The Gauss-Newton method is an iterative method used to solve nonlinear least squares problems. The method is named after Carl Friedrich Gauss and Adrien-Marie Legendre, who both independently developed the method in the early 19th century, and Joseph Louis Lagrange, who later generalized the method.

To understand the Gauss-Newton method, it’s helpful to first understand the concept of least squares. In statistics, least squares refers to the method of finding the line of best fit for a set of data points by minimizing the sum of the squares of the differences between the observed values and the values on the line. This is often used in regression analysis, where the goal is to find the line that best describes the relationship between two variables.

The Gauss-Newton method is a way of solving nonlinear least squares problems by iteratively refining the estimates of the parameters of the model. The method starts by making an initial guess at the parameters of the model, and then uses the Jacobian matrix to iteratively update the estimates until they converge to the solution.

The Jacobian matrix is a matrix of partial derivatives that describes the sensitivity of the model’s output to changes in the model’s parameters. The Gauss-Newton method uses the Jacobian matrix to compute the gradient of the objective function (i.e., the sum of the squares of the differences between the observed values and the values on the line), and then uses this gradient to update the estimates of the model’s parameters.

As an example, let’s say we have a set of data points that we want to fit with a quadratic function of the form:

y = a*x^2 + b*x + c

We can use the Gauss-Newton method to find the values of a, b, and c that minimize the sum of the squares of the differences between the observed values and the values on the line.

To start, we need to make an initial guess at the values of a, b, and c. Let’s say our initial guess is a=1, b=1, and c=1. We then compute the predicted values of y for each data point using this initial guess. For example, if one of our data points is (x=2, y=3), our predicted value of y would be 12^2 + 12 + 1 = 7.

Next, we compute the Jacobian matrix. This is a 3×3 matrix with the partial derivatives of y with respect to a, b, and c. For the quadratic function above, the Jacobian matrix would be:

[ 2x^2 x 1]

[ 4x^3 2x 1]

[ 6x^4 3x^2 1]

We then compute the gradient of the objective function, which is the sum of the squares of the differences between the observed values and the predicted values of y. The gradient is a 3×1 vector with the partial derivatives of the objective function with respect to a, b, and c.

Finally, we use the gradient and the Jacobian matrix to update our estimates of a, b, and c. The updated estimates are the current estimates plus the inverse of the Jacobian matrix times the gradient. We repeat this process until the estimates converge to the solution.

In summary, the Gauss-Newton method is a powerful iterative technique for solving nonlinear least squares problems. It starts by making an initial guess at the model’s parameters and then uses the Jacobian matrix and the gradient of the objective function to iteratively update the estimates until they converge to the solution.