A generalized minimal residual method

Let us assume now that the operator AA is an absolutely positively determined (APD) operator, which satisfies the conditions 

4 hc prool of this iheorttm is similar to the proof of Theorem 10. 

44ie main itrohltun with Iht iiractical application of this tnctlK)d is tluit, in the case of an ill- o.sod inverse problcin, the required condition for the o[)crator L —. -L I does not hold. Tht refort. oni should us(t the rcgulari/ation technique, which will Ix discussed later. 

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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). 

Once the set of L scaled moments of inertia F have been evaluated, the molecular structural parameters are derived by means of a standard least-squares fitting of the F s. This is found to provide the best averaging of small residual vibrational effects. Eor a linear friatomic molecule XYZ, the four moments of inertia would be analyzed for the parameters dYx and d z (see Table II). Importantly, the method employs a minimal set of isotopic substitution data compared to the mass-dependence method. It is, however, necessary to select the parent such that all isotopic substitutions satisfy either Am, > 0 or Am, < 0 for all atoms i. This minimizes residual vibrational effects. For the general case, there are moments 7 and I (a = a,b,c) associated with each axis, and these are used to calculate the corresponding Pa, Pt, Pc and the la, Ic- The moments of inertia 7 are then analyzed by least squares for the structural parameters. Table XXIII compares several structures for OCS. Results for SO2 are summarized in Table XVIII. It is apparent that the structures compare most favorably with the sfructures. Similar results are found for other molecules. 

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