The discrete L-curve rule terminates with failure or a maximum-curvature grid point

The input grid has finite length $q$. Each positive parameter gives an invertible matrix $A^{T}A+{\lambda }_{j}I$ because $z^T(A^{T}A+{\lambda }_{j}I)z=\mathrm{\parallel }Az{\mathrm{\parallel }}^2+{\lambda }_j\mathrm{\parallel }z{\mathrm{\parallel }}^2>0$ for $z\mathrm{\ne }0$. Thus the solution and residual calculations in [F1] are defined; a finite number of scalar norm, positivity, logarithm, determinant, and arithmetic operations determine the retained points and curvatures. Every loop terminates after at most $q$ iterations.

If fewer than three points remain, [F1] returns the prescribed failure status. Otherwise the finite nonempty set of assigned curvature values has a maximum; the finite set of indices attaining it has a least original grid index. The tie rule in [F1] therefore returns a maximizer and terminates.