-Wo) dA/dW
Then dA/dL = W?ÿ?ÿ and?ÿ dA/dW = L to be evaluated at the initial point
A ~ LW+ (Li-Lo)Wo + (Wi-Wo) Lo
You can see that the linearization is only accurate somewhat near the initial point because it omits the other term in the expansion, (L1-L0)(Wi-W0)
?ÿNow @mathteacher can tell me if I got it right.
The only?ÿlinearization of equations I remember doing was with matrices.?ÿ I'd have to look up how to do it again but I thought it was just the derivative of the equation.
After you get a set of equations of the form above, you can turn them into a matrix * vector = vector equation.?ÿ The matrix elements are the coefficients of the variables, which are the derivatives as shown above.
?ÿ
Correct, @Bill93. Geometrically, this is the equation of a tangent plane to a hyperboloid. Since the hyperboloid is curved and the plane is straight, the approximation is exact at the tangent point but falls off in accuracy as the plane and the curved surface separate.
Perhaps, since two points were given, it would be better to use midpoints of L and W instead of one of the end points. That would be L = 102.5, W = 154, and A = 15,795. That's probably what I would do in a non-surveying application. Not exact at either end, but exact at the midpoint and, on average, closer over the entire domain, which is what least squares does.
In AP Calculus, we never went beyond two dimensions. In today's world of instantaneous exact calculations, I had to spend some time explaining why such approximations were important to know. In the old days, when we had to interpolate logs or trig functions, this was exactly the math that we were doing. My justification, with some chalk expended, was that these approximations let you get close by using only your head and that's valuable for a quick check of an answer.
Here's a University of Pennsylvania link that shows it well from a pure math perspective. ?ÿ https://www.math.upenn.edu/~rimmer/math115/ch15sc4.pdf
While using the midpoint would get you a little closer, what they are leading up to is the Least Squares solution method where?ÿ Lo, Wo is the current best estimate, you don't actually have an L1, W1,?ÿ and you are going to let the math adjust all the estimates to get a better set (vector).
Thanks everyone. For this exercise, we needed to use the L1 and W1. I'll take a look at the links.
I like the program. They are still using Dr. Elithorp's videos from 2009. The material has not changed, so that is okay. Some of the videos are not quite as easy to see on some of the whiteboard exercises, but overall, it is a good program. Byron Calkins is running the program now, and doing a good job. He is spread pretty thin, but does as good a job as you could ask. They are trying to get a second full time instructor, and are seeking ABET Accreditation.
This is my 3rd semester. I already have an AAS, so I am essentially taking the last two years, plus a few extra classes that are required. Some of the classes refer to older books that would be pretty helpful to have, especially if Dr. Elithorp's videos are still being used. Byron is trying to get updated videos done, but he is spread pretty thin.
Make sure you are fresh on your math. That is the hardest part for me. I graduated in 1996 with my AAS, so my algebra and calculus are rusty.
I do not regret my decision to enroll.