December 2009 - mathfun@mailman.xmission.com

Re: [math-fun] Solving polynomial equations with roots, etc.
by Dan Asimov 02 Dec '09

02 Dec '09

As interesting as this discussion could be, for the record I assert that anything said about this issue is total B.S. unless there is a precise definition made of the term "God" and of the term "knowing". --Dan << Date: Mon, 30 Nov 2009 19:44:17 -0800 Subject: Re: [math-fun] Solving polynomial equations with roots, etc. From: "Stephen B. Gray" <stevebg(a)roadrunner.com> On a different subject, for anyone to comment on. Everyone knows that sqrt(2), pi, and all irrational numbers have no decimal strings that repeat an infinite number of times. That's not necessarily the same as the decimal expansion being totally without any pattern. That is, is there any way to predict the next digits of say sqrt(2) or pi without doing one of the usual computations? In other words, is it possible to "know" the entire decimal expansion of any "ordinary" irrational? Is anything known about patterns in "regular" irrational expansions? I'm excluding numbers invented for the sole purpose of being irrational or transcendental and with an obvious pattern like .101001000100001.... or .123456789101112..... ), etc.) I know about the question of "normal" expansions but that has little to do with my question. >> Any info will be appreciated. _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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[math-fun] numbers with predictable digit patterns
by rcs＠xmission.com 02 Dec '09

02 Dec '09

This might be an example of what SBG is looking for. The infinite product prod(1 - z^n) shows up in the theory of partitions. It's also a theta function. When you multiply it out, it turns out that most of the terms z^k have a coefficient of 0. When k is a pentagonal number j(3j+-1)/2, the coefficient of z^k is +-1, depending on whether j is even or odd. The power series begins 1 - z - z^2 + z^5 + z^7 - z^12 - z^15 + z^22 + z^26 - ... If you evaluate this at z = 1/2, and write the value in binary, you get a predictable pattern. It's not periodic, but it is predictable; and it's relatively cheap to compute single bits without reference to what's come before. This number is clearly irrational, and most likely transcendental, but I don't know if transcendence is proved. The non-zero portions may be thin enough that you can show it's not a quadratic irrational. Rich

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[math-fun] "Shanksing" CF convergents
by rwg＠sdf.lonestar.org 02 Dec '09

02 Dec '09

Model each three consecutive terms of the numerical sequence a_n as three points on the curve c+f*d^-n and form a new sequence b_{1,n} from these values of c. Likewise form b_{2,n} from b_{1,n}, etc. Then b_{k,n} is an array of estimates of lim_{n->oo} a_n: (d31) b := if k = 0 then a k, n n 2 b b - b k - 1, n - 1 k - 1, n + 1 k - 1, n else ------------------------------------------- b - 2 b + b k - 1, n + 1 k - 1, n k - 1, n - 1 Note that this is 0 for all n if the preceding sequence progressed geometrically, and infinity if it progressed arithmetically. These estimates tend to improve gradually with n and dramatically with k up to some point where noise and ripple in the a_n sequence gets amplified to the point of swamping everything. But let a_n be a sequence of (noise-free) continued fractions, e.g., (c34) a[n] := (cfexpand(d24[n]),%%[1]/%%[2])[1] %% 1 (d34) a := (cfexpand(d24 ), ---) n n %% 1 2 (c24) apply('matrix,makelist([3,7,15,1,k],k,2,8)) [ 3 7 15 1 2 ] [ ] [ 3 7 15 1 3 ] [ ] [ 3 7 15 1 4 ] [ ] (d24) [ 3 7 15 1 5 ] [ ] [ 3 7 15 1 6 ] [ ] [ 3 7 15 1 7 ] [ ] [ 3 7 15 1 8 ] with some term progressing arithmetically. Then the first acceleration (c35) apply(matrix,makelist(cf(b[1,n]),n,2,6)) [ 3 7 15 1 6 1 15 7 ] [ ] [ 3 7 15 1 8 1 15 7 ] [ ] (d35) [ 3 7 15 1 10 1 15 7 ] [ ] [ 3 7 15 1 12 1 15 7 ] [ ] [ 3 7 15 1 14 1 15 7 ] produces a sequence with that same term progressing twice as fast. (c36) apply(matrix,makelist(cf(b[2,n]),n,3,5)) [ 3 7 15 1 18 1 4 2 1 1 1 2 ] [ ] (d36) [ 3 7 15 1 22 1 4 2 1 1 1 2 ] [ ] [ 3 7 15 1 26 1 4 2 1 1 1 2 ] (c37) apply(matrix,makelist(cf(b[3,n]),n,4,4)) (d37) [ 3 7 15 1 46 1 1 3 3 1 3 ] What's going on? Each of these progressions is a ratio of arithmetic progressions. Given (c39) a[n] := (a*n+b)/(c*n+d) a n + b (d39) a := ------- n c n + d then the first "accelerant" is (c40) ratsimp(b[1,n]) 2 a c n + a d + b c (d40) ------------------- 2 2 c n + 2 c d or, dividing through by 2 c, (c41) opsubst("+" = lambda([[l]],2*c,%%*multthru(1/%%,apply("+",l))),%) a d b a n + --- + - 2 c 2 (d41) ------------- . c n + d Remarkably, only one coefficient has changed. After n accelerations, (c43) subst(b(0),b0,differenceq(b(n) = a*d/2/c+b(n-1)/2,b(n),b(0) = b0)) a d a d - b(0) c (d43) b(n) = --- - ------------ c n c 2 which clearly approaches a*d/c, whereat a_n approaches (c44) factor(subst(a*d/c,b,a[n])) a (d44) -, c i.e., we progressed geometrically through the sequence of arithmetic progressions. --rwg See http://en.wikipedia.org/wiki/Shanks_transformation , but note there are usually better ways to accelerate sums. Also, the Gregory's series for pi=4 atan 1 is discussed in HAKMEM--it is unclear whether this process converges in the long run. Things get *very* shaky after a few dozen iterations. many iterations.

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Re: [math-fun] Solving polynomial equations with roots, etc.
by rcs＠xmission.com 01 Dec '09

01 Dec '09

Date: Tue, 01 Dec 2009 20:49:57 -0800 Subject: Re: [math-fun] Solving polynomial equations with roots, etc. From: "Stephen B. Gray" <stevebg(a)roadrunner.com> Henry Baker wrote: > Isn't this what the Borwein expansions are all about? > > At 06:07 PM 12/1/2009, you wrote: >> from SBG ... > >> Date: Mon, 30 Nov 2009 19:44:17 -0800 >> Subject: Re: [math-fun] Solving polynomial equations with roots, etc. >> From: "Stephen B. Gray" <stevebg(a)roadrunner.com> >> >> On a different subject, for anyone to comment on. >> >> Everyone knows that sqrt(2), pi, and all irrational numbers have no >> decimal strings that repeat an infinite number of times. That's >> not necessarily the same as the decimal expansion being totally >> without any pattern. That is, is there any way to predict the next >> digits of say sqrt(2) or pi without doing one of the usual >> computations? In other words, is it possible to "know" the entire >> decimal expansion of any "ordinary" irrational? Is anything known >> about patterns in "regular" irrational expansions? >> >> I'm excluding numbers invented for the sole purpose of being >> irrational or transcendental and with an obvious pattern like >> .101001000100001.... or .123456789101112..... ), etc.) I know about >> the question of "normal" expansions but that has little to do with >> my question. >> >> Any info will be appreciated. >> >> Steve Gray >> > If by the Borwein expansions you mean the formula for computing hex digits without knowing the preceding ones, yes, I know about that. I also know about the contined fractions having simple patterns. Assuming that the "value" of the irrational is given only by the decimal expansion, is it possible to "know" the exact value by knowing the pattern? This is part of a theological issue which actually means very little to most people. (And it may be meaningless.). (I'm not a believer.)

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Re: [math-fun] Solving polynomial equations with roots, etc.
by rcs＠xmission.com 01 Dec '09

01 Dec '09

from SBG ... Date: Mon, 30 Nov 2009 19:44:17 -0800 Subject: Re: [math-fun] Solving polynomial equations with roots, etc. From: "Stephen B. Gray" <stevebg(a)roadrunner.com> On a different subject, for anyone to comment on. Everyone knows that sqrt(2), pi, and all irrational numbers have no decimal strings that repeat an infinite number of times. That's not necessarily the same as the decimal expansion being totally without any pattern. That is, is there any way to predict the next digits of say sqrt(2) or pi without doing one of the usual computations? In other words, is it possible to "know" the entire decimal expansion of any "ordinary" irrational? Is anything known about patterns in "regular" irrational expansions? I'm excluding numbers invented for the sole purpose of being irrational or transcendental and with an obvious pattern like .101001000100001.... or .123456789101112..... ), etc.) I know about the question of "normal" expansions but that has little to do with my question. Any info will be appreciated. Steve Gray

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