Chapter 6: LSA by Computer Algebra

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Chapter 6: LSA by Computer Algebra. 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

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Chapter 6: LSA by Computer Algebra
• 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)

Variation of Coordinates via Series Expansion

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
Fig. 3-13Example:

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

Hence f1 as a function of the unknown coordinates xis

(6.4)

• 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 **

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.
• http://www.wolfram.com/products/mathematica/trial.cgi
• CAS calculators
Resection example:
• 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
Generic procedure
• 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
Potential applications
• 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.