Hi Chi, and Michael,
Thanks for the thoughtful and detailed explanation, and for the
boost in confidence. As Chi said, I did investigations on the radius
of convergence, and include some of my results below. This was back
in June 2003, so your interpretation of the results is as good as
mine, but it looks like the radius is about 20 deg in theta and >200
MeV in momentum (but not very circular).
I agree with Chi about the merits of using the Newton-Rhapson
method in jumping over local minima to the real zero. Or think of it
this way: we are not just looking for any minimum in X^2, but for
where X=0 exactly. This is not quite correct, because I used a
modification of the Newton Rhapson method to solve the root of a
function into an 18 dimensional data space, where the solution maps
back onto some 4-dimensional subsurface 'image(P)'. In other words,
we have to project the hits onto the image of the track parameter
space to find a solution, and this is responsible for the small, but
non-zero X^2. I used a least-squares projection, 'J (J^t J)^{-1}
J^t', where J is the Jacobian. I think you can convince yourself
that the function 'P -> image(P)' is bijective using simple
geometric arguments; however, the more difficult case to consider is
if the projection is injective. It boils down to whether the
distance of the projection is smaller than the radius of curvature of
'image(P)'. Otherwise there will be multiple solutions (shortest
distance to the surface), and any reconstruction method will fail,
unless extra input is given with regards to the projection. Henry
Juengst from CLAS <juengst@moose.jlab.org> would be a good person to
talk to about that (multiple scattering matrices).
It may be worthwhile to go back and do a systemic check of the
convergence with real data, not MC. But first I would try to answer
questions like: why does it happen in one sector and not the other,
and not in Monte Carlo? Why do we also see the same thing in
straight-tracks, for which the image subsurface is linear? I suggest
first systematically accounting for geometrical and bfield effects,
making sure the time2distance function is correct and self-consistent
(for the local variations in reconstruction), and then returning to
convergence issues when we have more sensitivity.
For more details, see my thesis, http://blast.lns.mit.edu/
PUBLIC_RESULTS/THESES/Crawford_thesis_revA.pdf pp. 92-95 (also pp.
161-163), and Numerical Recipes, sec. 9.7, p. 383, http://
library.lanl.gov/numerical/bookcpdf/c9-6.pdf, for the complete
algorithm except for the projection.
--Chris
_______________________________________
TA-53/MPF-1/D111 P-23 MS H803
LANL, Los Alamos, NM 87545
505-665-9804(o) 665-4121(f) 662-0639(h)
_______________________________________
* check convergence of newton fitter
void TBLRecon::FitDump() const
...
cout<<" [";
for (int i=0;i<18;i++) cout<<fTracks[itrk]->fFit->GetData()[i]
<<" ";
cout<<"]"<<endl;
}
+ bugs
TBLFitTrack:
move 'fSum2/=2' after loop; was weighting inner point by 2^-18
add
nw_conv.C:
forgot 'newt.SetPidGeneric(1)'
fit in wc only:
not adjusting phi correctly
+ convergence
- proton
track: proton [0.343377 68.6376 172.486 -0.474141 0 -1.31836]
dp = (.70-.04)/2 = .32 MeV
dt = (93 -18)/2 = 38 deg (whole range of wc)
df = (196-163)/2 = 17 deg (whole range of wc)
dz = (200-(-14))/2 = 105 cm (whole range)
0 theta 20 40 60 80 100
120 140
+---------+---------+---------+---------+---------+---------
+---------+-----
0.00: ..................................................................
..........
0.02: .......xxx.x.x...xxxxxxxx....x.....x............x.......xxx..xx.x.
..........
0.04:
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxx.xx.xxxxxxx.....xxx..xxxx.xx.xxx.x..xx..xx
....
0.06:
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx..xxxxx...x..xxx.x..x..xxx...x.x..
xx.x
0.08:
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx...x.x..x..x......x....x...x...
x...
0.10: .xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx........xx..x..x.x.xx..x
..x...x...
0.12: .x.xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx........xx.x...x..xx...
..........
0.14: .x..xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx...x...x.......x..x.x
..........
0.16: ..x.x.xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx........x..x..........
....xx....
0.18: ...xxx.xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx.......xx...x..x...xxx
..........
0.20: ....xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx.......x...x.......x.
..........
0.22: ......xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx..x....x.....x..x..x.
.x.......x
0.24: ......xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx.......x.......x.....
..x.x.....
0.26: .......xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx...........x........
.....xx...
0.28: ..........xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx.........x.xx...x...
..........
0.30: ........xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx.......x.x.x..x.....
.......x..
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0.36: ...........xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx.......x............
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......x..x
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......x...
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0.74: ......................................x...........................
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........x.
0.98: ..................................................................
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1.00: ..................................................................
..........
- zoom
0 theta 10 20 30 40 50
60 70 80 90 100 110 120
p +---------+---------+---------+---------+---------+---------
+---------+---------+---------+---------+---------+---------+
0.00: ..................................................................
.......................................................
0.02: ..............o.ooo..oo...o......oo.ooo.ooooooooo........oo.......
.o..oo.o.......o..............o..o............ooooo..o.
0.04:
oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo..ooo...ooo.
oooooooooo.........oo.oooo.o.oo.ooo.oo..ooo...ooo
0.06:
oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
ooo..o..ooo.o.o.oo......oo....oooooo..o.....oo...
0.08:
oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
ooo.ooooo..o.o..o...o.....oo....o..o.....o....oo.
0.10: ..oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
oooooooooooooooooo.......o.o......ooo..o..o..o.oo..ooo.
0.12: ..o...oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
oooooooooooooooooooo...............oooo...o......oo..o.
0.14: .oo..o..oooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
ooooooooooooooooooooo.......o......ooo...o.o..........o
0.16: ...oo..oo..ooooooooooooooooooooooooooooooooooooooooooooooooooooooo
oooooooooooooooooooooo.....o.........oo.....o..........
0.18: ....oo.oooo.ooooooooooooooooooooooooooooooooooooooooooooooooooooo
oooooooooooooooooooooo.o............ooo......oo..o..oo.
0.20: .......ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
ooooooooooooooooooooooo...............oo......o........
0.22: .........o.oooo.oooooooooooooooooooooooooooooooooooooooooooooooooo
oooooooooooooooooooooooo....o.........oo.........oo....
0.24: ............oooooooooooooooooooooooooooooooooooooooooooooooooooooo
oooooooooooooooooooooooo..............o....o...o...o.oo
0.26: ..............oooooooooooooooooooooooooooooooooooooooooooooooooooo
ooooooooooooooooooooooooo..............o.....o..o......
0.28: ...............o.o.ooooooooooooooooooooooooooooooooooooooooooooooo
ooooooooooooooooooooooooo..............o....oo..o.oo...
0.30: ................oooooooooooooooooooooooooooooooooooooooooooooooooo
ooooooooooooooooooooooooo..............ooo..o...oo....o
0.32: .................o.ooooooooooooooooooooooooooooooooooooooooooooooo
ooooooooooooooooooooooooo.....o.......o.oo.........o...
0.34: ..................oooooooooooooooooooooooooooooooooooooooooooooooo
oooooooooooooooooooooooooo.............................
0.36: ...................o.ooooooooooooooooooooooooooooooooooooooooooooo
oooooooooooooooooooooooooo..............o..............
0.38: ....................oooooooooooooooooooooooooooooooooooooooooooooo
oooooooooooooooooooooooooo............o...........o....
0.40: .....................ooooooooooooooooooooooooooooooooooooooooooooo
oooooooooooooooooooooooooo....................o........
0.42: ......................oooooooooooooooooooooooooooooooooooooooooooo
oooooooooooooooooooooooooo.........o......o...o..o.o...
0.44: .......................ooooooooooooooooooooooooooooooooooooooooooo
oooooooooooooooooooooooooo.o...........o......oo.......
0.46: .......................ooooooooooooooooooooooooooooooooooooooooooo
oooooooooooooooooooooooooo...................o......o..
0.48: ........................oooooooooooooooooooooooooooooooooooooooooo
oooooooooooooooooooooooooo...........................o.
0.50: ........................oooooooooooooooooooooooooooooooooooooooooo
ooooooooooooooooooooooooooo............................
- electron
v: 4 35.493 (0) [0.723538 48.8462 15.4827 1.07487 0 -15.5684]
h: [-28.5758 -29.0597 -29.5443 -35.687 -36.2214 -36.7572 -50.0634
-50.6159 -51.1701 -59.3749 -59.9799 -60.5864 -74.4219 -75.0391
-75.6577 -85.8569 -86.5232 -87.1907]
d: [-28.7046 -29.0748 -29.5742 -35.6063 -36.17 -36.6448 -50.0209
-50.6582 -51.2903 -59.3552 -59.9918 -60.6023 -74.2662 -74.8876
-75.7872 -86.0127 -86.3443 -87.3331]
0 theta 20 40 60 80 100
120 140
p +---------+---------+---------+---------+---------+---------
+---------+-----
0.00: .................................................................
..........
0.02: ...................o..............................................
..........
0.04: ..................................o...o.o.........................
..........
0.06: .............oo........o........oooooooo..........................
..........
0.08: ...........o.........o.oo....ooooooo.o............................
..........
0.10: ...........o............o.oooooooo.oo.............................
..........
0.12: .....................ooo..oooooo.oo.oo............................
..........
0.14: ......................oooooo..ooooo.o.............................
..........
0.16: ...................ooooo.oooooo.ooo..o............................
..........
0.18: ..............o...ooooooooooooooooo.oo............................
..........
0.20: .............o...oooooooooooooooooo..oo...........................
..........
0.22: ............o.ooooo.ooooooooo.oooooo.o............................
..........
0.24: ..............oo.oo.ooo.oooo.oooooooo..o..........................
..........
0.26: ...........o..oooooooo.ooooooo.oooo...............................
..........
0.28: .............ooooooooooooooo.ooo...o..............................
..........
0.30: ...........oooooooooo.ooooooooooo..o..............................
..........
0.32: .........o..oo...ooooo.oooooooooo..o..............................
..........
0.34: ..........ooo.o.ooooo.oooooooo.oo.o..............................
..........
0.36: ...........ooo...oooooooooooooo.oooo..............................
..........
0.38: ...........ooo.o.oooooooooo.oooooo.o...o..........................
..........
0.40: .........o.ooo.o.oooooooooooooooo..o..............................
..........
0.42: .........o.oo..o.oooooooooooo.ooo..o..............................
..........
0.44: .......o.o.oo....ooooo.oooooo.oooooo..............................
..........
0.46: .........o.ooo..oooooo.oooooo.ooo.................................
..........
0.48: .......ooo.oo...oooooo.oooooo.ooo.o...............................
..........
0.50: ........oo..o..ooooooo.oooooo.ooo.................................
..........
0.52: .....oo.oo.oo.o.ooooooooooooo.ooo.................................
..........
0.54: .........o.oo.oooooooo.oooooo.ooo.................................
..........
0.56: ........oo.oo..ooooooo.oooooo.o...................................
..........
0.58: .....o.....oo..ooooooo.oooooooo.o.................................
..........
0.60: .........o.ooo.ooooooo.oooo..oo.o...o.............................
..........
0.62: .....oo.o..ooo.oooooooooooo..oo.o...o.............................
..........
0.64: .......ooooooo.oooooooooooo..oo.o...o.............................
..........
0.66: ...oo..ooo.ooo.oooooooooooo.........o.o...........................
..........
0.68: .....o.ooo.ooo.oooooooooooo..o..o...o.............................
..........
0.70: .....o..oo.oooooooooooooooo..o........o...........................
..........
0.72: .....o..oo.oooooooooooooXoo..o..o.....o...........................
..........
0.74: .....o.ooo.oooooooooooooooo..o..o..o..............................
..........
0.76: ........ooooooooooo.ooooooo..o..o..o..............................
..........
0.78: .......ooo.oooooooooooooooo..o..oooo..............................
..........
0.80: .......ooo.oooooooooooooooo..o.ooooo..............................
..........
0.82: ..o......o.oooooooooooooooo..o.ooooo..............................
..........
0.84: .......ooo.oooooooooooooooo.oo.ooo.o..............................
..........
0.86: .......ooo.oooooo..o..oooooooo.ooooo..............................
..........
0.88: ....o..ooo.ooo.oo.oo.ooooooooo.oooo...............................
..........
0.90: ....o..ooo.oo...o.oo.ooooooooo.oooo...............................
..........
0.92: ....oo.ooo.oo...o.oo.ooooooooo.oooo...............................
..........
0.94: ...oo....o.ooo..o..o.ooooooooo.o.oo...............................
..........
0.96: .......oo..ooo..o.oo.ooooooooo.oo.o...............................
..........
0.98: .......ooo.ooo..oooo.ooooooooo....o...............................
..........
1.00: ........oo...o..oooo.ooooooooo....o...............................
..........
+ start at wc
- proton
dp = (1.44-.06)/2 -> double upper limie
dt = (110-24)/2 -> shifted up
df = (173) but if p=.30: (179.6-168.5) -> bug?
dz = (200--12) -> same
- electron
dp = (1.52-0.10) -> good
dt = (95-18) -> but with scattered holes???
df = (17--3.5) -> much better than proton
dz = (140-10) -> with many scattered holes???
0 theta 20 40 60 80 100
120 140
p +---------+---------+---------+---------+---------+---------
+---------+-----
0.00: ..................................................................
..........
0.02: ................++.+.+.+.+..++.....+....
+...................................
0.04: ...........++.++..+.+++.+.......+........+.........+......++.+++
+++++++..++.
0.06: ...+.....+.++.+++.+++.+++++.+...............++.+++++++++++++++++
++++.+++..++
0.08: ....++...++.+++.++....+..++...++..+..+++++++++++++.+++.++++++++
+.+.++......+
0.10: ....++.++.+.+..++.++++.+.++++...+.+++++++++++++++++++++++++++..
+...+..+...+.
0.12: ...+..+...++.+++.+++...++++.+++++++++++++++++++++++++++++++++.
+......+...+++
0.14: .+.........+++.++.++++++++++++.+++++++++++++++++++++++++++.+.++
+.+...+.+++++
0.16: ....+......++.++.++++++++++++++.++++++++++++++++++++++++++.+.++.
+......+++++
segfault (bgrid)
- electron w/smaller phi
dp = (1.48-.24)
dt = (50-12)
df = (192-173)
dz = (174-140)
+ nonconvergence
run 1070, event 8, nw 0
v={ 0.768798, 27.6792, -169.675, x=-59.1049, y=-10.7683,
z=110.499 }
small problem range, z=103.993 -> 104.102
right in middle of large converge region
initial derivative is perpendicular to minimum
echo;root -l nw_conv.C Newt.PrintFlag=4 +o2 +z +plotd
echo;root -l nw_conv.C +o2 NewtDer.dP=0.02 NewtDer.dTheta=0.002
NewtDer.dPhi=0.01 NewtDer.dZ0=0.1
fixed it!
echo;root -l nw_conv.C +o2 NewtDer.dZ0=0.1
also works
echo;root -l nw_conv.C +o2 NewtDer.dZ0=0.1 +z
another bad line
120.000: 4 54.75 [0.768818 29.0105 -170.307 -0.577733 0 1.49155]
122.000: 4 474108.51 [0.768798 29.1969 -170.309 -0.552348 0 20.0135]
124.000: 4 54.75 [0.768793 29.0102 -170.307 -0.577737 0 1.49055]
echo;root -l nw_conv.C +o2 NewtDer.dZ0=0.5 +p +t +f +z
converges: .22<p=.76<1.48, 12<t=29<50, 173.5<f=170<192.5,
-30<z=110<186
still problem w/phi
NewtDer.dPhi=0.02
didn't help
todo: test stability of derivative (w/decreasing p,th,phi,z)
- electron w/smaller phi
converges: .22<p=.76<1.48, 12<t=29<50, 173.5<f=170<192.5,
-30<z=110<186
0 theta 20 40 60 80 100
120 140
p +---------+---------+---------+---------+---------+---------
+---------+-----
0.16: ......^.......^.....^^.^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^.^....^^^^^.
^..^.:....
0.18: .....^.^.........^^^...^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^......^^..^^
^.^.......
0.20: ..........^...^^.^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^.^.........^...^
.^^.^:....
0.22: ...^^^^.....^.^.^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^............^..
.....^.^..
0.24: ....^..^....^..^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^.................
.........:
0.26: ...^.^.^....^.^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^.^^................^.
..^....:^.
0.28: ...^......^^..^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^...................
....^.....
0.30: .^..^^.^.^^..^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^....................
..........
0.32: .......^^^..^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^.....................
..........
0.34: .....^.^..^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^......................
..^.....:.
0.36: ...^...^^..^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^.......................
..........
0.38: ........^..^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^........................
.......^..
0.40: ..^...^.....^^^^^^^^^^^^^^^^^^^^^^^^^^^^^.........................
......:...
0.42: ......^....^^^^^^^^^^^^^^^^^^^^^^^^^^^^^..........................
..........
0.44: ...^^.^^.^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^...........................
........^.
0.46: .^......^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^............................
..........
0.48:
^......^.^^^^^^^^^^^^^^^^^^^^^^^^^^^^...................................
....
0.50: .^^...^.^^^^^^^^^^^^^^^^^^^^^^^^^^^^..............................
..........
0.52:
^^....^..^^^^^^^^^^^^^^^^^^^^^^^^^^.....................................
....
0.54: ......^.^^^^^^^^^^^^^^^^^^^^^^^^^^................................
.........:
0.56: .....^..^^^^^^^^^^^^^^^^^^^^^^^^^.................................
......^...
0.58: .....^..^^^^^^^^^^^^^^^^^^^^^^^^..................................
..........
0.60: ........^^^^^^^^^^^^^^^^^^^^^.^...................................
..........
0.62: .^...^.^.^^^^^^^^^^^^^^^^^^^^^^...................................
..........
0.64: .....^.^^^^^^^^^^^^^^^^^^^^^^^....................................
..........
0.66: .^...^.^^^^^^^^^^^^^^^^^^^^^^.....................................
..........
0.68: .....^.^^^^^^^^^^^^^^^^^^^^^......................................
..........
0.70: .....^.^^^^^^^^^^^^^^^^^^^^^......................................
.........^
0.72: .....^^.^^^^^^^^^^^^^^^^^^^.......................................
..........
0.74: .^...^^^^^^^^^^^^^^^^^^^^^........................................
.....^....
0.76: .^..^^^^^^^^^^^X^^^^^^^^^^........................................
..........
0.78:
^....^^^^^^^^^^^^^^^^^^^^...............................................
.^..
0.80:
^....^^.^^^^^^^^^^^^^^^^^........................^......................
....
0.82:
^.......^^^^^^^^^^^^^^^^................................................
....
0.84: .^....^^^^^^^^^^^^^^^^^^..........................................
..........
0.86: ....^.^^^^^^^^^^^^^^^^^...........................................
.........^
0.88: ....^^^^^^^^^^^^^^^^^^^...........................................
.........
0.90: ......^^^^^^^^^^^^^^^^............................................
..........
0.92: ....^^^^^^^^^^^^^^^^^^............................................
..........
0.94: .^..^.^^^^^^^^^^^^^^^^............................................
..........
0.96:
^...^^^^^^^^^^^^^^^^^...................................................
....
0.98:
^^..^^^^^^^^^^^^^^^^^...................................................
....
On Feb 27, 2006, at 21:21:39, Chi Zhang wrote:
>
> Hi Michael,
>
> This subject has been discussed in early 2003 when Chris
> painstakingly convinced us (Tim, Tong, me and some others?) that
> the Newton method locates the global minimum. It is also documented
> in Numerical Recipy that "Newton method tends to pass right by the
> local minima". It s only a couple of pages. I still remember that I
> first made the comment that maybe we are looking at local minima
> and Tong got exited and called for emergency software meetings
> where Chris drew a lotta stuff on the board. :)
>
> I think Chris is the best person to comment on this as the
> implementer of Newton for BlastLib. But my understanding is that
> Newton is not a minimization method on any chi^2, but a "root"
> finding method on chi itself. Please see Numerical Recipy for a
> description of the algorithm.
>
> One extremely simplified example would be like this. Say if the
> track parameters(p,theta,phi,z) are so wrong that the track impact
> all wire planes on the upstream side of the hits. Compare to the
> other extreme that all impact points are on the down stream side of
> the hits. These two cases clearly defines chi=+(a big value) and
> chi=-(a big value) case. So if there is only one paramter to fit
> and only one hit to fit from, Newton will find the exact parameter
> s.t. chi=0. Higher dimension will be harder but to sweep from
> +very_big to -very_big, Newton won't even pause around any local
> minima unless its a root of the equation chi=0. if this thing has
> redundant roots, then good luck finding any algorithm.
>
> I believe Chris took great care to treat the exceptional cases and
> the version he implemented is in fact a Newton-Ralpson with Back
> Track.
>
> I do not believe the problem is in software, we get every clean and
> nice in Monte Carlo yet everything breaks down in data. It amuses
> me that we decided we can do what CLAS and BarBar did with 18
> layers of wires while those two have 36 and 52 layers in a much
> denser layout. The pain of not having the extra grips to
> calibration and debugging just hurts.
>
> I do not know how to produce those plots. I believe Chris prepared
> chi2 graphs for us three years ago when we had not even start
> tagging anything yet. I remember codes in Chris' Newton Fitter that
> dumps a great deal of info for diagnose purpose. I can not
> remember, but I doubt we retained all those codes when we moved to
> v3 for run time concern. We thought it was a resolved issue anyway.
>
> Chi
>
> On Mon, 27 Feb 2006, Michael Kohl wrote:
>
>> Hi Chris, Chi,
>>
>> is it in principle possible that the applied method of the Newton
>> fitter ends up in a local minimum of the chi2 function, i.e. the
>> solution found for p may not be a valid root?
>> Or, could it happen that multiple roots exist for p close to each
>> other and the one usually found may not be the one one was looking
>> for?
>>
>> It would be great if you could explain me the algorithm and its
>> pitfalls that is used in the Blast library. It would be even
>> better if we could talk about this in the analysis meeting on
>> Wednesday.
>>
>> An important plot would be the graph of chi2 versus p for the
>> given "solution" of the other variables and where p is being
>> varied over some sufficiently large range, in order to see wether
>> the found solution of p indeed minimizes chi2 globally but not
>> locally. Can you provide such a plot for typical elastic events
>> where the momenta are off by 30-50 MeV?
>>
>>
>> Best regards
>>
>> Michael
>>
>>
>>
>> +-------------------------------------+--------------------------+
>> | Office: | Home: |
>> |-------------------------------------|--------------------------|
>> | Dr. Michael Kohl | Michael Kohl |
>> | Laboratory for Nuclear Science | 5 Ibbetson Street |
>> | MIT-Bates Linear Accelerator Center | Somerville, MA 02143 |
>> | Middleton, MA 01949 | U.S.A. |
>> | U.S.A. | |
>> | - - - - - - - - - - - - | - - - - - - - - -|
>> | Email: kohlm@mit.edu | K.Michael.Kohl@gmx.de |
>> | Work: +1-617-253-9207 | Home: +1-617-629-3147 |
>> | Fax: +1-617-253-9599 | Mobile: +1-978-580-4190 |
>> | http://blast.lns.mit.edu | |
>> +-------------------------------------+--------------------------+
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