The generalized minimal residual method

Xot( that the la.st formula is derived by analogy with the Itasie foiinula (4.Id) of the miniuiiil rc sidual method. 

Note that this technique is equivalent to the Lanczos method, described above. Certainly, after making s substeps, we arrive at a new residual on the (n +1) iteration 

The iterative process (4.92) is terminated when the misfit reaches the given level t o  

The advantage of the generalized MRM and Lanczos methods over the original minimal residual method is that now the iteration coefficients arc determined as 

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Within the above scheme, we implemented the generalized minimal residual (GM-RES) method [52], which is a robust linear solver that ensures convergence of the iterative solution. 

Thus, the Euler equation has a unique solution, ma, which can be obtained by the minimal residual method, MRM, or by the generalized MRM. We noted in the beginning of this section that the solution of the minimization problem (4.99) is also unique. Thus, we can conclude that it is equal to mo. In other words, we have proved that minimization of the Tikhonov parametric functional (4.99) is equivalent to the solution of the corresponding Euler equation (4.100). 

Despite the important advances in fast LC, food matrices are very complex, and although in general multi-residue methods with minimal sample manipulation are demanded, sample extraction and clean-up treatments must be carefully developed to also reduce the total analysis time. The most recently introduced sample treatment methodologies for pesticide residue analysis have also been addressed, with QuEChERS being the most popular one for its easy application and good results. However, other alternatives, such as online SPE or the use of more selective methods such as MIP, are also being applied for the analysis of pesticides. 

Carroll and Ruppert (1988) and Davidian and Gil-tinan (1995) present comprehensive overviews of parameter estimation in the face of heteroscedasticity. In general, three methods are used to provide precise, unbiased parameter estimates weighted least-squares (WLS), maximum likelihood, and data and/or model transformations. Johnston (1972) has shown that as the departure from constant variance increases, the benefit from using methods that deal with heteroscedasticity increases. The difficulty in using WLS or variations of WLS is that additional burdens on the model are made in that the method makes the additional assumption that the variance of the observations is either known or can be estimated. In WLS, the goal is not to minimize the OLS objective function, i.e., the residual sum of squares,... 

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