Relaxation factor optimum

The optimum value of the relaxation factor k can be obtained from the largest eigenvalue of s. This eigenvalue may, in turn, be obtained approximately from the relative magnitude of vectors oik) and oik+1). Improvements discussed in the next section, and amplified in the next chapter, significantly alter the method. Trial-and-error choice of k is therefore preferable and probably necessary. 

Theorem The successive overrelaxation method with optimum relaxation factor converges at least twice as fast as the Chebyshev semi-iterative method with respect to the Jacobi method, and therefore at least twice as fast as any semi-iterative method with respect to the Jacobi method. Furthermore, as the number of iterations tends to infinity, the successive overrelaxation method becomes exactly twice as fast as the Chebyshev semi-iterative method. 

In the preceding expression, we note that (w - 1 -i- Q)/w corresponds to the eigenvalues and so we have 0 = 1 h- w(k - 1). To find I0 l, we realize that this value is determined by the largest and smallest eigenvalues for ll0F/3xll as well as the relaxation factor. The optimum w occurs when... 

The pulse width is an important factor in the measurement of pulsed spectra. The optimal pulse-width may be estimated21 from the equation cos a = exp(— TJT), in which a is the pulse width (in degrees), Tt the spin-lattice relaxation-time (in s), and T the pulse-repetition time (in s). For monosaccharides in 20% aqueous solution, values of the protonated carbon atoms are22 1 s at 30°. Using 8 k of computer memory for the acquisition, and a sweep width of 5-6 kHz, T becomes 0.6-0.8 s, and the equation gives an optimum pulse-width of 60°. In Fig. 1 is shown a series of spectra measured at different pulse-widths, all other variables being kept constant. The best s/n is seen to correspond to a 63° pulse. If, 3C-n.m.r. spectra are recorded for very concentrated solutions, or impure samples, the Tj values may become small, and, in such cases, a 90° sample pulse will be optimal. 

In purchasing a new spectrometer, you should specify the desired frequency(ies) which will be dictated by the magnet available and the nuclei and experiments desired. If at all possible, a spectrometer which can operate over a large frequency range (a factor of 25 or more) should be chosen. This will enable you to study a variety of nuclei under conditions of optimum magnetic field. It also permits the study of relaxation times and lineshapes as a function of frequency. (This is an important method in identifying relaxation mechanisms and in separating different contributions to the line-shape. See III.C.2. and VI.D.l.)... 

The optimum over-relaxation parameter is known only for a small class of linear problems and for select boundary conditions. The iteration matrix has eigenvalues each one of which reflects the factor by which the amplitude of an eigenmode of undesired residual is suppressed for each iterative step. Obviously the modulus of all these modes must be less than 1. The modulus of the factor with the largest amplitude is called the spectral radius and determines the overall long term convergence of the procedure for many iterative steps. If Pj is the spectral radius of the Jacobi iteration then the optimum value of X is known to be ... 

The use of Gd(iii) chelate complexes as the second spin label instead of N-labelled nitroxides ° or Cu(ii) complexes offers important advantages. Spectroscopic separation of the signals from nitroxide radicals and Gd(iii) centres relies to a lesser extent on the difference in resonance fields for the two types of paramagnetic species. The main factors that allow spectroscopic separation are the 2-3 orders of magnitude difference in longitudinal relaxation time (at optimum repetition rate for Gd(iii) species nitroxide radicals are nearly completely saturated) and the difference in the transition moments between high-spin [S = 7/2) Gd(iii) centres and low-spin (S = l/2) nitroxide radicals. 

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