The objective function for minimization

Larch's minimize function and its use of objective functions is discussed here. This is a really neat part of Larch because it allows the user to do almost any kind of fit to almost any kind of data. The objective function that implements the MBACK fit is this:

def match_f2(p):
    """
    Objective function for matching mu(E) data to tabulated f"(E) using the MBACK
    algorithm or the Lee & Xiang extension.
    """
    s      = p.s.value
    a      = p.a.value
    em     = p.em.value
    xi     = p.xi.value
    c0     = p.c0.value
    eoff   = p.en - p.e0.value

    norm = a*erfc((p.en-em)/xi) + c0 # erfc function + constant term of polynomial
    for i in range(MAXORDER):        # successive orders of polynomial
        j = i+1
        attr = 'c%d' % j
        if hasattr(p, attr):
            norm = norm + getattr(getattr(p, attr), 'value') * eoff**j
    func = (p.f2 + norm - s*p.mu) * p.theta / p.weight
    if p.form.lower() == 'lee':
        func = func / s*p.mu
    return func

There are several aspects of this function worthy of discussion.

1. It takes a single argument, which is the params Group from the mback function. That Group contains all of the parameters, constants, and arrays needed to evaluate the fit.

2. It is not registered in the call to registerLarchPlugin. This means that match_f2 is not a symbol that will be readily available to the user of the Larch command line. That's OK. The user is expected to interact with the mback function, which uses this function.

3. The return value of the function is the array to be minimized. The Minimizer's leastsq function will alter the values of the Parameters in the input Group until the sum of squares of the return array is as small as possible.

4. The evaluation of the return array is slightly different than the evaluation of the normalization function at the end of the mback function due to the presence of the theta and weight arrays. The weight array is used to adjust the significance of different regions of the data to the evaluation of the fit. This is also the purpose of the Lee and Xiang extension to the MBACK algorithm. It optionally weights the minimization function by shape of the μ(E) function, as seen in the next to last line. The theta array is used to exclude regions of the data (the beginning or end of the array or the regions around white lines) from the minimization function by multiplying the minimization function by 0.0 in those regions.

5. This function makes use of NumPy's vectorized calculations. Rather than iterating through the energy range (as a Fortran programmer might do), arrays are added and multiplied together, relying on NumPy conventions to do the iterations correctly and efficiently. As we will see in the next chapter, employing vectorization is a huge win in terms of execution time.