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Differential Equations Solutions 161

Differential Equations Solutions 161 - to compute the...

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171 0 2 4 6 8 10 0 2 4 6 8 10 nz = 34 S(r,r) after eigenpartition ordering 0 2 4 6 8 10 0 2 4 6 8 10 nz = 25 chol(S(r,r)) after eigenpartition ordering Figure 27.6. Results of reordering using eigenpartitioning. Solving Laplace equation on box, with n = 15625 Algorithm storage time residual norm Cholesky 28565072 1.02e+02 6.98e-14 Cholesky, R-Cuthill-McKee 16773590 3.79e+01 6.10e-14 Cholesky, minimum degree 8796896 4.08e+01 4.39e-14 Cholesky, approx. mindeg 7549652 3.08e+01 3.66e-14 (There were too many recursions in eigenpartition method specnd .) All algorithms produced solutions with small residual norm. On each problem, the approximate minimum degree algorithm gave factors requiring the lowest stor- age, preserving sparsity the best, and on the last two problems, it used the least time as well. (Note that local storage used within MATLAB’s symrcm , symmmd , symamd , and the toolbox specnd was not counted in this tabulation.) It is quite expensive
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Unformatted text preview: to compute the eigenpartition ordering, and this method should only be used if the matrices will be used multiple times so that the cost can be amortized. To complete this study, it would be important to try diFerent values of n , to determine the rate of increase of the storage and time as n increased. To judge performance, several hardware parameters are signi±cant, including computer (Sun Blade 1000 Model 1750), processor (Sun UltraSPARC-III), clock speed (750 MHz), and amount of RAM (1 Gbyte). The software speci±cations of importance include the operating system (Solaris 8) and the MATLAB version (6.5.1). Benchmarking is a difficult task, depending on the choice of hardware, software, and test problems, and our results on this problem should certainly raise more questions than they answer....
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