SCULIB_SKYFUNC

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SCULIB_SKYFUNC_1

calculate f(x) and its Jacobian for NAG non-linear fitting routine E04UPF when fitting ETA_TEL, B and TAU in SKYDIP analysis

Description:

E04UPF is a NAG routine that is used to fit a theoretical sky-dip curve to the measured data by varying ETA_TEL, B and TAU. This routine is called by E04UPF to calculate the M sub-functions fi(x) at the current solution vector X, and/or their Jacobian matrix. See the NAG manual if you need to know more about E04UPF.

The solution vector is composed as follows:-

x(1) = ETA_TEL
x(2) = B
x(3) = TAU

For each of the M measured airmasses the subfunction f(x) is:-

f (x) = J_{}

theory (Airmass(i)) - J_meas (Airmass(i))

J_{}

noise (Airmass(i)) (74)

where:-

J_{}

theory = (1-

η_{}

tel) J_tel + $η_{}$

telJ_atm (1-B $e x p (- τ A i r m a s s)) (75)$

J_{}

atm = J_amb X_G (76)

X_{G} = 1 + \frac{h 1 \times h 2}{J_{}}

amb

e x p (- τ \times A i r m a s s (i) / X_G c o n s t) (77)

The corresponding Jacobian has component (i,j):-

\frac{\partial f (x)}{\partial x (j)} = \frac{\partial}{\partial x (j)} J_{}

theory(Airmass(i))

\times \frac{1}{J_{}}

noise(i) (78)

leading to:

\frac{\partial f (i)}{\partial x (1)} = - J_{}

tel+J_atm - B J_tel

e x p (- τ A (i))

J_{}

noise(i) (79)

\frac{\partial f (i)}{\partial x (2)} = - η_{}

tel J_atm

e x p (- τ A (i))

J_{}

noise(i) (80)

\frac{\partial f (i)}{\partial x (3)} = η_{}

tel

\partial J_{}

atm $\partial τ - η_{}$

tel B $e x p (- τ A (i)) (\partial J_{})$

atm $\partial τ - J_{}$

atm A(i) $J_{}$

noise(i) (81)

where

\partial J_{}

atm

\partial τ = J_{}

amb $\partial X_{G}$ $\partial τ$ (82)

and

\frac{\partial X_{G}}{\partial τ} = - \frac{H_{1} H_{2} A (i)}{J_{}}

amb X_Gconst

e x p (- τ A (i) / X_{}

Gconst) (83)

The array USER is used to get the necessary ancillary information into the routine:-

USER (1) = J_TEL
USER (2) = J_AMB
USER (3) = not used
USER (4:M $+$ 3) = the measured airmasses
USER (M $+$ 4:2M $+$ 3) = the measured sky temperatures
USER (2M $+$ 4:3M $+$ 3) = the errors on the measured sky temperatures

SCULIB_SKYFUNC_1

Description:

Invocation

Arguments

See NAG manual description of OBJFUN parameter in E04UPF.

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