anthony@utcsstat.UUCP (Anthony Ayiomamitis) (05/21/85)
Subject: Least-squares problem
I need some help in solving the model given below. Is it the case
that an iterative solution is the only solution or can one derive normal
equations for MTC and G such that one can compute them directly?
If iteration is the only means, which methods are good/bad? Would
an 8-bit microprocessor such as that on the Apple lead to many numerical
errors (truncation etc)?
I tried getting the least-squares solution but ended up with two
equations (as expected) with two unknowns (as expected) but with a number
of terms involving various combinations and powers of MTC and/or G within
each of the two equations (that's where I am stuck!!).
Problem: Find normal equations for MTC and G s.t. the following is minimized:
S = SUM [Cd(i) - Cdhat (i)] ^2 where
Cdhat(n+1) = delta t * [k1*k2 + k1*G*t(n) - (MTC + Qu + k1*Vd(n))*Cd(n)]/Vd(n) +
Cd(n)
where: k1 = (MTC + Qu * Tr) / (Vb + Kr*t)
k2 = Cb(0) * Vb + Cd(0) * Vd(0)
Qu = constant
Tr = constant
Kr = constant
Vb = constant
Vd(0) = constant
Cb(0) = constant
Cd(0) = constant
with the initial condition Cd(1) = Cd(0). A sample data set is the following:
t Vd(n) Cd(n)
- ----- -----
0 1983 1.7
15 2304 5.0
30 2543 7.0
45 2938 9.2
60 3146 10.3
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