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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Taylor-series expansion of vector functions
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:
x = LS solutionfor coordinates, e.g.
x = [EU, NU]T in Section 3.5.2 (n = 2);
f1 = calculated angle A-U-B in Fig. 3-13, where
Hence f1 as a function of the unknown coordinates xis
x = k (6.7)
Fig. 6.1 Improving provisional coordinates by (approximate) x