Re: continued fraction program

Yes. I think so. Here is the result for 25.4/8:

1: 3/1 -55,118.110 ppm 2: 16/5 7,874.016 ppm 3: 19/6 -2,624.672 ppm 4: 54/17 463.177 ppm 5: 73/23 -342.349 ppm 6: 127/40 exact

You tell me if that is what you were wondering about.

Jon

That's interesting. So, 127/5 = 25.400000....

I wonder if this has any significance to the chosen exact ratio of

25.4mm/inch ?

Bob

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Yes, 25.4 mm per inch would be 254mm per 10 inches but can be reduced to

127mm in 5 inches.

RogerN

Dang. That was too easy and obvious. I was hoping for something deeper. Oh well.

;-)

Bob

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Yeah, that looks like it would do it.

So if number 4: 54/17 was close enough the compound gearing might be something like:

Multiply the fraction to get compound gearing, I'm doubling here.

2* 2*3*3*3 2* 17 = 6 * 18 2 * 17

So my final gears might be

36:12 and 36:34

So I'd have, from spindle to lead screw, a 34 driving a 36 attached to a 12 driving a 36 on the lead screw.

RogerN

That doesn't appear to be what I call a continued fraction. It seems much closer to the following (which is purely standard C - will run anywhere):

/* Find best rational approximation to a double */ /* by C.B. Falconer, 2006-09-07. Released to public domain */

#include #include #include #include #include #include

int main(int argc, char **argv) { int num, approx, bestnum, bestdenom; int lastnum = 500; double error, leasterr, value, criterion, tmp; char *eptr;

value = 4 * atan(1.0); if (argc > 2) lastnum = strtol(argv[2], NULL, 10); if (lastnum 1) { tmp = strtod(argv[1], &eptr); if ((0.0 >= tmp) || (tmp > INT_MAX) || (ERANGE == errno)) { puts("Invalid number, using PI"); } else value = tmp; } criterion = 2 * value * DBL_EPSILON; puts("Usage: ratvalue [number [maxnumerator]]\\n" "number defaults to PI, maxnumerator to 500"); printf("Rational approximation to %.*f\\n", DBL_DIG, value);

for (leasterr = value, num = 1; num < lastnum; num++) { approx = (int)(num / value + 0.5); if (0 == (int)approx) continue; error = fabs((double)num / approx - value); if (error < leasterr) { bestnum = num; bestdenom = approx; leasterr = error; printf("%8d / %-8d = %.*f with error %.*f\\n", bestnum, bestdenom, DBL_DIG, (double)bestnum / bestdenom, DBL_DIG, leasterr); if (leasterr ratapprx 6.175 Usage: ratvalue [number [maxnumerator]] number defaults to PI, maxnumerator to 500 Rational approximation to 6.175000000000000 4 / 1 = 4.000000000000000 with error 2.175000000000000 5 / 1 = 5.000000000000000 with error 1.175000000000000 6 / 1 = 6.000000000000000 with error 0.175000000000000 19 / 3 = 6.333333333333333 with error 0.158333333333334 25 / 4 = 6.250000000000000 with error 0.075000000000000 31 / 5 = 6.200000000000000 with error 0.025000000000000 37 / 6 = 6.166666666666667 with error 0.008333333333333 68 / 11 = 6.181818181818182 with error 0.006818181818182 105 / 17 = 6.176470588235294 with error 0.001470588235294 142 / 23 = 6.173913043478261 with error 0.001086956521739 247 / 40 = 6.175000000000000 with error 0.000000000000000

which indicates some errors in your program to me, or a misunderstanding.

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 [mail]: Chuck F (cbfalconer at maineline dot net) 
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... snip ...

My earlier answer was almost accurate, but I ran my program with the wrong input. Try:

[1] c:\\c\\ratapprx>ratapprx 3.175 Usage: ratvalue [number [maxnumerator]] number defaults to PI, maxnumerator to 500 Rational approximation to 3.175000000000000 2 / 1 = 2.000000000000000 with error 1.175000000000000 3 / 1 = 3.000000000000000 with error 0.175000000000000 10 / 3 = 3.333333333333333 with error 0.158333333333334 13 / 4 = 3.250000000000000 with error 0.075000000000000 16 / 5 = 3.200000000000000 with error 0.025000000000000 19 / 6 = 3.166666666666667 with error 0.008333333333333 35 / 11 = 3.181818181818182 with error 0.006818181818182 54 / 17 = 3.176470588235294 with error 0.001470588235294 73 / 23 = 3.173913043478261 with error 0.001086956521739 127 / 40 = 3.175000000000000 with error 0.000000000000000

So my comment about errors is nonsense.

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 [mail]: Chuck F (cbfalconer at maineline dot net) 
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25.4/8 is 3.175.

John

No problem. By the way, I only provide convergents. Just keep that in mind. The CF for 3.175 is [3;5,1,2,1,1]. This means exactly 6 convergents. Which is what you saw me post.

Jon

By the way, that program ALSO is a calculator program. In includes a few predefined values, like PI and E, but also all of the usual math functions like SIN, COS, COSH, EXP, LN, LOG, ABS, COTH, SQRT, and you name it. You can enter numbers, expressions, assignment statements, access the prior answer as part of the next formula, enter FP numbers in pure HEX format, use CF expressions such as [2;1,1,1]+[0;2,2,2] and get a return of [3;11,1] as your CF, which is 8/3+5/12 = 37/12.

Thought I'd get the point across that it doesn't just accept simple numbers. You can enter SQRT(7) and immediately see that it is the CF of [2;1,1,1,4,1,1,1,4,1,1,1,4,...] Just saying...

Jon

"Close" is not good enough; the Jameco cabinet mounting plate for 19" racks (PN 149500) now come with screws; i think they are made in China on Metric lathes. The screws look OK even against US ones, but they will not fit; forcing them will result in a virtual single piece of rack rail plus useless screw. And Jameco refuses to acknowledge the problem - much less fix it.

You'll find 127-tooth gears in many thread-cutting metalworking lathes.

Best regards, Spehro Pefhany

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For 25.4/8 the screen looks something like:

Actually, I use DOS drawing characters on the screen but you get the idea. Floating point is called out, as well.

The simple help message from ? is:

Jon

I thought it would be neat to write a program to calculate gear ratios and use exact and ratios from continued fractions. I wanted to enter the gears I had available in a table and see if I could get a program to select the gears for the correct ratio.

But then I got a lathe with quick change gears and now I have a CNC lathe that runs from EMC in Linux (linuxcnc.org). If I want to cut 10.123 threads per inch, no problem.

RogerN

I don't know what you mean by 'convergents' or 'CF'. ratapprx reports every value that reduces the error from the previous trial. Note the continuous reduction in error. Errors are absolute values.

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 [mail]: Chuck F (cbfalconer at maineline dot net) 
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CF (continued fraction) and convergent, if you look it up, is explicitly defined. See Wiki:

I think another term for it is approximant.

I list the sign of the relative error as to some people that may be an important consideration. They can freely ignore the sign, if they want to. It's not so easy to "put the sign back" if it isn't stated in the listing.

Jon

But I like convergent better. There's a reason.

There's a pair of recurrences related to successive numerators and denominators (I can refer you to some pages on this, easily) that is used to prove the fact that continued fractions, even in the case of irrationals such as pi, e, and various sqrt()s, MUST "converge" towards that particular value. The numerator sequence is often denoted by p(i) or P(i) and the denominator sequence is often denoted by q(i) or Q(i) in literature I've seen. It turns out that the even pairs are always less and the odd pairs always more, but in exactly this way:

p[0]/q[0] < p[2]/q[2] < ... < x < ... < p[3]/q[3] < p[1]/q[1]

This fact is developed through mathematical induction. I can describe it here, but most folks wouldn't care. (However, I did post something about it a week ago or so in sci.electronics.design.) The main point, though, is that the value is bounded by the even and odd convergents and trapped tighter and tighter as i goes towards infinity. Which, to me, makes the term very appropriate. Approximant just doesn't carry the meaning so well, to me.

Jon

25.4 is 254/10 AIUI the value 25.4mm was chosen for the inch in 1959 because it was close to and between the values the Brittish and Americans were using for the inch

prior to this US and Canadian miles were different sizes which was fun for surveyors.

LOL

"This left NGS in the position of not wanting to mandate which foot (U.S. Survey or International) a state should use. So, NGS left that decision to the individual states. Currently, NGS publishes SPCs for 7 states using the U.S. Survey Foot conversion factor, 1 state using the International Foot conversion factor, and 42 states using only meters, not feet, for SPCs. Based on STATE legislation we have or know about, 24 states have legislated the U.S. Survey Foot, 8 states have legislated the International Foot, and 18 states have no legislation on which conversion factor must be used."