Mersenne: Re: pi and 4
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Re: Mersenne: pi
At 10:50 AM 2/9/00 -0500, Jeff Woods wrote: You're bumping up against the Fundamental Theorem of Calculus here. Pi DOES have a precisely defined value, but you cannot express it in decimal form. You can express it as an infinite expansion, however. Q: What does this have to do with fundamental theorem of calculus? The fundamental theorem equates anti-derivitives with area. This has more to do with the proof that there are numbers which cannot be expressed as the ratio of integers (i.e. irrational numbers) due originally to pythagroeus (SP), as well as the proof of the irrationality of pi (due to Newton, I think). The first is proved *long* before the fundamental theorem, and is in some ways more fundamental. -Lucas _ Unsubscribe list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Re: Mersenne: Where's the flaw in my thinking?
This is not stupid. I am an electronic technician by training (not by trade it's a hobby now). Put number "a" into a shift register. Do the same with number "b". Set a b pointer to look at the right most bit of b loop start If b pointer points to a zero do nothing If b pointer points to a one add a to the accumulator If b pointer does not point to the left most bit shift b pointer and "a" left and go to loop start If b pointer points to the left most bit read the accumulator If a=3 and b=5 then a=11 b=101 first loop a=3 b will add 3 to the accumulator (because of the right hand one) second loop a=6 b will not add 6 to the accumulator (because of the zero) third loop a=12 b will add 12 to the accumulator (because of the left hand one) the accumulator has 3+12 which equals 15 which equals 3*5. You just took the specific case of a=b -Original Message- From: Jeremy Blosser [EMAIL PROTECTED] To: Mesenne Mailing List (E-mail) [EMAIL PROTECTED] Date: Friday, February 11, 2000 1:42 PM Subject: Mersenne: Where's the flaw in my thinking? Okay, I was sitting there the other day thinking about a non-FFT squaring algorithm... Say we have 14, which in binary is 1110... If we left shift this by the position of the 1, for each 1 in the binary representation, and add them together, we should get the square... So to square 14, we do this: 1110 3 == 111 + 1110 2 == 0111000 + 1110 1 == 0011100 + == 11000100 which is 196 So for each squaring, we have x left shifts and adds, where x is no larger that p. In any case... is this just me being dumb and missing that this is just a stupid way of squaring a number? _ Unsubscribe list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers _ Unsubscribe list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Re: Fwd: Re: Mersenne: Re: Base e arithmetic
Pierre Abbat wrote: Gosper's representation uses seven digits, the sixth roots of 1 and 0, in base 2.5+sqrt(-3/4). Where do I find out more about that? _ Unsubscribe list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Re: Mersenne: pi
On 11 Feb 00, at 16:51, Chip Lynch wrote: While I think the topic is stimulating and important, the Mersenne list probably isn't the best medium for it, unfortunately. Anyone recommend a few good links on the subject? (Pi, The Language of Mathematics, any of that) "The Joy of Pi" by David Blattner (1997) (published by Allen Lane, ISBN 0-713-99217-4) covers this in the sort of detail which has been discussed on this list - and some more - and includes the first million digits of the decimal expansion. I have no links with the author or the publisher except that a kind relative bought be a copy as a Christmas gift a couple of years ago. Regards Brian Beesley _ Unsubscribe list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Re: Mersenne: Where's the flaw in my thinking?
Jeremy Blosser wrote: Okay, I was sitting there the other day thinking about a non-FFT squaring I can't find any flaw. I have lost track of any comments on this by the Prime95 coders, and am curious what they have to say. Could be a time-saved criterion, because the MOD is taken automatically as a by-product of the IFFT, and I have read there is a cache and SIMD tuning in the assembly code which is hailed as "the most efficient FFT ever coded for the Pentium series," or the like. Comments? Best Regards, Stefanovic _ Unsubscribe list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Mersenne: multiprocessor FFT
With respect to the argument that an N-processor machine can't do better than a factor of N speed improvement: the argument sounds airtight, but it's not always correct. Instances of superlinear speedup occur, and not merely as a theoretical possibility. While I was consulting at Compaq (previously Digital) a customer reported such a case (I believe they feared the numbers indicated the presence of a bug in their program or in the operating system). This was on a multiprocessor Alpha system. Hennessy Patterson also noted an instance of it in their book (I'm referring to the 2nd edition). There was no bug: superlinear speedup can happen when each processor has its own cache, so the effective size of the cache grows as N is increased. It is most likely to occur in programs with good locality of reference where the problem size is considerably larger than the innermost local cache size. Knowing whether this really occurs for any particular processor will require actually making the experiment (though a really careful model could produce results which were likely to be accurate predictions). Achieving the effect also requires that the binding of processors to threads is sufficiently stable, either by explicit arrangement or by fortuitous properties of the OS and/or threads software. I have no time to recode the program myself, but the following is addressed to the author of MacGIMPS if he is reading this: I will soon have a 4-processor Daystar system bootable to BEOS, and I'm willing to have it serve as a guinea pig. Michael Yoder _ Unsubscribe list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
