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 -- {allegra,ihnp4,linus,decvax}!utzoo!utcsstat!anthony {ihnp4|decvax|utzoo|utcsrgv}!utcs!utzoo!utcsstat!anthony