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I am in nerd paradise. I have just used my new calculator to solve two nonlinear equations of two variables with Newton's multidimensional method.

Before I get to that:

I'm still trying to put the calculator through its paces, trying to figure out all the things it can and can't do (which I may talk about later), and just how to get the things it can do to work. I'm mostly done with that, but just today I learned how to use the "rectangular to polar" function (it unintuitively stores the results in the E and F memory), and I still don't know what the difference is between the CALC [ulate] and "=" buttons.

One function that I really couldn't figure out was the line copy function. Like most current pocket calculators, this one has one line for operations and one for the result. Plus, several previous lines are stored, in case you want to go back to them or edit them. Anyway, the line copy function puts a bunch of previous lines on one single line. The example the book uses is like this:


1 + 2 =

3

3 * 4 =

12

4 - 2 =

2

Then the line copy function is used, which results in the line

1 + 2 : 3 * 4 : 4 - 2

Pressing the equals key gives 3, 12, 2, in sequence.

Wow!, I thought. That must be the most useless function ever!

Then today I was thinking about the Ans [wer] key. With my older calculator, which didn't have the equation line, the Ans key was needed so you could do any calculations on the fly, without needing to plan ahead much.

So, say you wanted to multiply 2 * 16...

32.

Now you realize you need to take the fifth root of that number. Then you'd hit

5 (universal root button) Ans = 2.



BUT, due to my new calculator's equation line, the Ans button can be used for more than that! It can be used multiple times in one line (while retaining the same value as the previous answer. My old calculator updated the answer value after each calculation).

I quickly realized this could be used for Newton's method in one variable. Newton's method is

[new] X = [old] X - f(X)/f ' (X)

Where f is the function you want to equal zero.

Well, to use that on my calculator, all you have to do is initialize X by hitting some number and equals, then type in

Ans - f(Ans)/f ' (Ans)

where of course you have to enter the real function and figure out the derivative on your own.

But after you do that, all you have to do to iterate Newton's method is hit the equals button over and over!

This still isn't very useful for my calculator. It, after all, has a built in one variable equation solver using Newton's method, and that saves you the trouble of finding and typing the derivative, and hitting the equals button.

(It can be useful on other calculators that aren't as good as mine, though. For instance, it can be used with some recent TI calculators)

But then I got the idea of wondering if this could also solve two variable equations. Instead of using the Ans button, I could use the calculator's independent memories (it has A-F, and X and Y). Newton's method in one variable is a single step method. In two variables, it uses a bunch of steps, but I figured that would be a good use of the line copy feature!

I thought a good equation to test my calculator with was X = Log (X) [Note: It is my custom to write Log to mean Log to base e. I think writing Ln, unless I'm referring to the Ln button, is stupid] This equation has no real roots.

After thinking about it, I decided the best first step would be to raise e to both sides power, giving X = exp(X). If the variable would be X + i Y, this becomes

x + i y = exp(x)( cos (y) + i sin (y))

or

x - exp (x) cos (y) = 0

y - exp (x) sin (y) = 0


Two real (nonlinear) equations in two real unknowns.

I have to admit that I tried solving this equation with my calculator a few different ways before I found one that worked. The problem is that the final combined line can only have around 75 combined keystrokes. So, a straightforward typing in of the equations wouldn't work. This also means that equations that are too complicated probably would never work. But I eventually streamlined this particular problem.

Okay, to use Newton's method in multiple variables, I said the first expression (to the left of "= 0") was function "a" and the second expression was function "b". Then,

da / dx = c

da / dy = d

It just so happens that for this particular problem,

db / dx = -d

db / dy = c

making things easier for us. (Again, more complicated problems might not work)

This is what I typed

2 -> X
2 -> Y



(Just arbitrary seeds. However, all seeds won't work, if they're too far from the solution. I tried (5,5), and it lead to (0,0), which is not a solution, but generated a math error, because c = d = 0 there.)


X-eXcosY -> A

Y-eXsinY -> B

1-eXcosY-> C

eXsinY -> D

X + (BD - AC)/(C² + D² -> X

Y - (CB + DA)/ (C² + D² -> Y



And I entered everything just about literally as it's displayed. On this calculator, the juxtaposition BD works as B*D, so I suppressed the * to minimize key strokes. Also, you don't need to close parenthesis at the end of lines.

After I typed that in, I hit the "up" button enough times to take me to the first line ( X-eXcosY -> A ) and hit copy. After that, all you have to do is repeatedly hit "enter" to iterate.

Needless to say, this saves a lot of time compared to having to enter all of those formulas for each iteration.

The final answer, by the way, is

0.318131505 + i 1.337235701

That line copy method can also be used for infinite series.

Example, to use Leibniz's formula for pi



4 -> A

3 -> B


A - 4 / B + 4 / (B + 2) -> A

B + 4 -> B


Hit up to get to A - 4 / B + 4 / (B + 2) -> A , hit copy, then hitting equals over and over (slowly) calculates pi.

Of course, this is a terrible method for calculating pi (especially since all worthwhile calculators have pi permanently stored in memory anyway). But that's because Leibniz's formula converges very slowly. This is just to demonstrate proof of principle.

I imagine this could be useful if, for example, you have to solve a differential equation at a certain value, and it can only be solved as an infinite series, and it converges at a worthwhile rate, and no actual computer is nearby.

There are ways around that, even on 4 function calculators.

But the easiest way to do it with the fx-115MS is to use the fraction feature.

Say you want to know the remainder of 145695 when divided by 19.

Type in 145695 (fraction button) 19, hit equals, it evalutates to 7668 _| 3 _| 19 (ie., 7668 and 3/19). 3 is the remainder.

There are limitations, though: The whole number, numerator, and denominator, and the two fraction symbols of the result must take up 10 or fewer characters. This isn't that much of a limitation, though. I've seen TI calculators that only allow a 3 digit denominator, regardless of the rest of the number.




If you wanted to use the method of a four function calculator:

Example, find the remainder of 763549283 when divided by 9781. This problem has too many digits to use the fraction technique.

First, you just divide them. That gives 78064.54176....

Round it to 78064. Then type 763549283 - 9781 * 78064.

This equals 5299. That is the remainder.

I'm trying to learn to use a TI 89 Titantium. Anyway, long story short, an exam in a class I took a few years ago asked us to write a program to numerically calculate the energy levels of the Morse Potential (using the WKB approximation). It was a good exam problem, because it requires numerically finding roots, integrating, finding a root of a function that isn't explicitly defined, etc. Basically, putting a bunch of stuff together.

So, I tried doing the same thing on the TI 89, and found that it was easier to do, because the calculator has built in functions for root finding and integration. You don't have to use your own subroutines.

In fact, this was more accurate -- Exact, actually -- on the TI 89 than the results I got a few years ago (it turns out the Morse potential can be solved exactly [analytically as opposed to numerically] though our teacher didn't tell us that, and I doubt any student bothered look it up, though a Google search gives the result )

It might even run faster, though the old program's speed depends on the computer its operating on. The limitation in accuracy to the old program is in the integration routine. Maybe it could be fixed by using Gaussian integration rather than Simpson's or Bode's rule or something.

Anyway, this is the program:



Looking at it now, I see I could improve it in a few ways. For one thing, a comment says "Quantum number", but it doesn't refer to anything. In an older version of the program, I defined n after that line. Now, n is a parameter sent to the program.


I plan on trying to convert a bunch of numerical recipes programs to TI basic. I already converted gammaln.

Here's a much more general topic, which doesn't really have anything to do with math: Texas Instruments reliability.


I said before that a long time ago I had two TI calculators that didn't last me a year. Other TI lab calculators have also died (we also have a dead Sharp lying around somewhere in the lab; I'm surprised about that. My first calculator ever was a 4 function Sharp and it was built like a brick. It's probably still around and working even today, though it'd be at my mother's house), and someone at work told me they had some TI calculators for their house that died.

And check out the Amazon.com reviews. There are all kinds of reports of dying displays or other problems.

Anyway, there isn't much mystery as to why my two older TIs died. I opened them up to see what was going on. I think most TIs are screwed in at a few places, and the plastic is welded at a couple others, and you have to snap that. After I got them opened, I noticed something which really surprised me:

THE DISPLAY AND SOLAR CELL WEREN'T CONNECTED TO ANYTHING!!

They were just passively sitting there. The only thing that kept them connected to the rest of the calculator was the pressure of the front and back plates. After you got those off, the display could be lifted right out.

I tried a simple fix of one of my calculators by putting something inside of the case to apply extra pressure to keep everything in place, which sort of worked, but not very well.

I tried opening the lab TI 30XII to see if it was true for that model as well, but just removing the screws didn't do it, and I didn't want to snap the plastic as it isn't really my calculator.

The funny thing about that is, the TI 30XII is both solar and battery powered. So, you're never supposed to replace the battery? I guess the thing really is designed to be a disposable calculator!

My Casios are also both solar and battery powered, and I have opened them as well. Their screen is connected to the calculator's circuits with a data ribbon. And the solar cell is soldiered in. The new one, the 115ms, also has a little plate (it looks like either stainless steel or aluminum) behind the screen, as protection I guess. (It's been longer since I opened the older 115s, so I don't remember exactly what it's like inside) I have to confess the new one doesn't feel quite as "solid" as the old one, but I still have more confidence in it than TI stuff.

I notice that the TI 36X's box says "Built to last!" Well, we'll see.

Even my newer Casio calculator is a little quirky with log(1+1e-9) (the older Casio is fine), but I don't have that calculator with me right now and I don't want to post about it at length now anyway.



I've found a couple of good sites dedicated to calculators.

One, dedicated to old Hewlett Packard calculators, is the HP Museum.

One for Texas Instruments is Datamath. The Texas Instruments one has info for every single model of TI calculator ever, including variants. It also has some other trivia as well (I learned there that Casio made the first graphing calculator ever, in the '70's!).


It might seem kind of hard to believe there are people still dedicated to ancient calculators (it might be less hard to believe if you consider there are people still dedicated to videogame systems that are more than 20 years old).

But consider. When I bought my old Casio (which would have been around '96), it actually said "Complex" (rather than "Scientific") on it, to indicate that it could do math with complex numbers, and to distguish it from TI calculators that generally couldn't (I still think most TI calculators can't). And it couldn't really do complex math, just complex arithmatic (it couldn't find arccos(2), or even sqr(-1)).

Well, the HP 15C could do real complex math... and deal with 5 by 5 matrices... and had a solver... and a numerical integrater... and was programmable... all back in the early 1980's!

I made some posts above about how the Casio ffx-115ms had some impressive features, but if you take a long view of history, the only impressive thing about them is that they're in a calculator that costs only $15. Otherwise, the 15-c can do things the 115ms can't. (The 115ms isn't really programmable, but I've said about how it can do some simple iterative calculations)



Also, it seemed kind of impressive to me that the TI-89 can do symbolic math. It's like a mini-Maple, but more user friendly and portable. I kind of wish I had something like that when I did graduate level Electrodynamics. I never thought I'd see something like that in a handheld calculator.

Well, I guess I was wrong, because the HP 28S apparently could do symbolic math, again, back in the '80's!

Or, maybe it seems impressive