Sharp EL-W506T vs. Sharp EL-W516T

[Mjim]

The Sharp EL-W516x (and presumably the newer EL-W506T) seems to handle a pretty big number of divisions though; I tried a division count of 131,072 which worked fine, although you are probably going to want to go up the road and purchase some biscuits and boil the jug while you wait.

I'm guessing you meant the EL-506X, in which case I agree that being limited to n=100 seems pretty small unless you are dealing with small changes in area. The Casio fx-570MS, which also uses the Simpson method, can handle up to 2^n where you can choose a maximum value of n=9, or 512 divisions which is significantly better.

I'm off now on a Tangent about the Casio MS series, so please skip if uninterested!:
What though impresses me especially about the Casio MS series integration (despite not being as flash as the ES & EX series with it's faster and more accurate integration algorithm) is that it truncates the integration result to a certain number of significant figures depending on the number of the divisions and the function.

It really seems to be truncating the uncertain digits based on an internal error range calculation, but I find it hard to believe it uses the Simpson's rule error function since it requires calculating the 4th derivative of the function first and finding it's maximum....I mean if it was really calculating the error function, that is more impressive then taking the integral as far as I'm concerned

Simpson Error function is |Es| at the bottom of this linked page: http://tutorial.math.lamar.edu/Classes/C...grals.aspx

You can see this error function at work, take the function e^x (done on Casio fx-82MS temporarily upgraded to fx-570MS):

Interval [0,1]:
integrate(e^x,0,1,n=4) = 1.71828
integrate(e^x,0,1,n=6) = 1.71828183

Interval [0,2]
integrate(e^x,0,2,n=4) = 6.389
integrate(e^x,0,2,n=6) = 6.389056

With the Simpson error function it is always taking the absolute maximum of the 4th derivative, so by extending the interval to 2, we get a steeper interval and so higher maximum error (ie e^x when x=2, and since the derivative of e^x is just e^x, so will it's forth derivative be). This seems to be what is happening in the above examples, which explains why you have less digits over the larger interval despite using the same number of divisions. Regardless of how they did it, it amazes me that they managed to pull it off on a calculator that doesn't even have natural textbook entry.

In addition, if the confidence is too low it will throw a math error:
integrate(e^x,0,0.1,n=1) = 0.1052 (Casio auto-algorithm logic selects n=3)
integrate(e^x,0,1,n=1) = 2 (Casio auto-algorithm logic selects n=3)
integrate(e^x,0,2,n=1) = Math Error (Casio auto-algorithm logic selects n=5)

I can't say for certain what error checking algorithm is being used, but as long as the estimate gives an equivalent or higher error range back it means you can count on the digits you see. From a bit of testing:

integrate(e^(x^2),0,5):
Wolfram answer: 7,354,153,747.83713

64 Divisions:
Casio: 7,000,000,000
Sharp: 7,368,738,187
|Es| <= 26,106,748.11

Adding the error in the actual result is somewhere between:
7,342,631,439 - 7,394,844,935

The first 2 significant figures could be either 73 or 74 so the Casio gave the correct result back as you can rely only on the first significant figure.

512 Divisions:
Casio: 7,354,200,000
Sharp: 7,354,157,599
|Es|<= 796.7147252
7,354,156,803 - 7,354,158,396
Looks like the first 6 significant figures can be counted on with rounding: 735416. The Casio could of gone for another significant figure, but it still returns the correct answer.

Note: that |Es| gives the maximum possible error for a function using the Simpson rule given the number of divisions, and it's 4th derivative; not the actual error.

In addition it seems to automatically vary the number of divisions depending on the function and it's interval, so certain integrations will go faster. This is something Sharp needs themselves if they are sticking with the Simpson method (both the error and auto interval logic).

I guess with the ES and now EX series stealing the show from the MS with it's better integration algorithm (Gauss-Kronrod?, it's above my current math ability) and nicer display it probably doesn't matter anymore, but this is a rather impressive feat IMO. I also like that the MS is still the only series with the "Copy-Replay" function, or the ability to scroll up several functions ago and copy that function and all others that proceed for editing.

Look at me blabbing on about the Casio MS series on a post about a new Sharp.

Back to the Sharp:

I still prefer my Sharp EL-W516X over the Casio fx-115es Plus (only by a tiny bit, as I much prefer the equation solver, Integration and differentiation on the Casio), but mostly because of these 3 reasons:
1) The ability to save your Writeview history when powering off.

2) Being able to store relatively big equations in 4 memories and easily recall them (eg the |Es| error tests were done on the Sharp, since I can store both the 4th derivative maximum, the |Es| function to make use of that, as well as the Summation version of the function (though I could just of used the built in Sharp one!)).

3) Great Summation functionality, since you can specify the interval size (algebraically as well).

If it was the Casio fx-991ex instead, well, I'd probably go with that for it's much better display and faster processing, but I would miss terribly the ability to save my calculation history or formulas.