|Re: On horsepower|
Message #16 Posted by Rodger Rosenbaum on 25 Feb 2006, 2:23 p.m.,
in response to message #15 by Palmer O. Hanson, Jr.
My recollection of the article is that the reviewer didn't seem to understand that while the taking of the square root of two followed by squaring sequence yielded a 2 in the TI-59 display that did not mean that the value in the display register was exactly two. He also did not seem to understand that a similar effect can be obtained with the HP-41 by simply operating in fix 8 mode.
He never mentioned TI calculators in the entire article except to mention that TI used the same size magnetic cards with some of its calculators as the HP41 used. Let me quote from page 136 of the article:
The most fundamental defect in the architecture of the HP-41C, inadequate numerical precision, is a serious flaw indeed. Numbers are represented, both internally and in the display, with 10 decimal digits; there are no guard digits. As a result, inaccuracies are quite often introduced into the least-significant digit. For example, SQRT(2)^2 is evaluated by the calculator as 1.999999999. For operations on some data, the corruption goes still deeper and 2 or 3 digits become suspect. There is something absurd about the world's fanciest calculator not being able to give results accurate to more than seven or eight decimal places.
I already pointed out that he misunderstood some things. In the next paragraph, he says;
Actually, a subsidiary problem is more serious than that. Conditional tests on data are carried out on the full 10-digit representation. Consequently, a test that effectively asks "Is SQRT(2)^2 equal to 2?" will give a false result, wihch can lead a program astray.
Didn't he realize that this sort of problem, due to finite register size, occurs with every calculator that does floating point arithmetic? It is to be expected, and the calculator user must deal with it.
The author of that article wasn't comparing HP calcs to TI calcs specifically.
As to what number of digits are needed in the mantissa that clearly depends on the application. I admit that some applications are working with only a few digits. In inertial navigation we were working with at least seven digits as in accelerometer measurements good to a micro-g over a range from a micro-g to ten g's, etc. Slide rules were never good enough for processing that kind of data. In the early sixties we used Friden's a lot. If my memory is right we had eight or ten digits available with the Friden's.
In an earlier post, Ron Ross said:
The Hp41c was so superior to the Ti-59 in EVERY respect.
And you responded:
Back in January 1981 a complimentary article "The HP-41C: A Literate Calculator" appeared in BYTE. The author did comment "... The most fundamental defect in the architecture of the HP-41C, inadequate numerical precision, is a serious flaw indeed. ..."
My aim is to point out that the author of that article had a number of misunderstandings, and to suggest that the phrase "serious flaw indeed" is perhaps overstated. Are we to believe that having only 10 digits is a serious flaw, but that 3 more turn a serious flaw into a non-problem?
The nature of ill-conditioned problems is that you lose a certain number of digits, whether your calc has 10, 12, 13 or 14 digits. Any problem that loses 3 digits of accuracy on the HP-41 will almost certainly lose 3 digits on a TI calc too. Losing 3 digits out of 10, which would happen only rarely, still leaves 7 correct. That's enough for almost all practical problems. Hundreds of thousands of engineers and scientists found the 10 digits of the HP41 more than adequate.
I've said before that, all else being equal, having more digits is a good thing; sometimes, though, all else isn't equal. However, I've noticed that the recent TI calculators have gotten MUCH better with their math.
But comparing the HP41 with the TI59, or comparing the HP49 with the TI86 is like putting a welterweight in the ring with a heavyweight. It's just not fair in some respects and the outcome in those respects is just what you might expect. The TI86, which I own, can give a 14-digit square root; the HP48 can only give a 12-digit result. Just as we would expect. That's why I want to evaluate calculators on another basis--do they do what they should do? I plan to post more on this topic.