HW1 - Michael Mark 22838947#7 Rate of convergence of a...

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Michael Mark 22838947 1/22/2009 #7. Rate of convergence of a function n h g(1; h) e(h) e(h)/h e(h)/h^2 1 0.50000000 0.31204800 0.22825430 0.45650860 0.91301721 2 0.25000000 0.43005454 0.11024777 0.44099107 1.76396428 3 0.12500000 0.48637287 0.05392943 0.43143545 3.45148362 4 0.06250000 0.51366321 0.02663910 0.42622560 6.81960963 5 0.03125000 0.52706746 0.01323485 0.42351519 13.55248611 6 0.01562500 0.53370646 0.00659584 0.42213395 27.01657297 7 0.00781250 0.53700983 0.00329248 0.42143687 53.94391922 8 0.00390625 0.53865744 0.00164487 0.42108672 107.79819941 n h g(1; h) e(h) e(h)/h e(h)/h^2 1 0.50000000 0.51806945 0.02223286 0.04446572 0.08893143 2 0.25000000 0.53469172 0.00561059 0.02244235 0.08976940 3 0.12500000 0.53889637 0.00140594 0.01124751 0.08998006 4 0.06250000 0.53995062 0.00035169 0.00562705 0.09003280 5 0.03125000 0.54021437 0.00008794 0.00281394 0.09004599 6 0.01562500 0.54028032 0.00002198 0.00140702 0.09004929 7 0.00781250 0.54029681 0.00000550 0.00070352 0.09005011 8 0.00390625 0.54030093 0.00000137 0.00035176 0.09005032 Michael Mark 22838947 1/22/2009 Part 1 , f(x) = sin(x) and g(x; h) = (f(x+h)-f(x))/h As we can see from comparing the graphs below, the rate of convergence is not as quick as part 2 for this function. The rate of convergence for this function is 1/n since the function loo logarithmic in the graph. Given the error data from the table, this is true (constant). Part 2 , f(x) = sin(x) and g(x; h) = (f(x+h)-f(x-h))/(2h) Comparing the graphs from part 1 and part 2, the function for this part converges much more one in part 1 as seen from the graph. As n increases just a little bit, say past 1, the function already surpassed 0.5 as seen in the table. The rate of convergence is 1/(n^2) from error.
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  • Spring '08
  • Saigal
  • Convergence, Mathematical analysis, Rate of convergence, Aitken's delta-squared process, Michael Mark 22838947

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