simpsons rule question (1 Viewer)

[streetrcr]

hi

can someone plz explain simpsons rule with the odd and even method. my book only has the long method.

thanks

 

[jb_nc]

http://en.wikipedia.org/wiki/Simpsons_rule

 

[streetrcr]

i dont think it explains the odd and even method though, but im not sure coz i dont understand most of it

could someone explain it in simple terms to me?

 

[jb_nc]

That one?

Or this?

 

[streetrcr]

that is the method which is in my book, but i know there is a short cut method with odd and even. using that method above you need to do the equation for each function value which means using it multiple times and i always end up getting it wrong coz the equation is so long and messy.

 

[beentherdunthat]

Simple

h/3 [ (First + last value) + 2(sum of odd values) + 4 (sum of even)]

and h is the difference b/w the x1 and x2 values

eg. x1 = 0

x2=0.5

x3 =1

than h = o.5

k?

 

[streetrcr]

by first and last do u mean x1 and x3?

are the odds x1, x3 and evens x2?

am i on the right track?

 

[bewareoflys]

Hm.. I thought the Simpson Rule was

h/3 [ (First + last value) + 4 (sum of odd values) + 2 (sum of even) ]

and your on the right track .. correct me if im wrong though.. i ain't the best in maths either..

 

[beentherdunthat]

Hm.. I thought the Simpson Rule was

h/3 [ (First + last value) + 4 (sum of odd values) + 2 (sum of even) ]

and your on the right track .. correct me if im wrong though.. i ain't the best in maths either..

Nope. Remember 4 for even. 2 for odd or T-odd

Okies

This how it goes- Colours correspond with explanations

X Value x1 x2 x3 x4 x5 x6
0 0.5 1 1.5 2 2.5
f(x) 0 0.25 1 2.25 4 6.25 --- y=x^2 Lol i cbf'd to use a calculator so i guessed the y values hehe.

Edit: Damn thing didn't come out right. LOL. AH well .. you get my drift.

Anyway

First and last in this case - x1 + x6 = 5.25 +0 = 5.25

2 (odd) - Odd here is x3 and x5- value = 1+4=5 hence 2(5) = 10
4(even)- even here is x2 + x4 = value = 0.25 + 4 =4.25 hence 4(4.25) = 17

H value... is the CONSTANT difference b/w x values hence 0.5 ( see above where i bolded)

Finally-

0.5/3 ( 5.25 + 10 + 17)

and then you get your answer.

 

[iEdd]

It is 4 odd and 2 even. However, you must start at y₀.
Then you get this formula:
A~= h/3[y₀ + y_n-1 + 4(y₁ + y₃ + ...) + 2 (y₄ + y₄ + ...)]
And that's why it is y_n-1, because your nth term is labelled with a number less than it, which can be seen that the first term is y₀.

 

[PC]

You guys make it all too hard. Use a table!

Example 1:
Find an approximation for the area under the curve y = log₃x from x=4 to x=8 using 5 function values (4 strips).

x f(x) W P
4 1.2619 1 1.2619
5 1.4650 4 5.8599
6 1.6309 2 3.2619
7 1.7712 4 7.0850
8 1.8928 1 1.8928
----------
Sum = 19.3615

.: Area = 1/3 x h x Sum
= 1/3 x 1 x 19.3615
= 6.4538

Abbreviations:
x = value of x
f(x) = corresponding function value
W = "weighting", always goes 1, 4, 2, 4, 2, 4, ..., 1
P = "product" = f(x) x W

Then the area is just 1/3 x h x Sum, where h is the difference in the values of x.

BTW, Trapezoidal rule can be done the same way except that the weightings go 1, 2, 2, 2, 2, ..., 1 and the Area = 1/2 x h x Sum.

Easy ... and tidy!

 

[Buiboi]

It is 4 odd and 2 even. However, you must start at y].

its 2 odd and 4 even

 

[iEdd]

Okay, but only if you label the first value y₁.

 

[streetrcr]

kewl. think i get it now. thnx