1 Chapter 6: LSA Chapter 6: LSA by CAS  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 x near x 0 where x = x 0 +  x. Requires: (i) values of f & (various) f’, both evaluated at x 0, and (ii) small quantities  x: f(x)  f(x 0 +  x) = f(x 0 ) + + H.O.T. (6.1) H.O.T. = “Higher Order Terms” To approximate m multi-variate functions f 1 (x), f 2 (x),…, f m (x): view collectively as components of vector function f(x), then f(x) = f(x 0 +  x) = f(x 0 ) +  x + H.O.T. (6.2a) Define: A ij = (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 m observed data as closely as possible. Assume: m > n. Arrange observed data into column vector:  =

Apply least-squares (LS) condition:   f( x ) x = LS solution for coordinates, e.g. x = [E U, N U ] T in Section (n = 2); f( x ):  Calculated version of measured angles or/ and distances  Computed using values of the coordinates x (what’s the best x?)

f 1 = calculated angle A-U-B in Fig. 3-13, where Hence f 1 as a function of the unknown coordinates x is Fig (6.4) Example:

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

First obtain approx. solution (really: minimizes ||A  x – k|| 2 ) :  x = k (6.7) Solution improved to x new = x 0 +  x (6.8) This updated (still approximate) solution: provides a new (better) “x 0 “ Fig. 6.1 Improving provisional coordinates by (approximate)  x Use new x 0 to repeat procedure until convergence is met

Calculation of derivatives (6.2b) for matrix elements A ij : (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 x 0 ; also for k =  – f(x 0 ) Seek help from CAS tools Maple V, Mathematica (“Mtka”), REDUCE, DERIVE, MACSYMA, MuMath, MathCAD, etc. URL for free Mtka download (save-disallowed): CAS calculators

9 Resection example: 1.Download and install trial version of Mtka 2.Enter Program 6.1 given in book 3.Press Shift + Enter to run each line 4.Results should agree with Solver results in Ch. 3

10 Generic procedure 1.Define unknowns = x (n x 1 vector) 2.Put “observed data” into  (m x 1 vector) 3.Prepare computed version of  as f(x) (m x 1) 4.Prepare A ij = D[f i,x j ] (m x n) (symbolic) 5.Prepare a reasonable provisional solution, x 0 6. k =  – f(x 0 ); A -> A(x 0 ) (now numerical) 7.  x = (A T WA) -1 A T Wk 8.Update x 0 to x 0 +  x; repeat from step 6 until solution converges

11 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.