HP’s Solve & Integrate-Still ahead of its time.

[Matt Agajanian]

Hi all

Now that numerical integration and solver functionality is pretty much standard on TI & Casio models, are the 15C, 42S, 33S, 32S II, etc. routines & algorithms still considered more advanced than TI & Casio graphing & non-graphing solver & numerical integration functions?

[Steve Simpkin]

The HP-34C was HP's first calculator with root finding (Solve) and numerical Integration functions. Introduced with the HP-33C, HP-37E and HP-38C at the beginning of July 1979 (around the same time as the HP-41C), it was one of HP's last LED calculators.

One interesting bit of trivia for this model is that HP Marketing did not really want to make this calculator. Their marketing studies convinced them that their customers did not want a calculator with a "Solve" key. Even after HP committed resources to making the HP-34C, HP management did not want to fully document the new functionality.

In the following excerpt, Dr. William Kahan recalls his role in creating the Solve and Integrate algorithms used in the HP-34C and then effectively outmaneuvering HP middle management to convince them to make the HP-34C.

"HAIGH: So you’re talking about the 85 at the moment, and the solve key, is that relevant to what became the 15C you mentioned earlier? Or is that a different machine?
KAHAN: It got into the 34C first, and then the 15C; it got into the 34C in a way that is going to make an interesting story. I came up to Corvallis one day while we were working on some of these calculators and found the group with whom I worked—Stan Minz’s group—deeply depressed because they had been told that they were losing money. And this had to be paradoxical, because their financial and other calculators were the only ones that were selling— the financial calculators were selling in numbers. And none of the other groups were selling anything much. They were all behind schedule, you see. So how could it be that this division was losing money when it was the only one selling anything? Well, that was very depressing. Next time I came up, there was a picnic scheduled on the grounds. Finally, the accounting people had figured out what was going on. You see, they charged all the overhead expenses of this establishment—the fab line producing semiconductor chips, not just for the Corvallis group but for other parts of HP—and everything else was charge pro rata to sales. Okay? These guys were the only ones selling, so they were the only ones who were bearing all the overhead cost for everything. They were paying for the lights and everything else for everybody, losing money. They were the only ones who had an income that could pay for some of this stuff. So suddenly they turned into heroes, and Stan had scheduled a picnic for Friday afternoon on the grounds to celebrate. He had spiked the punch, which was very definitely against Hewlett-Packard policy. But I can mention this because I don’t think he works for Hewlett-Packard anymore, hasn’t for a long time, so I can mention this. And he was in an extremely jolly mood. As you know, I am teetotal, so I can observe this stuff with a certain equanimity. “Professor Kahan,” he said, “you’ve been nagging us about the damn solve key for the longest time. I tell you what, when I went to college, I had an enormous amount of trouble with integration. Now, if you can tell us how to get an integrate key onto a calculator,” said he, figuring it was impossible, “if you can figure out how to do that, then I’ll let you have the solve key, too.” And so a couple of weeks later, in collaboration a bit with Dennis, we came up with what I thought was a really good integrate algorithm.
HAIGH: And actually, it’s the integrate key that you describe in a paper in the HP Journal. [W. Kahan, “Handheld Calculator Evaluates Integrals,” Hewlett Packard Journal, August 1980, 23- 32].
KAHAN: Yes. I describe the solve key, and I describe the integrate key. But I certainly don’t describe the way they came into the systems, you see. That we will talk about when we talk about Robert Barkin. Anyway, that was a promise he now lived to regret, because he had no support from the marketing people that any such enterprise would pay, and he didn’t want to go out on a limb. However, he had something else. There was a young woman he’d been compelled to employ because of Hewlett-Packard’s affirmative action and nondiscrimination policies. He didn’t want her, and he was stuck with her. But suddenly he saw an out. He now knew what she could do. She could work with me on the design of a calculator with solve and integrate key which, as it happens, he hadn’t the slightest intention of producing.
HAIGH: And was the solve key one of the things you saw as a crucial feature to realize this personal ambition of a calculator that would support everything that the students needed?
KAHAN: Absolutely. Well, that had to be in there because that deals with what are called implicit functions, you see. You’ve got a function. The function is defined as a solution of an equation. So absolutely, and the integrate key was needed too. But you see, I told you when these ideas came into my mind. This was 1978. Initially, I had in mind that the solve key was needed for the business world. That’s who I really intended it for. It was only when he prodded me into producing an integrate key, too, that I said, “Well, with solve and integrate—this could be a calculator for engineers.” So it’s 1978 or thereabouts, and I worked with this young woman. I regret that I have forgotten her name. Her name probably figures somewhere in these documents about the contributors to these calculators and maybe if I read through them, I’d find her name and say, “Yes, that’s the one.” The way we worked was that each of us had an aluminum box connected by an umbilical cable to one of these calculators, like the one I’ve shown you in my hand, the 34C, except originally that case design was intended for other calculators first. And there were little paper labels over the keys, and we could re-label the keys. And inside were the actual guts of the calculator except that the ROM was replaceable. What she would do would be to microcode up a calculator, put the microcode in the ROM, see that it worked on her box, and send another copy of the microcode to me by UPS. I put the microcode in my aluminum box, and I’d see what she had. I’d look it over. I’d say, “You know, we really ought to do this-or-that instead.” So I’d send back a letter, and also the old ROM; these ROMs were erasable with ultraviolet light and somewhat expensive. So there was this exchange back and forth, and since nothing else seemed to be happening, there’s no deadline or anything, we also improved the programmability. We decided to make it easier to program and so on.
But still, nothing was happening. I didn’t know at first why Mintz said, “Okay, we’re going to build this now.” I was told by the guys up in this group that what had happened was that Mintz had seen his engineers clustering around her desk, and had wondered about that. First, she wasn’t pretty, and second, she was married. What the hell are they doing there? He discovered what they were doing was using that box. For example, if they had to design a transistor circuit and figure out what the bias should be, and there are some resistors that then have to be done to get the right bias, you’ve got to solve some equations. They’re only slightly transcendental. There is no closed form. Of course, what you can do is you write a Fortran program. You go to the HP3000. You compile it. You get your result and so on. Everything’s fine. But, they could just go to this box and key the thing in, and they got the solution almost instantly. It took much less time than writing a Fortran program. And what’s more, writing a Fortran program, there is, as I explained to you, a significant probability that you’re going to have to run it more than once. I mean, when I got back to Toronto from Stanford, I made some changes, which got the run rate down to four. You’d run four times in order to get one job. And I don’t think the HP 3000 could have been any better than that, although HP had some good computer guys.
He also saw the guys who were designing the power supply, and it has a little transformer in it. It uses saturable core in order to get the oscillations to come out. It’s got hysteresis loop. You wonder, “What is the efficiency?” Well, it’s a certain amount of power dissipated, because every time you go around the hysteresis loop, you lose a little bit of energy. So they have to integrate that in order to figure out how much energy they’ve lost, you see. It’s a simple little integral. It’s not all that difficult except that it can’t be done in closed form. You’ve got to do it numerically. Instead of writing a program in Fortran to go to the HP 3000, they’d key in this little thing, click click-click. They wait for a little while, and out comes an answer. Not only that, the answer comes out with an uncertainty. You’re told, “The integral is equal to this, plus or minus that.” And where does the plus or minus come from? Well, it depends on how well you know the thing you’re integrating. And they had some idea of how well they knew what they were integrating, so they had not only the power dissipation but, also, how uncertain you could be about it—which was really rather more than you could get from the Hewlett-Packard 3000.
The penny dropped in Stan Mintz’s mind. That’s how the HP-34C was born. They agreed to do it, and then like a thunderclap, they were appalled when I said, “You know, we’re going to have to put some guidance into the manual because people who use these keys, especially the integrate key, they can fool themselves. These things cannot be foolproof. There will be situations where people will get misleading answers, and they need a little bit of guidance about that.” “Kahan, you just told us to do this stuff, and now you tell us that you’re going to get wrong answers! I mean, all this time, we’ve been listening to you tell us how to get the right answer, invariably, every time!” Well, the difficulty was then that I’d have to go to the manual writers. But there was a Hewlett-Packard policy which said, “We are professionals, and we sell to professionals. We tell them what the device does, and they figure out how to use it. We’re not writing tutorial material in our manuals.” And I tried to explain, “Look—this time you’ve got to put some tutorial material in the manuals. You really must. Otherwise, folks are going to fool themselves.”
Well, the managers wouldn’t do it, but I had persuaded Barkin, and I can’t remember the name of the other guy. It’s probably in there somewhere. There were two guys who were writing the manuals, and I persuaded them. I think persuasion is the wrong word. This was a case of subversion. I subverted them and got them to do something that their managers had told them not to do. The manual writers listened to my arguments and decided that I was right, and their managers were wrong. And that’s a dangerous decision, you know. You can get fired for that. They wrote the two extra chapters into the manual, which said something about the solve key and something about the integrate key, and a little bit to warn you. And I had written up some more stuff, which ultimately got into the Hewlett-Packard Journal. And the managers were outraged. They had said explicitly, “Don’t do that.” And now their guys had done it. And they said, “Take it out.” And they were told, remember, “If we take it out, it’ll delay the appearance to market.” You’ve heard that story before. That was it. The managers had just been blackmailed by time-to market, so they left it in. Then afterwards, they did a survey, and they discovered that the customers loved this stuff. In fact, the customers would often say they had bought the calculator because they’d been told that there was advice in the manual about these problems, which was advice they actually needed not only for the calculator but also when they solved similar problems on the big machines. And so when I came up with these articles, they were perfectly happy to print them in the Hewlett Packard Journal. I was told by the editors some years later that they had had more requests for reprints of these articles than for all their others put together.
KAHAN: Okay, so they put this, you might call it tutorial material, into the manual. I believe some of the manuals are now available online, and so you can look at it yourself. Well, of course, the particular writing staff manager was pissed, and the marketing people were, of course, annoyed about a violation of policy; when they interviewed customers, though, they had these little questionnaires—you buy something; they’ve got a questionnaire, you send it back. They would see that people actually liked that material and, in some cases, had bought the calculator because the material was there, so they felt confident as purchasers that they could use it. I think ultimately there was some award from the Willamette Valley Chapter of the Technical Writers’ Association that went to Barkin and company for writing what was considered an exceptionally good manual. That was the first of two occasions when I heard about an award for an HP manual. So there were guys who looked at me sidelong at HP, but there wasn’t any hostility. They figured I had done the right thing. The calculator was extremely successful, as such calculators go."

Dr. Kahan's full oral history PDF document can be found at: https://drive.google.com/file/d/1Jlg9EWQ...zwcol/edit

[Csaba Tizedes]

(10-29-2023 03:14 PM)Matt Agajanian Wrote:  (...) routines & algorithms still considered more advanced than TI & Casio (...)?

HP15C Manual

Open at page 200 and try that sine integral with any CASIO.
I don't believe engineers at CASIO were not ashamed during these decades...

[Matt Agajanian]

(10-29-2023 06:13 PM)Csaba Tizedes Wrote:  

(10-29-2023 03:14 PM)Matt Agajanian Wrote:  (...) routines & algorithms still considered more advanced than TI & Casio (...)?

HP15C Manual

Open at page 200 and try that sine integral with any CASIO.
I don't believe engineers at CASIO were not ashamed during these decades...

This rings a bell. Several months ago I tried that integral with my
TI 30X Pro MathPrint
TI 36X Pro
Casio fx-115ES Plus (editions 1 and 2)
Casio fx-991ES Plus 2nd Edition
Casio fx-991EX Classwiz and
Casio fx-991CW Classwiz (yes, I actually bought one)
Sharp EL-W516T & X

With the Casio & Sharp models, I needed to put 10^-9 (or smaller) as the lower limit so I could obtain that 1.6054 result. Clearly, the 34C algorithm knows how to integrate better than the TI, Sharp, & Casio which use Simpson's method. I think the 34C uses Romberg method if I'm not mistaken.

[carey]

(10-29-2023 07:43 PM)Matt Agajanian Wrote:  Several months ago I tried that integral with my
….
Casio fx-115ES Plus (editions 1 and 2)
Casio fx-991ES Plus 2nd Edition
….
With the Casio & Sharp models, I needed to put 10^-9 (or smaller) as the lower limit so I could obtain that 1.6054 result. Clearly, the 34C algorithm knows how to integrate better than the TI, Sharp, & Casio which use Simpson's method.

The Casio User's Guide for the fx-115ES PLUS and fx-991ES PLUS 2nd Edition says they use the Gauss-Kronrod method.

[brouhaha]

(10-29-2023 07:43 PM)Matt Agajanian Wrote:  I think the 34C uses Romberg method if I'm not mistaken.

The 34C, 15C, and the Advantage Pac for the 41 all use the Romberg method, with a transformation for non-linear sampling. The 15C Owner's Handbook and the 15C Advanced Functions Handbook, and Kahan's HP Journal article are the best references.

I suspect that the Saturn-based calculators that have numerical integration do the same, but I haven't looked.

[Valentin Albillo]

.
Hi, Eric,

(10-29-2023 09:51 PM)brouhaha Wrote:  I suspect that the Saturn-based calculators that have numerical integration do the same, but I haven't looked.

Correct, from HP Journal July 1984:

"INTEGRAL proceeds by computing a sequence of weighted sums of the value of the integrand for selected values of the integration variable, IVAR. These sums are accumulated in an extended-precision, eight-level modified Romberg scheme."

Regards.
V.

[dm319]

Putting aside the maths for a second, I am so impressed with the UI of solve. The HP-12c has such an optimal layout, and genius to use a press followed by numeric input to input, whereas other presses will solve. The same goes for soft buttons on the HP-42s which seems to emulate that set up.

Actually, my only criticism of the UI is that I wish the solver could put all pointless stuff it outputs into y, z and t, somewhere else! I have it set to NSTK on my DM42, but it is still a pain, and means the HP-12c is oddly more elegant when solving multiple TVM problems.

[MBS]

(10-29-2023 07:43 PM)Matt Agajanian Wrote:  This rings a bell. Several months ago I tried that integral with my TI 30X Pro MathPrint
[...]
Clearly, the 34C algorithm knows how to integrate better than the TI, Sharp, & Casio which use Simpson's method.

I recently got a TI 30X Pro MathPrint. It solves the integral accurately, taking about 0.25 seconds. The HP-28s with 10^-5 tolerance takes about 3 seconds.