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How should I create the function that provides the residuals for fitting of impedance data using scipy.optimize.least-squares?

I would like to use the Levenberg-Marquardt algorithm implemented in the least-squares function of Scipy's optimize module to fit measured impedance data. However, I'm a little confused about how to provide the residuals.
In the example shown in the documentation, it's clear that you should only provide the subtraction between the experimental and calculated values. So instead of creating a function that returns , we should write one that returns an array of .
While this is straight forward for normal data, I'm not sure how to do this for impedance data. Each impedance measurement (Z) has a real and a complex part, so the objective function becomes where wt is the weight factor, w is the frequency (independent variable) and a is the set of parameters to find. The problem is that I don't know how to correctly provide the residuals to the least-squares function as I don't think would work.

The last example in the scipy.optimize.least_squares documentation shows how to deal with complex residuals. Namely, if you have complex input and output:
def f(z):
return z - (0.5 + 0.5j)
you can wrap the function to expand to two real numbers:
def f_wrap(x):
fx = f(x[0] + 1j*x[1])
return np.array([fx.real, fx.imag])

Scalar minimization using scipy (`minimize` vs `minimize_scalar`)

I have a polynomial function for which I would like to find all local extrema. I can evaluate the polynomial via P(x) and to its derivative via d_P(x).
My first thought was to use minimize_scalar, however this does not seem to be able to take advantage of the fact that I can evaluate the derivative. Alternatively, I can use the more general minimize function and provide the gradient.
Is there a rule of thumb about which method will work better, or is this something where I should test out both methods and see what works better. Since the function I am optimizing is a polynomial (well behaved) I wonder if it really matters so much which I use, but if someone has a more background that would be great.
In particular, P(x) is the (unique) polynomial of degree n which alternatively attains a value of 1 or -1 on a set of n-1 points.
Here is a sample of the P(x) scaled so that P(0)=1. Note that the y axis is plotted on a symlog scale.

Since you have a continuous scalar function, the documentation of minimize_scalar suggests a more discrete optimization approach. Since it doesn't use gradient information you won't have trouble with noise/discontinuities/discreteness in your objective. However, if you use minimize in conjunction with a gradient based method then you will have trouble with convergence for noise/discontinuities/discreteness.
If the objective function is fist order continuous then both minimize and minimize_scalar should yield the same solution for a given bound.

Computing intersection of a function with a specific interval using scipy

I'm stuck trying to get functions that are existent in scipy (or sympy) for the following task:
Suppose we are given the following function:
f(A,B,C) = k1-A*sin(B*k2-C)
for each of the axis A,B,C of the space we have a specific interval, like [a_lb, a_ub], [b_lb, b_ub], [c_lb, c_ub], [d_lb, d_ub].
Which functions of scipy can be used to compute if the space encompassed by the boundaries is intersected by the given function? I thought of like e.g. computing the Hessian matrix.
Thank you for hints
Best regards

If I understand correctly, what you are looking for is an answer to whether f(A,B,C) bounded in the domain [a_l,a_u]x[b_l,b_u]x[c_l,c_u] has a value within [d_l,d_u]. You can try using scipy.optimize.minimize for this.
If you run scipy.optimize.minimize on f with the bounds [a_l,a_u]x[b_l,b_u]x[c_l,c_u], you should get the minimal value of f in the domain. Similarly, minimizing -f will give you the maximal value of f in the domain. f intersects the given boundary if and only if the interval [fmin, fmax] intersects the interval [d_l,d_u].
Note that scipy.optimize.minimize is a non-linear optimization and therefore requires an initial guess. The middle point of the domain box is a natural choice, but since the non-linear optimization may encounter a local minimum (or not converge), you may want to try several other initial guesses as well. scipy.optimize.minimize has many (optional) parameters so I recommend you read its documentation and play with them to fine-tune your usage to your needs.

Intelligent fitting with python

Is there a more intelligent function than scipy.optimize.curve_fit in Python?
I also need to define a function to fit data with.
I've spend ages trying to fit data with it. I can fit only basic functions and fitting two lines with piecewise function is impossible while the y-axis has low values like 0.01-0.05 and x-axis values like 20-60.
I know I have to plug in initial values, but still it takes too much time and sometimes it does not work.
EDIT
I added graph where are data I fitted and you can see the effect of changing bounds in scipy.optimize.curve_fit.
The function I fit with is this one:
def abslines(x,a,b,c,d):
return np.piecewise(x, [x < -b/a, x >= -b/a], [lambda x: a*x+b+d, lambda x: c*(x+b/a)+d])
Initial conditions are same everytime and I think they are close enough:
p0=[-0.001,0.2,0.005,0.]
because the values of parameters from best fit are:
[-0.00411946 0.19895546 0.00817832 0.00758401]
Bounds are:
No bounds;
bounds=([-1.,0.,0.,0.],[0.,1.,1.,1.])
bounds=([-0.5,0.01,0.0001,0.],[-0.001,0.5,0.5,1.])
bounds=([-0.1,0.01,0.0001,0.],[-0.001,0.5,0.1,1.])
bounds=([-0.01,0.1,0.0001,0.],[-0.001,0.5,0.1,1.])
starting with no bounds, end with best bounds
Still I think, that this takes too much time and curve_fit can find it better. This way I have to almost specify the function and it seems like I am fitting by changing parameters not that curve_fit is fitting.

Without knowing what is exactly the regression algorithm in Python it is quite impossible to give a definitive answer. Probably the calculus is iterative and requires initial guesses, which are probably derived from the specified bounds. So, the bounds have an indirect effect on the convergence and the results.
I suggest to try a simpler algorithm (not iterative, no initial guess) coming from this paper : https://fr.scribd.com/document/380941024/Regression-par-morceaux-Piecewise-Regression-pdf
The code is easy to write in any computer language. I suppose this can be done with Python as well.
The piecewise function to be fitted is :
The parameters to be computed are a1, p1, q1, p2 and q2.
The result is shown on the next figure, with the approximate values of the parameters.
So that, no bounds are required to be specified and as a consequence no problems related to bounds.
NOTE : The method is based on the fitting of a convenient integral equation such as shown in the above referenced paper. The numerical calculus of the integral is subjected to deviations if the number of points is too small. In the present case, they are a large number of points. So, even scattered this is a favourable case for the practical application of this method.

1.Algorithms behind curve_fit expect differentiable functions, thus it can go south if given a non-differential one.
For a more powerful interface to curve fitting, have a look at lmfit.

Python: Plotting a power law function with exponential cutoff

I have a graph between 2 functions f and g.
I know it follows a power law function with exponential cutoff.
f(x) = x**(-alpha)*e**(-lambda*x)
How do I find the value of exponent alpha?

If you have sufficiently close x points (for example one every 0.1), you can try the following:
ln(f(x)) = -alpha ln(x) - lambda x
ln(f(x))' = - alpha / x - lambda
So depending on where you have your points:
If you have a lot of points near 0, you can try:
h(x) = x ln(f(x))' = -alpha - lambda x
So the limit of the function h when x goes to 0 is -alpha
If you have large values of x, the function x -> ln(f(x))' tends toward lambda when x goes to infinity, so you can guess lambda and use pwdyson's expression.
If you don't have close x points, the numerical derivative will be very noisy, so I would try to guess lambda as the limit of -ln(f(x)/x for large x's...
If you don't have large values, but a large number of x's, you can try a minimization of
sum_x_i (ln(y_i) + alpha ln(x_i) + lambda x_i) ^2
on both alpha and lambda (I guess It would be more precise than the initial expression)...
It is a simple least square regression (numpy.linalg.lstsq will do the job).
So you have plenty of methods, the one to chose really depends on you inputs.

The usual and general way of doing what you want is to perform a non-linear regression (even though, as noted in another response, it is possible to linearize the problem). Python can do this quite easily with the help of the SciPy package, which is used by many scientists.
The routine you are looking for is its least-square optimization routine (scipy.optimize.leastsq). Once you wrap your head around the way this general optimization procedure works (see the example), you will probably find many other opportunities to use it. Basically, you calculate the list of differences between your measurements and their ideal value f(x), and you ask SciPy to find the parameters that make these differences as small as possible, so that your data fits the model as well as possible. This then gives you the parameter you are looking for.

It sounds like you might be trying to fit a power-law to a distribution with an exponential cutoff at the low end due to incompleteness - but I may be reading too far into your problem.
If that is the problem you're dealing with, this website (and accompanying publication) addresses the issue: http://tuvalu.santafe.edu/~aaronc/powerlaws/. I wrote the python implementation of the power-law fitter on that page; it is linked from there.

If you know that the points follow this law exactly, then invert the equation and put in an x and its corresponding f(x) value:
import math
alpha = -(lambda*x + math.log(f(x)))/math.log(x)
But the if the points do not exactly fit the equation you will need to do some sort of regression to determine alpha.
EDIT: Ok, so they don't fit exactly. This is getting beyond a Python question, but there may be something in numpy that can handle it. Here is a numpy linear regression recipe but your equation can't be rearranged into a linear form, so you'll have to look into non-linear regression.