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CHAPTER 5 PROBLEMS

GERALD AND WHEATLEY, 6TH EDITION
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31.  For h=0.1:

f(0.42) = .04346
f(0.22) = .01205

Plugging these into the center difference formula (the second line in
the bottom box on p373), I get 

.04346 - .01205
---------------	=  .15705 = f'(0.32)
  0.42 - 0.22

For h=0.05:

f(0.37) = .03384
f(0.27) = .01811

...which yields f'(0.32) = .15730

To combine these two answers using equation 5.21 on page 370, you need
to know (1) which of the two is more accurate and (2) what the
exponent n is.  Here, n=2 and the h=0.5 answer is more accurate, so
the extrapolated answer is

better answer = .15730 + 1/3 (.15730 - .15705) = .15738

You can continue to extrapolate, improving the answer with every step,
as shown in table 5.6.  For this particular problem, that would entail
evaluating f(.295) and f(.345), using those to compute f'(0.32) using
the center difference formula, and then using eqn 5.21 to extrapolate
between the h=0.5 and h=0.25 answers with n=2.  After that, you can
extrapolate the two EXTRAPOLATED answers using eqn 5.21 with n=4.  The
answer then is 0.15728, as shown in the back of the book.

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40.

a. h=0.1:

0.1
--- [1.543 + 2(1.669) + 2(1.811) + 2(1.971) + 2(2.151) + 2(2.352)
 2         + 2(2.577) + 2(2.828) + 3.107]
 
= 1.7684

b. h=0.2:

0.2
--- [1.543 + 2(1.811) + 2(2.151) + 2(2.577) + 3.107]
 2
 
= 1.7728

c. h=0.4:

0.4
--- [1.543 + 2(2.151) + 3.107]
 2
 
= 1.7904

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Repeat problem 40 (just the h=0.1 part), but using Simpson's 1/3 rule:

The number of points between 1.0 and 1.8 is 9, which divides nicely
into the three-point panels that this rule uses, and so we can just
use the corresponding composite rule [eqn 5.35]:

0.1
--- [1.543 + 4(1.669) + 2(1.811) + 4(1.971) + 2(2.151) + 4(2.352)
 3         + 2(2.577) + 4(2.828) + 3.107]

= 1.76693

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Repeat problem 40 (just the h=0.1 part), but using Simpson's 3/8 rule:

This one's not so easy because the points in the table don't divide
neatly into 4-point panels.  Because of this, you have to use some
combination of different methods -- e.g., two panels of Simpson's 3/8
rule and one of Simpson's 1/3:

3(0.1)
------ [1.543 + 3(1.669) + 3(1.811) + 2(1.971) + 3(2.151) + 3(2.352)
  8         + 2.577]  

  0.1
+ --- [2.577 + 4(2.828) + 3.107]
   3

= 1.76695