Mjordan2nd

The error when using a rule for approximating a definite integral is given by the following equations:



I have no idea how these were derived, but these were the equations given to us. We were told that we'd learn how they were derived in a numerical analysis class. This is all fine and good, but I'm not sure exactly how I'm supposed to pick the x value to calculate the coefficient k to find the largest error that there can be. The book offers a very convoluted explanation of this, and it almost seems to arbitrarily pick an x for the calculation of k at one of the limits of integration. This seems to make sense, but I'm not sure exactly how it picks whether to use the a value as x or the b value. If anyone could explain this to me, or link me to a site which has a decent explanation of this, I would greatly appreciate it.

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pegasus001

If i recall good you dont take arbitrary x-es. Your function is given in a table like

So you can find the midpoint easily. I`ll open a bit the book of my numerical analisys for you.

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Rashakil

That coefficient should be the maximum absolute value of the given derivative on the interval [a,b].

What the book is saying is that there is some value of x where the error uses _exactly_ that value f''(x) (or f''''(x), depending on the formula). But you can't really find that x, unless you can integrate exactly and do some unnecessary magic; the important thing is that it's less than the max-norm of the function f'' (or f'''') on the interval [a,b]. So if you use the maximum possible |f''(x)| or |f''''(x)|, you'll be overestimating the error, which is a good thing (since underestimating the error is a bad thing).

pegasus001

So these errors can be applied only when you dont have a polinomial function ie of the type y = ax^n + ..... Because when you have the function in that way there is no need to find the error couse there is no error. So when you are given the function in a table manner than you use the lagrange formula or the newton formula to find a polinomial function by using the finite differences. Example :
So ask your teacher for the function and then apply these things.

Good luck.

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You never test the depth of a river with both feet.
The believer is happy. The doubter is wise.
Free speech carries with it some freedom to listen.
The next generation will always surpass the previous one. It`s one of the never ending cycles of life.

Rashakil

No, polynomial functions will have errors too when approximating their integrals...

grimpirate

K basically represents the maximum y value of the function on the interval [a, b] of the particular derivative. In the midpoint and trapezoidal K would be the maximum value of the second derivative on that interval. In Simpson's rule it would be the maximum value of the fourth derivative in that interval. So basically you use the stuff you learned from taking the derivative and analyzing the graph of an equation to determine the maxima of a function on a given interval. Here's a small example:
We can deduce without any fancy calculus analysis that the largest value the second derivative will have on that interval occurs at x = 1. This is because any x larger than 1 will produce a y value smaller than the one which occurs at x = 1 (you can check this with a table of values or by doing the necessary maxima calculations). Therefore, K = |f''(1)| = 2

Now if you were given a number of subdivisions you could easily find the error for both the trapezoid and midpoint approximations. Hope that helps.

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pegasus001

What does these formulas do exactly?

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You never test the depth of a river with both feet.
The believer is happy. The doubter is wise.
Free speech carries with it some freedom to listen.
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Mjordan2nd

Thanks guys, that helped a lot!

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