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Chapter 6: LSA by Computer Algebra

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- CAS: Computer Algebra Systems
- ideal for heavy yet routine analytical derivation (also useful for numerical/programming tasks);
- independent method to check spreadsheet results

- Mathematics involved:
Taylor-series expansion of vector functions

- Analytical, calculus-based theory of LSA

Taylor Series Expansion

Taylor’s Theorem: gives approximation of f(x) at xnearx0

where x = x0 + x. Requires:

(i) values of f & (various) f’, both evaluated at x0, and

(ii) small quantities x:

f(x) f(x0 + x) = f(x0) + + H.O.T. (6.1)

H.O.T. = “Higher Order Terms”

To approximate m multi-variate functions f1(x), f2(x),…, fm(x): view collectively as components of vector function f(x), then

f(x) = f(x0 + x) = f(x0) + x + H.O.T. (6.2a)

Define: Aij = (6.2b)

Resection w/ redundant targets: measured: many (m) angles

Objective: obtain the best set of (n) coordinates (i.e. E, N) for unknown station(s), that will fit the mobserved data as closely as possible.

Assume: m > n.

Arrange observed data into column vector:

=

- Apply least-squares (LS) condition:
- q f(x)
x = LS solutionfor coordinates, e.g.

x = [EU, NU]T in Section 3.5.2 (n = 2);

f(x):

- Computed version of measured angles or/and
distances

- Computed using values of the (best) coordinates x

- Computed version of measured angles or/and

Fig. 3-13

f1 = calculated angle A-U-B in Fig. 3-13, where

Hence f1 as a function of the unknown coordinates xis

(6.4)

- How to find the best solution x?Utilize the fact: x = x0 + x
- x0 = (any) approximate solution. Thus
- f(x0 + x)
- Apply 6.2(a)(b):f(x0) + x + H.O.T.
- Hence, x – [ – f(x0)] + H.O.T. 0 (6.5)
- Note: xis the only unknown in this problem
- Rephrasing (6.5):
- Minimize || x – k + H.O.T. ||2 , where k – f(x0)
- (weighted problem, weight matrix w)
- ** If we modified a problem very slightly (dropping H.O.T.) then
- the solution should only differ slightly **

- First obtain approx. solution (really: minimizes ||Ax – k||2):
x = k (6.7)

- Solution improved to xnew = x0 + x (6.8)
- This updated (still approximate) solution: provides a new (better) “x0“

Fig. 6.1 Improving provisional coordinates by (approximate) x

- Use new x0 to repeat procedure until convergence is met

- Calculation of derivatives (6.2b) for matrix elements Aij:
- (i = 1 to m, j = 1 to n)
- By hand: lengthy (m can be >> 1; n also) & error-prone
- Symbolic expression to be numerically evaluated repeatedly by substituting x0; also for k= – f(x0)
- Seek help from CAS tools
- Maple V, Mathematica (“Mtka”), REDUCE, DERIVE, MACSYMA, MuMath, MathCAD, etc.
- URL for free Mtka download (save-disallowed):
- http://www.wolfram.com/products/mathematica/trial.cgi
- CAS calculators

- Download and install trial version of Mtka
- Use program enclosed in CD-ROM
- Open the file resection.mb with Mtka
- Press Shift + Enter to run each line
- Results should agree with Solver results in Ch. 3

- Define unknowns = x (n x 1)
- Put “observed data” into q (m x 1)
- Prepare computed versions of qas f(x) (m x 1)
- Prepare Aij= D[fi,xj] (m x n) (symbolic)
- (Reasonable) provisional solution = x0
- k= – f(x0); A -> A(x0) (numerical now)
- x = (ATWA)-1ATWk
- Updatex0tox0+Dx; repeat from step 6 until solution converges

- Recovering missing parameters of a circle using (4 or more) observed points
- Locating the center, major & minor axes of an ellipse by observed points
- Parameters of a comet trajectory using observed data
- Etc.