Residuals Euclidean norm

Convergence to the solution of a nonlinear equation system is often checked by means of an appropriate merit function, such as the Euclidean norm of the residuals. 

It is also advisable to look at the linear system (7.68) from a different perspective. It can be viewed as the solution with the minimum Euclidean norm of the residual vector of the following overdimensioned system ... 

A graphic that can be produced to better describe the actual situation is the plot of b against y-y, where symbolizes the Euclidean vector norm [4CM16], As Figure 5.14 discloses, an /.-shaped curve results with the best model occurring at the bend, which reflects a harmonious model with the least amount of compromise in the trade-off between minimization of the residual and regression vector norms. The regression vector norm acts as an indicator of variance for the... 

The solver must stop when an acceptable accurate solution is found. A stop criterion must thus be defined by a sufficientfy small residual value or preferably a norm of the residual [166]. In the non-preconditioned CG-method it is natural to use the 2-norm since the euclidean inner product (r, r) is already calculated as part of the algorithm. This is not the case neither in the preconditioned version of the CG-solver nor the BCG-methods. In these methods extra computations are thus required to calculate the stop criterion. For that reason, the less expensive infinity-norm is frequently used as stop criterion for these solvers. One possible criterion is that the norm of the residual must fall below a specific value jrjm < e. However, this criterion is difficult to use when employing the p-norm since this norm is grid dependent. Besides, the tolerance e has to be fit to the system under consideration since the residuals... 

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