University of Florida/Egm6341/s10.team2.niki/HW5

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Problem 2:Continuation of proof of trapezoidal error[edit]


Statement[edit]

P. 27-1

Continue the proof of Trapezoidal Rule Error to steps 4a and 4b and determine and

Solution[edit]

From steps 3a and 3b we get the expression

(Summary p 26-3)

(Summary p 26-3)

Step 4a[edit]

where,


          
Step 4b[edit]

where,


          


Selecting such that

,we get


Summary[edit]

(Summary p 26-3)

Author[edit]

--Egm6341.s10.team2.niki 21:38, 23 March 2010 (UTC)


Problem 7: Understanding the derivation of the proof of Trapezoidal Error[edit]


Statement[edit]

P. 28-2

Redo steps in the proof of the Trapezoidal Rule error by trying to cancel terms with odd order derivatives of "g"

Solution[edit]

We begin with equation (5) on P. 21-1 which is the result of transformation of variables on equation (1) P. 21-1

(Prob 4 HW4) (P. 21-1)

(5) on P.21-1

From Prob 5 HW4, we can express the above equation as:

(1)

Step2a[edit]

Integrating the term withing the square brackets by "Integration by parts" Prob 7 HW4 we can rewrite (1) as follows

(2)

In order to eliminate terms with even powers of we need to remove terms with odd derivatives of .Therefore, the boundary term in eqn (2) above must be set to zero by selection of .

We have from eqns (1 and 2)P. 21-3

(1)p21-3

(2)p21-3

Setting gives and hence we get

(3)

So following this method, the next term to be eliminated will have P4(t)

(4)

Setting and solving we get .Continuing on these lines to we get the eqn (2) in the form

(5)

(6)

(7)

manipulating the terms yields,

(8)

(9)

Transforming g(t) back to f(x) we get [see [prob 6 HW4]

(10)



To see the difference in the two approaches we must compare the equations from the two methods. From (1) P. 27-1 we have

(11)

It is seen that the first term in eqn 10 is a summation as against the term of eqn (11) which is dependent only on the end points.