In this blog post, I will show you how to use scipy.optimize.least_squares to solve nonlinear least squares problems in Python. Nonlinear least squares problems are optimization problems where the objective function is a sum of squared residuals, and the residuals depend on some nonlinear function of the parameters. For example, suppose we have some data points (x, y) and we want to fit a curve of the form y = a * exp(b * x) + c, where a, b and c are the parameters to be estimated. This is a nonlinear least squares problem because the residuals are y – (a * exp(b * x) + c), and they depend on the exponential function of the parameters.

To solve this problem using scipy.optimize.least_squares, we need to define a function that takes the parameters as input and returns the residuals as output. For example:

def fun(p, x, y): a, b, c = p return y - (a * np.exp(b * x) + c)

Then, we need to provide some initial guess for the parameters, and optionally some bounds or other options. For example:

p0 = [1.0, 1.0, 1.0] # initial guess bounds = ([0.0, 0.0, 0.0], [np.inf, np.inf, np.inf]) options = {'max_nfev': 100}

Once you have obtained the solution using scipy.optimize.least_squares, you can further analyze and visualize the results. Plotting the fitted curve alongside the original data points can provide valuable insights into the quality of the fit. Additionally, you can examine the optimal parameters to gain a deeper understanding of the relationships within your data. Nonlinear least squares optimization is a powerful tool for curve fitting and parameter estimation in a wide range of scientific and engineering applications, and with the flexibility and robustness of scipy.optimize.least_squares, you can efficiently tackle complex nonlinear optimization problems in Python.

Finally, we can call scipy.optimize.least_squares with the function, the initial guess, the data points, and the optional arguments. The function returns an OptimizeResult object that contains information about the solution, such as the optimal parameters, the cost