GaussNewtonalgoritmen

The Gauss‑Newton algorithm is an iterative numerical method used to solve non‑linear least‑squares problems. It was introduced by Philipp Ludwig Ernst Gauss and Carl Friedrich Gauss independently in the 19th century, but it was Carl Friedrich Gauss who first described the method in his 1833 dissertation. The algorithm is particularly effective for fitting model functions to noisy data when the model is differentiable with respect to its parameters.

A non‑linear least‑squares problem seeks parameters \(x\) that minimize the sum of squares of residuals \(r_i(x)=y_i - f_i(x)\). The objective function is \(S(x)=\sum_i r_i(x)^2\). The Gauss‑Newton method linearises the residuals around the current estimate \(x_k\) using the Jacobian matrix \(J(x_k)\), whose elements are \(\partial r_i / \partial x_j\). The parameter update is then obtained by solving the normal equations \((J^T J)\Delta x = J^T r\) and setting \(x_{k+1}=x_k+\Delta x\). This eliminates the second‑derivative term present in the full Newton method, making the algorithm computationally cheaper while retaining quadratic convergence near a solution.

The algorithm converges rapidly when the residuals are small and the model is reasonably close to linear in the parameter domain. However, if the residuals are large or the Jacobian is poorly conditioned, the method can diverge or converge slowly. To address these issues, variants such as the Levenberg‑Marquardt algorithm incorporate a damping term that blends the Gauss‑Newton direction with a gradient‑descent step. The Gauss‑Newton algorithm is widely used in fields such as robotics, computer vision, and econometrics for tasks like bundle adjustment, calibration, and parameter estimation in dynamic systems.