Linear optics measurement and correction - I 115
respectively (note that J
T
i J i is a scalar). In the original form, the coefficient
λ needs to serve the role of a scaling constant with the dimension of p
−2
i
for the i’th parameter. Since the parameters may have different dimensions,
using a single coefficient for all parameters may not be appropriate. Multiplying each term in the cost function by a scaling constant, e.g,
1
σ 2
p i
, would be
necessary (with σ pi being the expected level of uncertainty of parameter p i ).
On the other hand, the λ coefficient in the scaled (i.e, modified) L-M method
is a dimensionless number independent of the parameters or the least-square
problem. Therefore, it is more convenient to use the scaled L-M method.
It is important to note that for the sake of preventing the final solution from
taking large excursions to the under-constrained directions, the λ coefficient
cannot be allowed to decrease to near zero [50]. If λ is too small, the constraints
disappear and the solution may again acquire large ∆K for little reduction
of χ
2 . Based on the acceptable level of
∆K
k , for any given storage ring, a
minimum value of λ may be specified; or simply a constant λ can be used.
For the scaled L-M method, λ = 0.001 is often a good starting point for the
initial trial of λ for the orbit response matrix fitting problem.
Constraints over combinations of parameters:
Constraints can also be applied in ways other than adding individual ∆K
terms to the χ
2 definition, as was done in Eq. (4.37) or through the L-M methods. For example, if we know a combination of quadrupole parameters are not
constrained well by the optics data, a constraint can be imposed to require
the stray of the solution in the corresponding direction to be the minimal.
A common under-constrained combination is for the gradient of two adjacent
quadrupoles to go in opposite directions, if the betatron phase advances between the quadrupole magnets are small. Suppose quadrupole i and i + 1 are
such a pair of neighboring magnets, the term
w
2
σ 2
K
(L i ∆K i − L j ∆K j )
2 ,
can be added to the χ
2 definition to prevent large excursions in the direction
represented by L i ∆K i −L j ∆K j = 0, where L i and L j are the effective lengths
of the two magnets, respectively. The residual vector and the Jacobian matrix
are then similarly extended, with the difference being that the rows of the
Jacobian extension, W, consists of the coefficients that define the direction,
e.g., the row for the above constraint term (say, the n’th row of W) would be
W n: =
w n
σ K
(0, · · · , L i , 0, · · · , 0, −L j , 0, · · · ).
A general under-constrained direction can be described as
Nq
i=1
c i ∆K i = 0,
or c
T ∆K = 0,
(4.43)
respectively (note that J
T
i J i is a scalar). In the original form, the coefficient
λ needs to serve the role of a scaling constant with the dimension of p
−2
i
for the i’th parameter. Since the parameters may have different dimensions,
using a single coefficient for all parameters may not be appropriate. Multiplying each term in the cost function by a scaling constant, e.g,
1
σ 2
p i
, would be
necessary (with σ pi being the expected level of uncertainty of parameter p i ).
On the other hand, the λ coefficient in the scaled (i.e, modified) L-M method
is a dimensionless number independent of the parameters or the least-square
problem. Therefore, it is more convenient to use the scaled L-M method.
It is important to note that for the sake of preventing the final solution from
taking large excursions to the under-constrained directions, the λ coefficient
cannot be allowed to decrease to near zero [50]. If λ is too small, the constraints
disappear and the solution may again acquire large ∆K for little reduction
of χ
2 . Based on the acceptable level of
∆K
k , for any given storage ring, a
minimum value of λ may be specified; or simply a constant λ can be used.
For the scaled L-M method, λ = 0.001 is often a good starting point for the
initial trial of λ for the orbit response matrix fitting problem.
Constraints over combinations of parameters:
Constraints can also be applied in ways other than adding individual ∆K
terms to the χ
2 definition, as was done in Eq. (4.37) or through the L-M methods. For example, if we know a combination of quadrupole parameters are not
constrained well by the optics data, a constraint can be imposed to require
the stray of the solution in the corresponding direction to be the minimal.
A common under-constrained combination is for the gradient of two adjacent
quadrupoles to go in opposite directions, if the betatron phase advances between the quadrupole magnets are small. Suppose quadrupole i and i + 1 are
such a pair of neighboring magnets, the term
w
2
σ 2
K
(L i ∆K i − L j ∆K j )
2 ,
can be added to the χ
2 definition to prevent large excursions in the direction
represented by L i ∆K i −L j ∆K j = 0, where L i and L j are the effective lengths
of the two magnets, respectively. The residual vector and the Jacobian matrix
are then similarly extended, with the difference being that the rows of the
Jacobian extension, W, consists of the coefficients that define the direction,
e.g., the row for the above constraint term (say, the n’th row of W) would be
W n: =
w n
σ K
(0, · · · , L i , 0, · · · , 0, −L j , 0, · · · ).
A general under-constrained direction can be described as
Nq
i=1
c i ∆K i = 0,
or c
T ∆K = 0,
(4.43)