chapter8 - i book 2005/4/7 22:31 page S-68#89 i i i Answers...

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± ± “book” — 2005/4/7 — 22:31 — page S-68 — #89 ± ± ± ± Answers to Selected Exercises for Chapter 8 Section 8.1 (page 671) 1. (a) [ a, b ]=[0 , 1], p ( x )= 3 / (2 + x ), q ( x )=0 , r ( x 2 / (2 + x ) 3 (b) 1 5 6 2 7 6 17 20 2 23 20 19 22 2 25 22 1 w 0 w 1 w 2 w 3 w 4 = 0 8 729 1 125 8 1331 0 3. (a) [ a, b ]=[1 , 2], p ( x )=2+ 1 x , q ( x 1 1 x , r ( x (b) 1 27 20 151 80 13 20 4 3 91 48 2 3 37 28 213 112 19 28 1 w 0 w 1 w 2 w 3 w 4 = 2 e 0 0 0 10 e 3 5. (a) [ a, b , 3], p ( x 1 x , q ( x 1, r ( x )=10 (b) 1 5 6 7 4 7 6 7 8 7 4 9 8 9 10 7 4 11 10 1 w 0 w 1 w 2 w 3 w 4 = 20 2 . 5 2 . 5 2 . 5 100 7. Maximum N Absolute Error Error Ratio rms Error Error Ratio 4 0.00181295 0.00153506 8 0.00044549 4.070 0.00034992 4.387 16 0.00011276 3.951 0.00008416 4.158 32 0.00002816 4.004 0.00002068 4.071 64 0.00000704 4.001 0.00000513 4.033 128 0.00000176 4.000 0.00000128 4.016 256 0.00000044 4.000 0.00000032 4.008 S-68
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± ± “book” — 2005/4/7 — 22:31 — page S-69 — #90 ± ± ± ± Selected answers for Section 8.2 S-69 9. Nw N ( 1 2 )( w N ( 1 2 ) w 2 N ( 1 2 )) / ( w 2 N ( 1 2 ) w 4 N ( 1 2 )) 4 0.59796574 4.044 8 0.59576579 4.011 16 0.59522172 4.003 32 0.59508608 4.001 64 0.59505219 4.000 128 0.59504372 256 0.59504160 11. Maximum N Absolute Error Error Ratio rms Error Error Ratio 4 0.00000000 0.00000000 8 0.00000000 0.300 0.00000000 0.322 16 0.00000000 2.500 0.00000000 2.550 32 0.00000000 0.046 0.00000000 0.047 64 0.00000000 0.861 0.00000000 0.915 128 0.00000000 0.404 0.00000000 0.440 256 0.00000000 0.285 0.00000000 0.273 The truncation error terms that were dropped to derive the numerical method involve the third and fourth derivatives of the exact solution. For this problem the exact solution is a quadratic function, whose third and fourth derivatives are identically zero. The ±nite di²erence method therefore produces the exact solution for any N . 13. Maximum N Absolute Error Error Ratio rms Error Error Ratio 4 0.01215773 0.01025484 8 0.00304571 3.992 0.00238352 4.302 16 0.00076185 3.998 0.00057609 4.137 32 0.00019049 3.999 0.00014170 4.065 64 0.00004764 3.999 0.00003514 4.032 128 0.00001191 4.000 0.00000875 4.016 256 0.00000298 4.000 0.00000218 4.008 15. (a) u (36 inches) = 0 . 00175240 inches (b) u (36 inches) = 0 . 00228059 inches 17. u (0 . 5) = 0 . 346685 Section 8.2 (page 683) 3. (a) [ a, b ]=[0 , 1], p ( x )= 3 / (2 + x ), q ( x )=0 , r ( x 2 / (2 + x ) 3
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± ± “book” — 2005/4/7 — 22:31 — page S-70 — #91 ± ± ± ± S-70 Answers to Selected Exercises for Chapter 8 (b) 1 5 6 2 7 6 17 20 2 23 20 19 22 2 25 22 22
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This note was uploaded on 10/13/2009 for the course MATH 471 taught by Professor Anna during the Spring '09 term at University of Michigan-Dearborn.

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chapter8 - i book 2005/4/7 22:31 page S-68#89 i i i Answers...

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