Least squares (${\min }_x\frac{1}{2}{\sum }_j{R}_j(x{)}^2$)

The least squares approach chooses parameters by minimizing the sum of squared residuals. A linear least-squares problem has affine residuals, whereas nonlinear least squares permits nonlinear dependence on the parameters. The Euclidean norm makes the objective $\frac{1}{2}\Vert R(x){\Vert }_2^2$; the factor one-half simplifies its derivatives without changing its minimizers.

Here the residual map $R$ may be nonlinear. Its Jacobian matrix $J=DR$ gives $\nabla f=J^{T}R$ and ${\nabla }^{2}f=J^{T}J+{\sum }_j{R}_j{\nabla }^2{R}_j$. The Gauss-Newton algorithm drops the second term to obtain an approximation using only first derivatives of the residuals. Full column rank of $J$ makes $J^{T}J$ positive definite; at a zero-residual solution this approximation equals the exact Hessian.

For nonlinear least squares $f(x)=\frac{1}{2}\Vert R(x){\Vert }_2^2$, with residual Jacobian matrix $J=DR$, the method approximates the Hessian by $J^{T}J$. The exact Hessian is $J^{T}J+{\sum }_j{R}_j{\nabla }^2{R}_j$. Full column rank of $J$ makes the approximation positive definite. At a zero-residual solution with full-rank Jacobian, $J(x_k{)}^{T}J(x_k)\to {\nabla }^{2}f(x_{\ast })$, so any convergent sequence of full Gauss-Newton steps satisfies the Dennis-Moré condition and converges superlinearly. A nonzero residual at the solution leaves a generally nonvanishing discarded Hessian term; the same conclusion does not follow automatically.