LeastSquaresAnsätze

LeastSquaresAnsätze refers to a set of techniques used in numerical analysis and optimization to find the best approximate solution to an overdetermined system of linear equations. These systems arise when there are more equations than unknowns, meaning an exact solution that satisfies all equations simultaneously may not exist. The core idea is to minimize the sum of the squares of the differences between the observed values and the values predicted by a model.

The problem can be formulated as finding a vector x that minimizes the norm of the residual vector, ||Ax - b||², where A is a matrix representing the system of equations, x is the vector of unknown variables, and b is the vector of observed values. The solution to this minimization problem is typically found by solving the normal equations, which are given by AᵀAx = Aᵀb. The matrix AᵀA is square and often invertible, allowing for a direct calculation of the least-squares solution x = (AᵀA)⁻¹Aᵀb.

In cases where AᵀA is not invertible or ill-conditioned, alternative methods like Singular Value Decomposition (SVD) are employed to compute the least-squares solution. SVD provides a more robust approach and can also identify solutions with desirable properties, such as minimum norm. LeastSquaresAnsätze are widely applied in fields such as statistics, machine learning, signal processing, and engineering for tasks like curve fitting, regression analysis, and parameter estimation.