Re: [math-fun] [EXTERNAL] Re: Computing pi (or anything else) to N digits
Maple seems to like "radical number": http://www.maplesoft.com/support/help/Maple/view.aspx?path=type/radnum "A radical number is defined as either a rational number or I, or a combination of roots of rational numbers specified in terms of radicals. A sum, product, or quotient of these is also a radical number." At 07:54 AM 10/17/2012, Schroeppel, Richard wrote:
Radical numbers?
-----Original Message----- From: math-fun-bounces@mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Michael Kleber Sent: Tuesday, October 16, 2012 6:57 PM To: math-fun Subject: [EXTERNAL] Re: [math-fun] Computing pi (or anything else) to N digits
On Mon, Oct 15, 2012 at 10:45 PM, Henry Baker <hbaker1@pipeline.com> wrote:
"Transcendental" means not the root of any finite polynomial with integer coefficients.
http://en.wikipedia.org/wiki/Transcendental_number
Is there a name for a number which isn't algebraic for a _solvable_ Galois polynomial -- i.e., a number which can't be constructed by rational & root operations?
I think the most common description would be "[not] solvable/expressible by radicals". I don't know of a dedicated term for either state.
--Michael
This appears to exclude recursively built stuff like sqrt(1+sqrt2). --Rich -----Original Message----- From: Henry Baker [mailto:hbaker1@pipeline.com] Sent: Wednesday, October 17, 2012 10:37 AM To: math-fun Cc: rcs@xmission.com; Schroeppel, Richard Subject: Re: [math-fun] [EXTERNAL] Re: Computing pi (or anything else) to N digits Maple seems to like "radical number": http://www.maplesoft.com/support/help/Maple/view.aspx?path=type/radnum "A radical number is defined as either a rational number or I, or a combination of roots of rational numbers specified in terms of radicals. A sum, product, or quotient of these is also a radical number." At 07:54 AM 10/17/2012, Schroeppel, Richard wrote:
Radical numbers?
-----Original Message----- From: math-fun-bounces@mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Michael Kleber Sent: Tuesday, October 16, 2012 6:57 PM To: math-fun Subject: [EXTERNAL] Re: [math-fun] Computing pi (or anything else) to N digits
On Mon, Oct 15, 2012 at 10:45 PM, Henry Baker <hbaker1@pipeline.com> wrote:
"Transcendental" means not the root of any finite polynomial with integer coefficients.
http://en.wikipedia.org/wiki/Transcendental_number
Is there a name for a number which isn't algebraic for a _solvable_ Galois polynomial -- i.e., a number which can't be constructed by rational & root operations?
I think the most common description would be "[not] solvable/expressible by radicals". I don't know of a dedicated term for either state.
--Michael
Hello, there is a way to compute with the ruler and compass some numbers like arctan(1/2)/Pi, bit by bit, but the complexity is growing like the square of each iteration unfortunately, https://cs.uwaterloo.ca/journals/JIS/compass.html best regards, simon plouffe
There was an old trick for calculating floating point log_2(x) back in assembly language days: Assume 1.0<=x<2.0. Square it 7 times, to convert mantissa bits (bits to the right of the binary point) into exponent bits. Decant the 7 exponent bits into the nascent logarithm; zero them out in the x^128 value, bringing it back into the 1.0 - 2.0 range. Rinse, repeat, till you've got 28 bits of logarithm. (The PDP6 had 36-bit words.) A similar trick essentially reproduces SP's idea: Iterate: cos 2x = 2 (cos x)^2 - 1 Keeping track of the sign of cos(2^N x) will read out x/pi in binary. (I learned these from Gosper.) Rich ----- Quoting Simon Plouffe <simon.plouffe@gmail.com>:
Hello,
there is a way to compute with the ruler and compass some numbers like arctan(1/2)/Pi, bit by bit, but the complexity is growing like the square of each iteration unfortunately,
https://cs.uwaterloo.ca/journals/JIS/compass.html
best regards,
simon plouffe
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This algorithm might not be so bad for computing logs in parallel on a "GPU": Graphical Programming Unit, e.g., from AMD or nVidia. These GPU units have a lot of processing power, but relatively small amounts of memory; memory access latency might also cost many processing cycles. At 02:12 PM 10/18/2012, rcs@xmission.com wrote:
There was an old trick for calculating floating point log_2(x) back in assembly language days: Assume 1.0<=x<2.0. Square it 7 times, to convert mantissa bits (bits to the right of the binary point) into exponent bits. Decant the 7 exponent bits into the nascent logarithm; zero them out in the x^128 value, bringing it back into the 1.0 - 2.0 range. Rinse, repeat, till you've got 28 bits of logarithm. (The PDP6 had 36-bit words.)
A similar trick essentially reproduces SP's idea: Iterate: cos 2x = 2 (cos x)^2 - 1 Keeping track of the sign of cos(2^N x) will read out x/pi in binary. (I learned these from Gosper.)
Rich
* rcs@xmission.com <rcs@xmission.com> [Oct 19. 2012 05:12]:
There was an old trick for calculating floating point log_2(x) back in assembly language days: Assume 1.0<=x<2.0. Square it 7 times, to convert mantissa bits (bits to the right of the binary point) into exponent bits. Decant the 7 exponent bits into the nascent logarithm; zero them out in the x^128 value, bringing it back into the 1.0 - 2.0 range. Rinse, repeat, till you've got 28 bits of logarithm. (The PDP6 had 36-bit words.)
A similar trick essentially reproduces SP's idea: Iterate: cos 2x = 2 (cos x)^2 - 1 Keeping track of the sign of cos(2^N x) will read out x/pi in binary. (I learned these from Gosper.)
Rich
[...]
Just for the record: http://www.jjj.de/predict.txt (dated 1995-November-12). I came up with this in a lecture (by Hanspeter Herzel, about nonlinear dynamics). When I posted the text to some math newsgroup the answer was roughly "This is known under the name <something>" where "This" apparently meant that the cos() functional equation is the one at the tip of the Mandelbrot antenna; but the bits-producing trick was apparently ignored. Earlier references are welcome. Equivalent techniques are possible for other periodic functions and I recall having seen a nice paper about this, but cannot find it right now. Regards, jj
participants (5)
-
Henry Baker -
Joerg Arndt -
rcs＠xmission.com -
Schroeppel, Richard -
Simon Plouffe