Leastsquaresimprove

Leastsquaresimprove is a term used to describe a class of procedures aimed at improving a numerical solution produced by a least-squares solver. It refers to refinement steps that reduce the residual r = b − Ax and improve the accuracy of x. It is not a single standardized algorithm, but a generic label used in texts and software.

Core idea: begin with an initial least-squares solution x0, compute residual r0 = b − Ax0, and perform corrective updates by solving a smaller system for delta to produce x1 = x0 + delta, repeating until convergence.

Common techniques include iterative refinement using normal equations or QR/SVD-based corrections; reweighting to handle outliers; regularization (ridge or Tikhonov) to stabilize ill-conditioned problems; and robust least squares to downweight large residuals. In nonlinear contexts, leatsquaresimprove may refer to Levenberg–Marquardt type steps or similar update schemes within nonlinear least-squares frameworks.

Implementation notes: leatsquaresimprove can be integrated into existing solvers as an optional refinement stage; it is typically efficient if residuals are small and A is well-conditioned, and may require higher-precision arithmetic to gain accuracy.

Limitations and caveats: improvement may stall if the problem is ill-posed or inconsistent; in highly ill-conditioned or noisy data the gains are limited; there is no universal guarantee of convergence or improvement in all cases.

See also: least squares, iterative refinement, Gauss-Newton, Levenberg–Marquardt, QR factorization, SVD, regularization.