Re: Exact or Least-Squares Solution 5 Equations
- From: spellucci@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx (Peter Spellucci)
- Date: Thu, 25 Oct 2007 09:40:27 +0000 (UTC)
In article <1193280435.133367.29540@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
monir <monirg@xxxxxxxxxxxx> writes:
Hello;
I've tried a number of techniques to solve 5 equations in 5 unknowns
with no avail!! I hope someone might have either come across a
similar problem or has the appropriate analytical tools and would be
kind enough to share his/her expertise.
Here's a brief discription of the problem.
5 UNKNOWNS: B, C, D, F, G
SQRT { (x2-F)^2 + (y2-G)^2 } = { a + B.t2 + C.t22 + D.t23 } Exp(m.t2)
SQRT { (x3-F)^2 + (y3-G)^2 } = { a + B.t3 + C.t32 + D.t33 } Exp(m.t3)
SQRT { (x4-F)^2 + (y4-G)^2 } = { a + B.t4 + C.t42 + D.t43 } Exp(m.t4)
SQRT { (x5-F)^2 + (y5-G)^2 } = { a + B.t5 + C.t52 + D.t53 } Exp(m.t5)
SQRT { (x6-F)^2 + (y6-G)^2 } = { a + B.t6 + C.t62 + D.t63 } Exp(m.t6)
All other quantities in the above equations are known.
Q1: What are the exact expressions for B, C, D, F, G ??
Q2: In the event that such exact expressions are defficult/impossible
to derive (I certaily hope not!), then what are the least-squares
regression formulas ??
Your expert help would be greatly appreciated.
Monir
this system looks a little bit strange:
on the right hand side we have a linear part in the three unknowns.
if you can pick here a submatrix of the 5 by 3 matrix formed by
[ t2 t22 t23 ]
...............
[ t6 t62 t63 ]
which has rank 3 you could solve the corresponding 3 equations for B,C,D
in terms of the sqrt... involving the unknowns F and G, insert in the other
two equations, with only F,G remaining as unknowns and then solving these
using e.g. Newtons method.
if this is impossible, then the system might have no solution at all and you need
to use nonlinear least squares minimizer to get a "quasisolution"
for this I would work in two stages:
first squaring both sides, then building the difference LHS-RHS, squaring again
and summing up you get a sum which is quartic in the 5 unknowns, miminimize
this by an appropriate code (e.g. ELSUNC) , then going back to the
orignal equation, building RHS-LHS squaring and summing up, doing the
same with the solution from the first step as an initial guess.
this because the least squares solution of the original system will not be
identical to that of the squared system (the optimal sum of squares will not be
zero in most cases) .
a quartic in 5 unknowns can also be quite hard to minimize, but the problem
involving the square roots might be even harder.
hth
peter
.
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