Calculator Accuracy & Usefulness

[Wes Loewer]

I teach both math and physics. So while I stress sig figs in physics, in math class I'm fond of proclaiming that in math, we are unencumbered by reality. :-)

[Maximilian Hohmann]

(05-25-2015 02:59 PM)Claudio L. Wrote:  (GPS receivers are again a good example).

Certainly. But nobody ever used a pocket calculator for the calculations involved.

[Garth Wilson]

Quote:Reading this thread it gives the impression that in engineering there's no need for precision. Far from reality!
Engineers don't only use "measured" quantities. We also use complex algorithms with thousands of computations.

True, but not every application that uses thousands of computations needs a lot of digits to prevent errors from accumulating. And when you do, say you do an FFT on a set of 2K samples taken with an 8-bit (ie, 256-step) A/D converter. Six digits is enough.

Quote:Surveyors are a good example

There will always be applications that need a lot of digits, like surveying and financial calculations with interest. I would say very little of engineering is that way though.

Quote:In electronics, engineers need to count 'ticks' of clocks that are gigahertz in frequency. Lots of digits there.

Crystal accuracy is usually in the range of 5 to 7 sig figs. I have an 8-digit frequency counter with a crystal oven and a trimmer to calibrate it against WWV. The specifications say you can get the error down to 0.1ppm, but by luck I was able to get .01 (although I didn't check to see if that went out the window within the next day or week). Crystals are the most precise components we can get, or at least I can't think of any components more precise off the top of my head. Analog-to-digital and digital-to-analog converters probably come next (if you get the ones with lots of bits, like 24, and even then, the accuracy of the voltage reference may not be nearly that good), but those are often applied to audio myths. For some great lessons and demonstrations on the "golden-ears" baloney, watch http://www.youtube.com/watch?v=BYTlN6wjcvQ . It starts out with a lecture at a conference and then goes to demonstrations from quality digital audio equipment in his studio that let him manipulate the exact amounts of different negative characteristics the golden-ears people said were major problems, and you can see if it is or not. In one part, he searched for the most offensive noise he could find, and superimposed in on a fine string quartet performance, and you couldn't hear it at all if it was more than 9 bits (ie, one part in 512) down from the desired program material. On my less-than-ideal PC speakers on the desk, I couldn't hear it until it was 7 bits down. Yes, it's on YouTube which compresses the audio and loses information, but he gives the URL where you can download the raw wave files if you want to, otherwise see what he does with various experiments right there.

An accomplished man I met on another forum wrote that he worked on Nimbus weather satellite. It had a 250 mW transmitter and their 85-foot dish with Maser amplifier could achieve autolock at -150 dbm at a range of 3000 miles. However, the launch vehicle suffered an early burnout and the orbit was degraded, causing the satellite to be lost to the free world for three days. No one at NASA Goddard or the DEW line or any tracking stations around the world were able to locate any evidence that it existed. They scanned the skies continuously in every sort of random and geometric pattern for days, but no cigar. Finally, he had an idea and whipped out his trusty circular pocket slide rule and came up with a reasonable approximation of what the orbit would look like with a 10-second premature shutoff and suggested to his boss that they point the antenna in a certain direction at a certain time. It worked. They found it. The boss wanted to know where he had studied astrophysics, but he didn't think the boss would appreciate knowing about his plastic slide rule so he simply shrugged it off.

[SlideRule]

(05-23-2015 11:42 PM)Matt Agajanian Wrote:  Further, how useful, essential, reliable, etc. were HP Spice/Spike, Woostocks, V'Gers (yes, Voyagers), 41s in real engineering disciplines with their 10-digit accuracy?

Uncertainty is always introduced with measurement and engineers primarily utilize quantities attained by measurement. We (engineers) simply tend to maintain a disciplined sense of humor concerning the physical world and any attendant nominal mensuration.

BEST!

SlideRule

[Dave Shaffer]

I haven't figured out how to quote part of a previous post, but in reference to

"Crystal accuracy is usually in the range of 5 to 7 sig figs. I have an 8-digit frequency counter with a crystal oven and a trimmer to calibrate it against WWV. The specifications say you can get the error down to 0.1ppm, but by luck I was able to get .01 (although I didn't check to see if that went out the window within the next day or week). Crystals are the most precise components we can get, or at least I can't think of any components more precise off the top of my head."

Check out atomic frequency standards! We use these for our radio astronomical observations. Current Hydrogen maser systems have stability/accuracy at 1e-14, and atomic standards in development are at 1e-17 and hope for 1e-18 !!! There are H masers at radio telescopes all over the world for Very Long Baseline Interferometry (VLBI).

VLBI geodesy approaches 1mm precision on intercontinental baselines (up to 10000 km or so long, or a part in 10 billion (1e10) or thereabouts), so the required calculations are generally done in quadruple precision. We measure 100 GHz frequencies to microhertz and 100 millisecond time delays to picoseconds.

A relatively cheap GPS time and frequency standard will easily give you 1e-9 or 1e-10 precision, so trade in your crystal system!

Cheap rubidium standards (surplus units are ~$100 on Ebay) will give you a few orders of magnitude better, with some care.

The H masers are more like a quarter to half million dollars, and need a nicely controlled thermal chamber (0.001 degree regulation) for best performance. As we discovered once upon a time, they are also magnetometers: the unit in the base of the 140' antenna at NRAO (Green Bank, W.Va.) changed frequency at the 1-e13 level when the metal telescope structure overhead (all 2600 tons!) moved around, changing its self magnetization effects as well as dragging the Earth's magnetic field around.

There are truly some cases where 15 digits or more are useful. (However, we don't use calculators for our analysis!!)

[Jim Horn]

While it is often said that slide rules got men to the moon, such calculation often required far more precision than any slide rule could provide. But what they did provide was a fast way to get an answer to ~3 or 4 digits of accuracy when the alternatives were far less available, slower, etc. So a 'slip-stick' could quickly give you an answer which, if it was one that was promising (i.e. it showed you *could* reach orbit / the moon / some other very difficult problem), the alternatives would then be used to determine the more exact (and needed) answer.

Many of us from those prehistoric days had our CRC Handbook of Mathematical Tables or similar listings of logarithms, trig functions, etc., often including logs of trig functions and other aids to computation. The good ones included interpolation constants for each entry so any value could have its function evaluated to high precision with nothing else but pencil and paper. Of course, some had access to computers running custom code to solve the hardest problem (high order Runge-Kutta numerical solutions to find lunar trajectories, for example). But before doing the gruesome work, slide rules were the standard way to get you in the right direction as well as checking the final answer of more exact calculation to ensure it wasn't way off due to some programming or input data error.

Around 1978 or so I took a US Air Force course on satellite orbital perturbation analysis. Needless to say, the final exam involved some really messy equations for which I was as prepared as any with my HP-67 programmable calculator. But for laughs, when sitting for the exam I pulled out my pocket Post / Hemmi 1461 Versalog II slide rule and said I was ready for the exam - which got quite a few chuckles. Of course, the '67 was used and blazed through the problems.

Right up until its batteries died (typical charge was about 2 hours of use) as I had studied so hard the night before I had forgotten to plug it in to charge. So the Post came out again and I was able to complete the test with it.

Mercifully, another student finished early and loaned me his calculator which I used to redo all the answers. None of the slide rule answers were wrong though the new answers carried more digits. But the second time (with the calculator) was far easier...

[Thomas Radtke]

<ot>

(05-25-2015 11:23 PM)Dave Shaffer Wrote:  I haven't figured out how to quote part of a previous post [...]

Only way appears to be deleting text from a fullquote.
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[Garth Wilson]

(05-26-2015 02:58 AM)Thomas Radtke Wrote:  <ot>

(05-25-2015 11:23 PM)Dave Shaffer Wrote:  I haven't figured out how to quote part of a previous post [...]

Only way appears to be deleting text from a fullquote.
</ot>

Put [quote] and [/quote] around the part you want to quote. (I embedded something else here to keep them from doing their job.) It would probably be good to have a sticky giving all the phpBB stuff you can do in a post-- sizes, colors, etc..

I am aware of atomic clocks, but they are not even in most engineering-type companies, let alone businesses or households.

[Claudio L.]

(05-25-2015 09:57 PM)Garth Wilson Wrote:  I would say very little of engineering is that way though.

Not as little as you may think. Structural or mechanical engineering problems can pile up nodes and degrees of freedom like it's nothing, and matrices grow huge even for simple problems.
Yes, you can design a crane on the back of a napkin with an approximated method and a slide rule, and get a decent approximation, but if you put the crane in modern design software, it creates a model with several hundred nodes and solves a large sparse matrix doing thousands of operations. If you forced the software to work with 3 or 4 digits, you'd get a much worse approximation than the slide rule. The moment you use modern "generic" methods that don't have specifically targeted simplifications, internal precision becomes more important to guarantee numeric stability.

In electronics, almost any PCB board with a fast clock needs to be ran through a finite element program to determine electro-magnetic interference between adjacent tracks.
It's not something perceptible to humans, like your audio examples, but a CPU running at GHz speeds can't tolerate that a bit gets randomly flipped on a track. In the past, this didn't matter because speeds were slower and components and tracks were much larger.
As things get faster and smaller, we need more numerical digits with more advanced methods to properly design, fabricate and even operate our engineering creations, and this is a lot of the engineering being done today.

[Garth Wilson]

(05-26-2015 04:24 PM)Claudio L. Wrote:  In electronics, almost any PCB board with a fast clock needs to be ran through a finite element program to determine electro-magnetic interference between adjacent tracks.
It's not something perceptible to humans, like your audio examples, but a CPU running at GHz speeds can't tolerate that a bit gets randomly flipped on a track. In the past, this didn't matter because speeds were slower and components and tracks were much larger.
As things get faster and smaller, we need more numerical digits with more advanced methods to properly design, fabricate and even operate our engineering creations, and this is a lot of the engineering being done today.

You're talking my language. In the 1980's, I worked in applications engineering at a company that made UHF power transistors, mostly for military communications and radar. I also do PCB layout, including mixed-signal, with low-frequency analog, 2.5GHz RF, chip antenna, digital, and switching power supplies on the same board, and the signals from one section cannot be getting into another or you'll have a mess. On the last one of this type that I laid out, the performance was far better than the client expected. They were using an ANT IC and the manufacturer recommended a particular antenna design printed on the board, but I proposed a particular chip antenna to replace it to get a more spherical radiation pattern and use less board space. The result was an RF range was 20 times what the ANT IC's manufacturer said we would get. They were blown away, but the client was also happy because now they could reduce the RF output power and make the battery last longer. I calculate for filters, transmission-line impedances, Smith-chart calculations, propagation delays, etc., and never need very many sig figs.

[Matt Agajanian]

(05-25-2015 02:59 PM)Claudio L. Wrote:  

(05-24-2015 09:28 AM)Tugdual Wrote:  We don't need very deep accuracy simply because the measurement systems already include this concept in the real life. You don't measure bacteria sizes in km and/or the distance in between galaxies in mm.

Reading this thread it gives the impression that in engineering there's no need for precision. Far from reality!
Engineers don't only use "measured" quantities. We also use complex algorithms with thousands of computations.
Just try an ODE (Runge-Kutta anyone?), do a shootout at various precisions and see where it takes you when you are thousands of tiny steps away from the boundary condition.
We can take the final result and only use a few digits, and we can round the input to a few digits, but all intermediate calculations NEED some precision or it'll be a disaster. Changing the third digit on the boundary condition might get you off only by 5% or so, but after a thousand steps, you'll be an order of magnitude off if you only use 3 digits at every step.

Surveyors are a good example: they use exclusively measured quantities, yet they were historically desperate to get more precision: multiple measurements with the tape, compensating for the temperature of the tape, compensating for the sag, compensating for the stretching of the tape due to axial stress, etc. are just examples of how much they needed that precision in the past. Today, they use GPS to get an accuracy within 1 cm anywhere on earth. Satellites are orbiting 20200 km above earth, and within 1 cm you need to operate with numbers of the order of 2 020 000 000 cm, that's 10 digits bare minimum right there.

In electronics, engineers need to count 'ticks' of clocks that are gigahertz in frequency. Lots of digits there. Transmission of signals needs accurate clocking (GPS receivers are again a good example).

Old engineers with slide rules didn't have to deal with this, or simply couldn't even if they wanted to. These examples are a consequence of the gradual increase in precision over the years, of which early calculators were a very important part.

Some areas of engineering need more precision than others, but for sure calculators with more precision (and later computers) enabled a LOT of progress.

Hello there.

The reason I asked the original question is because of accuracy discrepancies in older calcs as compared to accuracy in today's calcs. I mean and meant no disrespect to the sciences. I only want to clarify how useful, accurate, relevant are/were the computationx that calcs gave then and now.

[bshoring]

This may be a side issue, but don't the Topcat, Spice, HP-41's and Voyagers all have exactly the same level of accuracy, as well as the late Woodstocks (67, 27 & 29C) ?

Bob

[d b]

Yes. Go to
http://www.rskey.org/~mwsebastian/miscprj/results.htm
and scroll down to the group at 9.000417403

there is also other groups of HPs,
basicly the saturns, from the 87 to the 49g at 8.99999864267
at 9.0044076644, the classics and early woodies
the 9G & 30s at 9 flat
and all by their lonesomes, the 33s & 35s

Mike states somewhere that closeness to 9 is not the absolute accuracy of these machines. This is forensics only, to determine whether different calculators use the same algorithm and maybe the same ROM. It's the accuracy with the input of 9 degrees, which is the way he started doing this. The comparative ratings (precision?) would change with another beginning number.

[Matt Agajanian]

Which brings me to my original, yet modified question., plus a follow-up question.

With this wide range of calculation discrepancy in calculators, how can any calculator be trusted as yielding reliable answers? And, when using any calculator, how and where do you truncate the calculated result to yield a reliable answer?

[SlideRule]

(05-29-2015 01:14 AM)Matt Agajanian Wrote:  ... how can any calculator be trusted as yielding reliable answers?

PURE numbers have infinite precision, measurements do NOT. I would suggest this posture as a FIRST sieve in assessing a calculation as a reliable answer. Calculations in the Physical Science disipline necessitate approximation PRIOR to any number crunching, especially when the numerical inputs to the analytical expression are attained through measurement, since ALL measurements intoduce UNCERTAINTY. I trust the reliabilty of my calculator because I understand the limitations of my disipline.

[attachment=2114]

BEST!
SlideRule

[Matt Agajanian]

(05-29-2015 02:03 AM)SlideRule Wrote:  

(05-29-2015 01:14 AM)Matt Agajanian Wrote:  ... how can any calculator be trusted as yielding reliable answers?

PURE numbers have infinite precision, measurements do NOT. I would suggest this posture as a FIRST sieve in assessing a calculation as a reliable answer. Calculations in the Physical Science disipline necessitate approximation PRIOR to any number crunching, especially when the numerical inputs to the analytical expression are attained through measurement, since ALL measurements intoduce UNCERTAINTY. I trust the reliabilty of my calculator because I understand the limitations of my disipline.



BEST!
SlideRule

THANK YOU!!

THANK YOU!!

THANK YOU!!

This satisfies my lifelong pondering because it completely answers how calculator results are regarded and accepted within context, applicability and reason.

As often as I've heard that the sciences and other math-dependant professions demand accuracy, this answer fully clarifies how calculator results are regarded and understood.

[Dieter]

(05-27-2015 10:41 PM)Den Belillo (Martinez Ca.) Wrote:  Mike states somewhere that closeness to 9 is not the absolute accuracy of these machines.

That's also why I said here several times over the years. ;-) The essential point is that right after the first step of the "forensics" test the calculator is working with approximations. It does not calculate the cosine of sin(9), but that of 0,1564344650.

For those who are interested in the "perfect" result a calculator with n digits working precision should return, I once suggested a small program for the 34s. The latter is perfect for such tests since it features up to 30+ digit precision as well as a nice RSD function that rounds intermediate results to n significant digits, thus emulating a "perfect" calculator with that precision.

Here is the link to that post in the old forum. So a perfect 10-digit device should return 9,000417403, and one with 12-digit precision should yield 8,99999864267. And that's exactly what most HPs since the mid-seventies deliver.

Dieter