hi all,
last night i thought up of a new way to solve the reconstruction
problem, that may involve a fewer integrations through the magnetic
field. i haven't gone through the current code, but from what i've
heard, we minimize the sum of square deviations of X=(x1,y1, x2,y2,
x3,y3) from the measured wire chambers as a function of P=(px, py, pz,
z).
i am thinking of attacking it as a root-finding problem, instead of
minimization, and using a multidimensional form of newton's method.
(root-finders are more efficient, aren't they?) assume that the funtion
X(P) is one-to-one, then we can make a course table of the function on a
grid over P, and calculate the derivate dX/dP. then we start with P_0
from a quickfit routine, calculate this curve's deviation dX from the
measured X_0, and use the matrix dX/dP to come up with the new guess
P_1. it should quickly converge to P_n=f^{-1}(X_0).
since the problem is over-determined, we could use the singular value
decomposition (SVD) of dX/dP to pull dX back to dP, which would yield a
least squares solution. i am hoping that by using apropriate prescales,
it would be sensitive to relative errors.
if the B field is fairly uniform, then grid could be quite couse. how
fine it is just affects how rapidly the solution converges. most of the
calculations could be performed just once to get this grid of SVDs.
this could also be interpolated to points inbetween.
after i get my TPdf program done, i'd like to try this out. none of
the existing integration routines whould have to change, and it seems
that some of parts could come from numerical recipies or nag.
any comments on this method?
--chris
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