goofy frequencies

How about starting with 1/1 and looping on...

if the fraction is below the target, bump the numerator

if above, bump the denominator

Wouldn't that work? A PC could do some decent fraction of a billion iterations per second. Who needs elegance these days?

John

the

I think that's a point in the American Scientist article posted by The Phantom under a different thread:

The author summarizes at the end, this way:

"The fact is, the design of simple gear trains is no longer a computationally interesting problem, because computation itself has overwhelmed it. With so much calculating power at your fingertips, it's hardly worth the bother of being clever. You can solve gearing problems by brute-force, using methods that would have been unthinkable for Camus or Brocot, or even for Merritt, who was writing hardly more than 50 years ago. If you need to approximate some ratio, just have the computer try all pairs of gears with no more than 100 teeth. There are only 10,000 combinations; you can churn them out in an instant. For a two-stage compound train, running through the 100 million possibilities is a labor of minutes."

"The whirling gears of progress have put the gearmakers out of work."

Jon

That's what CBFalconer's brute force program does. For finding fractions with the small numerators and denominators this thread has been discussing, and using a modern computer, brute force is just fine.

However, when doing the computations on a slow machine, such as a pocket calculator, continued fraction methods may be enough faster to make the difference between a boring wait and immediate answers.

Elegance is its own reward.

Programmable-calculator routines are the ultimate in programming non-portability. Not so good as regards commenting, either.

John

Sounds good to me. I might even be able to implement that as a PSpice function ;-)

...Jim Thompson

This reminds me of a conversation my wife and I had with Dr. Fouts, some years ago. (Dr. Fouts operates the Chimpanzee and Human Communication Institute" at Central Washington University and our family is a regular contributer. His advisors were Allen and Beatrix Gardner.) We were talking about some of the work of other researchers into primate communications using methods other than sign language using body parts (hands, by and large.)

We talked about research using plastic, colored chips of various colors or shapes (tokens) used to facilitate communication in lieu of using hand-signs. And I discovered that Dr. Fouts was clearly angry about that form of research. He told me that it was cruel to the animals. Because, he said, "What happens when they don't have their plastic chips around? How do they communicate?"

I understood immediately what he meant. My daughter is profoundly autistic and I remember an event that happened many years ago and taught me a lesson I always keep in mind. She came over to me with a happy expression on her face and with some eagerness hand-guided me into the kitchen. She just started, I thought, to move my hand towards the refrigerator. But then she stopped and just stood there and wouldn't move. And slowly, I saw her face change. She grew increasingly 'dark.' And then she just threw herself onto the floor and started crying and sounded so sad and miserable.

I had no idea, at first. But then after some seconds it dawned on me that she wanted me to make something and was easger to get me into the kitchen to let me know. But that when she directly faced the terrible difficulty in letting me know what it was, the impossibility of it was there and the weight of it simply crushed her emotions and broke her.

I called out to my wife and asked, "Have you recently made anything for Athena that she liked, but might be 'a little complicated' to prepare?" Becky said, "Yeah! Some scrambled eggs and vegetables in the skillet."

As soon as I started drawing out the supplies for this, Athena looked upwards from the floor and noticed that I was getting "the right stuff" out. When she saw that, she got up and watched as I made her the same thing. She was so excited that the communication had worked successfully.

But the success isn't the point. It's what happens when someone knows exactly what they want to communicate and then realize they are in an impossible situation to express it. It is very painful. And it can lead to profound frustration and even anger, at times.

Returning to that time with Dr. Fouts, he said that chimps get frustrated and even angry when they have been taught to communicate and then find themselves in a situation where 'the tools of that communication' are unavailable. He felt it was cruel to teach an animal such boons in communication and help them develop skills with tools they cannot always have around. This is why he felt that sign language was the only humane way of proceeding. A chimp always has its body and if it learns to use what it naturally has for communication, then there is always some chance at success when the desire to communicate occurs. But if that language is tied ONLY to certain items that cannot possibly always be present, then deep frustration and possibly even anger (and violence) can be the result, at times. He felt it was not only harmful to the chimps, but also potentially dangerous to the researchers, to force them to rely solely upon external devices for their communications.

We don't always have a laptop with us, a fancy desktop computer with predictive branches, dual and quad cpus, etc. But everywhere we are, we have our fingers and some dirt to draw in. Your point about the pocket calculator reaches towards this point, as well. There are many times we still need to 'communicate' with others or ourselves about something. And it pays to know methods that don't require having a Cray (or equivalent) at hand. Better still, if all we need are our fingers and some dirt, or perhaps just some paper and pencil. Having knowledge reduces that frustration from a lack of tools. And, it "feels" wonderful, too, because of the freedom it provides.

Jon

We are well and truly spoiled by the computational power so readily available nowadays.

In a paper published in the IRE Transactions on Circuit Theory in 1958, the author describes a rather complicated algorithm for finding the roots of a

The algorithm was run on an IBM 650 computer and took 80 minutes, giving a result with about 4 or 5 accurate digits.

My HP50 calculator can solve the problem in 5 seconds, with the expenditure of about 1 watt-second of energy, returning 11 accurate digits.

On a PC, the solution is essentially instantaneous.

But....there is still a place for elegant, non-brute-force algorithms. A problem that would take several hours on a modern computer WITH a good algorithm just wouldn't have been possible to solve at all 50 years ago, and could take months (or years!) on a modern computer using a brute force method.

Indeed. It reminds me of the projected progress of computation: in maybe 20 years we will have the computational power to brute-force simulate the human brain. The rub? In 40 years, we'll actually know how to write that program. Heck, for all we know, we have enough computational power today to run a well-optimized brain program. But the complexity is so high, no team of software engineers can comprehend enough to do it.

Maybe in 60 years, we'll have the computational tools to ask a (then) modern computer, "write me a brain program which is optimized to run on a Core 2 Duo". Little more than a parlor trick using severely antiquated hardware; like porting Guitar Hero to the NES. (Incidentially, by this time humanity will have outlived its intellectual usefulness. Scary to imagine.)

Tim

I doubt it. Computers still aren't very good at simple stuff like speech recognition, face recognition, language translation, driving a car, or running a tennis-playing robot. I'm guessing that a human brain is far more complex than a gigawatt of computers can match, even

And Moore's Law is about to hit the wall.

John

fractions with

and using

pocket

Hell and gone off topic and off thread. But one of the more insightful posts i have read in a long time. The all too clear comparison and the clear demonstration of our humanity being hostage to our ability to communicate is very memorable.

What makes you say this? It seems like every few years somebody announces some kind of "quantum limitation" or some such, and the researchers just keep blithely breaking the known limits of what's possible.

My mind still boggles when I consider a processor with a clock that's faster than the output freq. of my microwave oven.

Thanks, Rich

It is not about the physical limits; the physical limits are quite far. The new technologies are not paying off, and this is what really sets the limit. The saturation point is reached. There is a whole bunch of already developed advanced technologies that are not deployed widely because of the economical reasons; X-ray lithography is one example.

If your processor clock would be x10 times faster then now, would it provide any better value to you? If so, how much are you willing to pay for that? I don't see much difference between the computers of today and the computers of 10 years ago.

Vladimir Vassilevsky DSP and Mixed Signal Design Consultant

I recall the good old days when papers in the IEEE Proceedings solemnly announced that 100nm was the end of the line for semiconductors. Now, IIRC, there are transistors made from graphene at the 1 nm level.

The physical limits aren't far at all. Transistors started getting worse instead of better at the 45 nm node.

Cheers

Phil Hobbs

Which means it is nearly time for a change in technology, probably to carbon.

Everybody's hoping so, but it's sort of like going from bicycles to airplanes in one go. The fabrication problems are completely different from silicon. The era of Mead-Conway style scaling is definitely over.

Cheers

Phil Hobbs

As the cost of a mask set hit $20e6 or so. Fewer and fewer designs can justify the expense of a billion transistors. So the fab equipment business will have a shrinking base of applications to spread their costs over.

John

Well, beyond carbon there has got to be a change to self assembling nanotech. After all, that's how Nature makes brains.

Memory is the obvious application. There's no letup in memory requirements, nor is there likely to be.