integral (not a homework question)

[Graphmastur]

Simply for the record. THIS IS NOT A HOMEWORK QUESTION. Okay, now on to the homework looking question...

Okay, I have this integral:
[attachment=3168:gamma(Z).tiff]
If I want to find the antiderivative (indefinite derivative?) of the function, how do I do it? This will allow me to solve it right? If not, how do I solve it quickly?

My overall goal is to solve it fairly quickly, or in a reasonable amount of time. Is this possible? I will be using a HUGE n value, I don't know what z is just yet, and t is the variable, so.... please help.

Also, ignore the "/; Re(z)>0" thing. I think that means it's just real numbers greater than 0, or something. Since I only want to use real positive numbers for z, that's no problem.

Also, I understand the derivative. Antiderivatives are still above my current level. So, how do I solve this?

Thanks in advance.

[wesley]

Well, that's pretty funny. That's no ordinary antiderivative you've got there. That is a gamma function. You learn about those in probability and statistics with applied calculus (calculus 2).

Also, yours has a limit. This is the most important thing. It doesn't mean you keep integrating for every value of n, that would take forever of course! (Considering the fact that n goes to infinite.) Thus, you simply substitute infinite with an arbitrary variable until you are done integrating, at which point, you will insert infinite for that variable you placed in the equation.

Also, the antiderivative is simply an integral, not an indefinite derivative.

This problem can be solved in a quick manner of time, but it takes some good thought and understanding the gamma function.

This is NOT, by any means an elementary calculus problem. If it isn't homework, do you mind sharing its source?

I didn't dive too deep into the gamma function in my class last year, but we went over it a little, so it is familiar.

[Graphmastur]

[wesley]

Oh, I see.

Why don't you just make an implementation with memorization. This would be a great deal better than recursion (as you most-likely know).

[wesley]

Ah, man!

I can't believe I got hooked on this problem!

Just to calculate that integral, you need to take just as many steps as multiplying all the numbers from 1 to n. The integral is recursive, you need to use integration by parts.

[thornahawk]

It looks to me like a not-too-docile (for numerical work, that is) integral, not in any event any more advantageous than (for instance) the defining infinite integral for the gamma function.

"This is a programming question, because I need to be able to implement it."

There are so much better methods for calculating a gamma function; the Stirling and Lanczos approximations being the most popular. You should look into those instead of looking for integrals.

If my words are not convincing enough, you might wish to peruse the following Mathematica output:


The entries of that table are of the form -log₁₀(|(f(n,z)-Γ(z))/Γ(z)|) (i.e. the negative common logarithm of the relative error), where n is the row heading, z is the column heading, and f(n,z) is the expression inside the limit. As you can see you only gain one digit of accuracy every time you increase n ten times. Much ado about nothing when there are ways to compute the gamma function more accurately with less effort.

thornahawk

[Graphmastur]

[thornahawk]

Hmm, I seem to have screwed the labeling up in that post; the rows are supposed to be the z values and the columns the n values for that integral. As you have seen it required a limit of 10⁸ (and approximately the same amount of function evaluations) to get merely 8 digit accuracy. Stirling and Lanczos, written properly, can give 12-14 digit accuracy with less effort.

thornahawk

[Graphmastur]

[thornahawk]

If you truly insist on this route (which I still think is a profligate waste given that I have already mentioned better alternatives which you still choose to ignore), I shall give another piece of evidence that might force you to reconsider.

Your integrand (the expression inside your integral) is of the following form:

(1-t/n)ⁿt^z-1

where n will be a large number in your calculations (since you can't take limits on the computer anyway) and z is the argument of the gamma function. Any numerical integration algorithm, from the simplest trapezoidal to the most convoluted adaptive quadrature scheme, will involve evaluating your integrand for a large bunch of numbers within the open interval (0,n).

Since you say you will be using Java's floating-point machinery, I assume you are already aware of concepts like overflow/underflow and machine epsilon. To be specific, if t/n ever becomes less than machine epsilon, your integrand is effectively the same as t^z-1. That impacts the accuracy of calculations in your left portion of the interval.

On the right portion of your interval (near n), you risk overflow/underflow in your computation. As an exercise, pick a large enough n and a fixed z, and try computing the integrand for values of t near 0 and near n. You'd be surprised.

As a final note, I wish to remind you that one way of reducing the burden of computing the gamma function is to reduce the argument to a preset unit interval, e.g. (0,1] or (1,2]; the functional equation for gamma takes care of arguments outside your preset interval.

thornahawk

[Graphmastur]

I will use one of the other methods, if they can give me completely accurate. Is this possible?

[wesley]

[thornahawk]

Think of gamma like the other more common transcendentals: sin(1) and ln(2) are never exactly computed, be it Java or some other programming language; these quantities are (usually) computed up to the number of significant figures the machine supports. Since the gamma function is rational only for nonnegative integers, you can only approximate it.

Which brings me to the question that should have been asked right off the bat in the very first reply: why exactly do you need a gamma function. Usually computing it only comes up in the context of computing a more complicated function in a lot of applications. What are you trying to compute, and how does gamma enter that picture?

thornahawk

P.S. I've alluded to P. Luschny's work in a number of my previous posts (including some of my posts related to computing the gamma function); do have a peek at his site.

[Graphmastur]

[thornahawk]

Frankly, your insistence on quadrature tells me you've never even attempted to graph the function you're trying to integrate. 'cause, a good look at the graph alone would have told me that even the most sophisticated adaptive quadrature scheme would fare just as badly as a simple Riemann sum. Trying to pretend a decidedly ill-behaved function can be adequately represented as a (piecewise) polynomial has never ended well.

I have no further advice to give; I think I have done my utmost here, and there seems to be no dissuading you. I can only hope whatever lessons you learn from attempting this exercise are not easily forgotten.

Cordially,

thornahawk