A Scalar Homotopy Method for Solving an Over ... - TechScience

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60 Copyright © 2009 Tech Science Press CMES, vol.53, no.1, pp.47-71, 2009The locus of evolution is plotted in Fig. 3, and the residual norm versus the evolutionsteps is shown in Fig. 4.Locus of EvolutionInitial Guess (0.5,0.5,0.5)Final Result (0.93054,1.2184,0.85109)0.90.850.80.75z0.70.650.60.550.51.50.910.80.70.60.5 0.5yFigure 3: The locus of evolution of Example 4.x13.5 Example 5The following example is given in Roose, Kulla, Lomb and Meressoo (1990):F i = 3x i (x i+1 − 2x i + x i−1 ) + 1 4 (x i+1 − x i−1 ) 2 ,x 0 = 0, x n+1 = 20. (40)Table 2: The numerical solutions of Example 5 with n=10.x 1 x 2 x 3 x 4 x 53.0832 5.3831 7.3952 9.2397 10.969x 6 x 7 x 8 x 9 x 1012.612 14.186 15.705 17.176 18.606
A Scalar Homotopy Method for Solving an Over/Under-Determined System 6110 110 010 -110 -2Residual Norm10 -310 -410 -510 -610 -710 -810 -910 -1010 -110 400 800 1200Numbers of EvolutionFigure 4: The residual norm for each root of Example 4.Initial values are fixed to be x i = 20, i= 1,. . . ,n. The convergence criterion is chosenas ε = 10 −10 . After 8768 steps, the solution is obtained. The parameter n is 10 andthe result is shown in Table 2. As compared with those reported by Spedicato andHuang (1997) for the Newton-like methods, the present scalar homotopy method ismore accurate and time saving, where the computational time is less than 0.1 secby using a PC586.3.6 Example 6Then, we consider an example similar to the one given by Krzyworzcka (1996):F 1 = (3 − 5x 1 )x 1 + 1 − 2x 2 ,F i = (3 − 5x i )x i − x i−1 − 2x i+1 , i = 2,··· ,9,F 10 = (3 − 5x 10 )x 10 + 1 − x 9 . (41)The initial guess is fixed to be x i = −0.1, i = 1,..,10. The convergence criterion isε =10 −10 . After 392 steps, the result obtained is reported in Table 3.As reported by Mo, Liu and Wang (2009) the Newton method cannot be applied forthis example, and their solutions obtained by the conjugate direction particle swarmoptimization method are different from the present solutions. For this example it
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