Integration error: AUTO vs APPROX

[johnhelt]

Dear all. This is my first posting, but I have encountered a strange problem on the TI89 (AMS 2.05) that I feel need the attention of the board.

The problem is in the definite integration of the following equation (from x=0 to x=0.4), which I had to solve at some point:

-12.02335*(x+1.66667)^2/(x^2+0.17238*x-0.43095)

Now, if you solve this equation from 0 to 0.4, with AUTO mode on, you should get something like -124.107.

If you then select approximate you should get 52.626. (you can ofcourse also be in AUTO mode and still solve the equation approximately simply by using diamond-enter).

I have calculated the integral numerically from various kinds of quadrature, and I get the latter result again and again. This means, the default solver in AUTO mode, somehow makes an error! Have you heard about this before? I find it rather concerning considering to be honest!

Cheers,
Tobias

[BlainUSM]

Well, my TI-89 has the same problem. The function you describe (let's call it f) is continuous, positive, and increasing on [0, 0.4]. Therefore, the correct answer should be positive and between .4*f(0) ~ 30 and .4*f(0.4) ~ 102. Mathematica produces 52.625... which is correct and consistent with what the calculator produces in approximate mode. Additional experimentation with similar problems will probably help to pinpoint the reason for the error in AUTO mode.

[BlainUSM]

Integration isn't really my specialty, so someone with solid integration skills may be better able to respond to your post.

For now, here's what I've found:

The TI-89 evaluates definite integrals, i.e. int(f(x),x,a,b) differently in AUTO mode and APPROXIMATE mode.

In AUTO mode, the calculator will first attempt to find an indefinite integral F(x) and then evaluate F(b)-F(a). If the calculator fails to find an indefinite integral, then the original input will be returned.

In APPROXIMATE mode, the calculator will simply apply numerical integration to approximate the integral.

(I figured this out by trying to integrate the Normal density function from negative 4 to 4. The indefinite integral of the Normal density curve is known to not have a closed form. Therefore in AUTO mode, the calculator should return the original input (which it does). However, in APPROXIMATE mode, the calculator returns an actual number (close to 1), which means that some kind of numerical integration is being used in APPROXIMATE mode.)

For your function, the error does not seem to arise solely from being in AUTO vs. APPROXIMATE mode, but rather from the fact that the integration function does not produce the correct answer. The previous claim is based on the finding that both correct and incorrect answers may be obtained in either mode:

In APPROXIMATE mode

Define f(x) = -12.02335(x + 1.66667)^2/(x^2 + .17238-.43095)

Define g(x) = int(f(x), x)

g(0.4)-g(0) = -124.107... (which is incorrect)

whereas

∫(f(x),x,0,0.4) = 52.625... (which is correct)


In AUTO mode

nInt(f(x), x, 0, 0.4) = 52.625...(which is correct)

while

∫(f(x),x,0,0.4) = -124.107... (which is incorrect)

Therefore, the mode doesn't appear to be the underlying cause of the problem.

Until TI offers a software fix, the TI-89's integration function ∫ should be checked with numerical integration (by using APPROXIMATE mode or nInt). What a bummer!

Tobias, thank you for the useful shortcut for performing calculations in APPROXIMATE mode (Green Key and then Enter). Your post brings up an important problem. I have sent a message to TI. Hopefully, a software fix will be released soon.