[math-fun] Bug in my published 34 yr old formula !?
https://books.google.com/books?id=oYGz3S_WTHUC&pg=PA261&lpg=PA261&dq=NYU+Com... pp 267-268. This is the formula which I've used regularly to convert an arbitrary composition of homographic functions (e.g., a sum) to a (non regular) continued fraction. To my bewilderment and distress, it just stopped working, even after I rederived it. The rarely used terminating case went undetected. Empirically, the final term should be x = c[m+1](y + d[m]/c[m]), not c[m+1](y + d[m]/d[m+1]). --rwg
Sounds like a career ender.
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Bill Gosper Sent: Monday, June 04, 2018 3:44 PM To: math-fun@mailman.xmission.com Subject: [math-fun] Bug in my published 34 yr old formula !?
https://books.google.com/books?id=oYGz3S_WTHUC&pg=PA261&lpg=PA26 1&dq=NYU+Computers+in+Mathematics+conference+1984&source=bl&ots =3efR2XdELO&sig=aBXmt4W35dl4dqcQg- uxRtMflYs&hl=en&sa=X&ved=0ahUKEwjvmLL31rrbAhUGj1kKHcgsBTEQ6AEIY TAH#v=onepage&q=NYU%20Computers%20in%20Mathematics%20conferen ce%201984&f=false pp 267-268. This is the formula which I've used regularly to convert an arbitrary composition of homographic functions (e.g., a sum) to a (non regular) continued fraction. To my bewilderment and distress, it just stopped working, even after I rederived it. The rarely used terminating case went undetected. Empirically, the final term should be x = c[m+1](y + d[m]/c[m]), not c[m+1](y + d[m]/d[m+1]). --rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
This is a case of transference across species --- from Comp. Sci. to Maths. --- of algorithmic gangrene, an infection well-known to software engineers. Any program, unused for a period sufficiently long to ensure that its author has forgotten exactly why it used to work, is certain to fail mysteriously when reactivated. The causative organism remains to be identified: conspiracy theorists are invited to provide more-or-less plausible suggestions. WFL On 6/5/18, David Wilson <davidwwilson@comcast.net> wrote:
Sounds like a career ender.
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Bill Gosper Sent: Monday, June 04, 2018 3:44 PM To: math-fun@mailman.xmission.com Subject: [math-fun] Bug in my published 34 yr old formula !?
https://books.google.com/books?id=oYGz3S_WTHUC&pg=PA261&lpg=PA26 1&dq=NYU+Computers+in+Mathematics+conference+1984&source=bl&ots =3efR2XdELO&sig=aBXmt4W35dl4dqcQg- uxRtMflYs&hl=en&sa=X&ved=0ahUKEwjvmLL31rrbAhUGj1kKHcgsBTEQ6AEIY TAH#v=onepage&q=NYU%20Computers%20in%20Mathematics%20conferen ce%201984&f=false pp 267-268. This is the formula which I've used regularly to convert an arbitrary composition of homographic functions (e.g., a sum) to a (non regular) continued fraction. To my bewilderment and distress, it just stopped working, even after I rederived it. The rarely used terminating case went undetected. Empirically, the final term should be x = c[m+1](y + d[m]/c[m]), not c[m+1](y + d[m]/d[m+1]). --rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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I prefer the term “bit-rot”. Jim Propp On Tuesday, June 5, 2018, Fred Lunnon <fred.lunnon@gmail.com> wrote:
This is a case of transference across species --- from Comp. Sci. to Maths. --- of algorithmic gangrene, an infection well-known to software engineers.
Any program, unused for a period sufficiently long to ensure that its author has forgotten exactly why it used to work, is certain to fail mysteriously when reactivated.
The causative organism remains to be identified: conspiracy theorists are invited to provide more-or-less plausible suggestions.
WFL
On 6/5/18, David Wilson <davidwwilson@comcast.net> wrote:
Sounds like a career ender.
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Bill Gosper Sent: Monday, June 04, 2018 3:44 PM To: math-fun@mailman.xmission.com Subject: [math-fun] Bug in my published 34 yr old formula !?
https://books.google.com/books?id=oYGz3S_WTHUC&pg=PA261&lpg=PA26 1&dq=NYU+Computers+in+Mathematics+conference+1984&source=bl&ots =3efR2XdELO&sig=aBXmt4W35dl4dqcQg- uxRtMflYs&hl=en&sa=X&ved=0ahUKEwjvmLL31rrbAhUGj1kKHcgsBTEQ6AEIY TAH#v=onepage&q=NYU%20Computers%20in%20Mathematics%20conferen ce%201984&f=false pp 267-268. This is the formula which I've used regularly to convert an arbitrary composition of homographic functions (e.g., a sum) to a (non regular) continued fraction. To my bewilderment and distress, it just stopped working, even after I rederived it. The rarely used terminating case went undetected. Empirically, the final term should be x = c[m+1](y + d[m]/c[m]), not c[m+1](y + d[m]/d[m+1]). --rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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I suspect the laws of mathematics may have changed over those 34 years. -tom On Tue, Jun 5, 2018 at 5:34 PM, James Propp <jamespropp@gmail.com> wrote:
I prefer the term “bit-rot”.
Jim Propp
On Tuesday, June 5, 2018, Fred Lunnon <fred.lunnon@gmail.com> wrote:
This is a case of transference across species --- from Comp. Sci. to Maths. --- of algorithmic gangrene, an infection well-known to software engineers.
Any program, unused for a period sufficiently long to ensure that its author has forgotten exactly why it used to work, is certain to fail mysteriously when reactivated.
The causative organism remains to be identified: conspiracy theorists are invited to provide more-or-less plausible suggestions.
WFL
On 6/5/18, David Wilson <davidwwilson@comcast.net> wrote:
Sounds like a career ender.
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Bill Gosper Sent: Monday, June 04, 2018 3:44 PM To: math-fun@mailman.xmission.com Subject: [math-fun] Bug in my published 34 yr old formula !?
https://books.google.com/books?id=oYGz3S_WTHUC&pg=PA261&lpg=PA26 1&dq=NYU+Computers+in+Mathematics+conference+1984&source=bl&ots =3efR2XdELO&sig=aBXmt4W35dl4dqcQg- uxRtMflYs&hl=en&sa=X&ved=0ahUKEwjvmLL31rrbAhUGj1kKHcgsBTEQ6AEIY TAH#v=onepage&q=NYU%20Computers%20in%20Mathematics%20conferen ce%201984&f=false pp 267-268. This is the formula which I've used regularly to convert an arbitrary composition of homographic functions (e.g., a sum) to a (non regular) continued fraction. To my bewilderment and distress, it just stopped working, even after I rederived it. The rarely used terminating case went undetected. Empirically, the final term should be x = c[m+1](y + d[m]/c[m]), not c[m+1](y + d[m]/d[m+1]). --rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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-- -- http://cube20.org/ -- http://golly.sf.net/ --
Or at least the fines and jail terms.
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Tomas Rokicki Sent: Tuesday, June 05, 2018 8:40 PM To: math-fun Subject: Re: [math-fun] Bug in my published 34 yr old formula !?
I suspect the laws of mathematics may have changed over those 34 years.
-tom
What is apparently not widely known is that proper commenting can serve as a prophylactic. --ms On 05-Jun-18 20:28, Fred Lunnon wrote:
This is a case of transference across species --- from Comp. Sci. to Maths. --- of algorithmic gangrene, an infection well-known to software engineers.
Any program, unused for a period sufficiently long to ensure that its author has forgotten exactly why it used to work, is certain to fail mysteriously when reactivated.
The causative organism remains to be identified: conspiracy theorists are invited to provide more-or-less plausible suggestions.
WFL
On 6/5/18, David Wilson <davidwwilson@comcast.net> wrote:
Sounds like a career ender.
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Bill Gosper Sent: Monday, June 04, 2018 3:44 PM To: math-fun@mailman.xmission.com Subject: [math-fun] Bug in my published 34 yr old formula !?
https://books.google.com/books?id=oYGz3S_WTHUC&pg=PA261&lpg=PA26 1&dq=NYU+Computers+in+Mathematics+conference+1984&source=bl&ots =3efR2XdELO&sig=aBXmt4W35dl4dqcQg- uxRtMflYs&hl=en&sa=X&ved=0ahUKEwjvmLL31rrbAhUGj1kKHcgsBTEQ6AEIY TAH#v=onepage&q=NYU%20Computers%20in%20Mathematics%20conferen ce%201984&f=false pp 267-268. This is the formula which I've used regularly to convert an arbitrary composition of homographic functions (e.g., a sum) to a (non regular) continued fraction. To my bewilderment and distress, it just stopped working, even after I rederived it. The rarely used terminating case went undetected. Empirically, the final term should be x = c[m+1](y + d[m]/c[m]), not c[m+1](y + d[m]/d[m+1]). --rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Poo-poo --- I blame global warming. Or dark matter. Or Copenhagen ... WFL On 6/6/18, Mike Speciner <ms@alum.mit.edu> wrote:
What is apparently not widely known is that proper commenting can serve as a prophylactic. --ms
On 05-Jun-18 20:28, Fred Lunnon wrote:
This is a case of transference across species --- from Comp. Sci. to Maths. --- of algorithmic gangrene, an infection well-known to software engineers.
Any program, unused for a period sufficiently long to ensure that its author has forgotten exactly why it used to work, is certain to fail mysteriously when reactivated.
The causative organism remains to be identified: conspiracy theorists are invited to provide more-or-less plausible suggestions.
WFL
On 6/5/18, David Wilson <davidwwilson@comcast.net> wrote:
Sounds like a career ender.
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Bill Gosper Sent: Monday, June 04, 2018 3:44 PM To: math-fun@mailman.xmission.com Subject: [math-fun] Bug in my published 34 yr old formula !?
https://books.google.com/books?id=oYGz3S_WTHUC&pg=PA261&lpg=PA26 1&dq=NYU+Computers+in+Mathematics+conference+1984&source=bl&ots =3efR2XdELO&sig=aBXmt4W35dl4dqcQg- uxRtMflYs&hl=en&sa=X&ved=0ahUKEwjvmLL31rrbAhUGj1kKHcgsBTEQ6AEIY TAH#v=onepage&q=NYU%20Computers%20in%20Mathematics%20conferen ce%201984&f=false pp 267-268. This is the formula which I've used regularly to convert an arbitrary composition of homographic functions (e.g., a sum) to a (non regular) continued fraction. To my bewilderment and distress, it just stopped working, even after I rederived it. The rarely used terminating case went undetected. Empirically, the final term should be x = c[m+1](y + d[m]/c[m]), not c[m+1](y + d[m]/d[m+1]). --rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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On Mon, Jun 4, 2018 at 12:44 PM Bill Gosper <billgosper@gmail.com> wrote:
https://books.google.com/books?id=oYGz3S_WTHUC&pg=PA261&lpg=PA261&dq=NYU+Com... pp 267-268. This is the formula which I've used regularly to convert an arbitrary composition of homographic functions (e.g., a sum) to a (non regular) continued fraction. To my bewilderment and distress, it just stopped working, even after I rederived it. The rarely used terminating case went undetected. Empirically, the final term should be x = c[m+1](y + d[m]/c[m]), not c[m+1](y + d[m]/d[m+1]). --rwg
It converted the sum 1 0 1 0 ( ) Product[ , {n, ∞}] == x x E 0 - 1 n to a CF: 1 + x + ContinuedFractionK[x (1 - n), n + x, {n, 2, ∞}] == x 1 + x - --------------------------------------------- 2 x 2 + x - ------------------------------------- 3 x 3 + x - ----------------------------- 4 x 4 + x - --------------------- 5 x 5 + x - ------------- 6 x 6 + x - ----- . . . == x E^x/(E^x - 1) I.e., the factorial generating function for BernoulliB[k,1], as the Bernoulli numbers *should* have been defined. (Wikipedia notates Subsuperscript[B,k,+].) But given a 2x2 product for Tan x, Product[ -1 + 2 k , {k, ∞}] -1 + -------- z 1 -1 + 2 k -2 + -------- z 1 1 == ---------- 1 - Tan[x] In[281]:= Dot @@ Table[%215[[1]], {k, 5}] // FullSimplify Out[281]= {{(945 - z (105 + z (420 + z (-45 + (-15 + z) z))))/z^5, 1 + 105/z^4 - 45/z^2}, {(945 + z (-1050 + z (-315 + z (150 + (5 - 2 z) z))))/z^5, (105 + z (-105 + z (-45 + z (10 + z))))/z^4}} In[283]:= Series[Divide @@ %% - 1/(1 - Tan@z), {z, 0, 5}] Out[283]= {O[z]^6,O[z]^6} it gives a nonsensical continued fraction. --rwg This problem arose trying to work around Mathematica's K notation, which lacks provision for terms with an alternating formula. I recently saw the notation a(k) --------- K b(k) + c(k) k ----- d(k) for a period 2, e.g.. Maybe it's time to just deprecate nonregular CFs in favor of ∏ Matrix[k], with some way to specify which elements to divide as k→∞, when the product doesn't converge elementwise. Julian & I once sought unsuccessfully an f[k]^-1 M[k] f[k+1] transformation that would assure elementwise convergence. (Which would do something weird to regular CFs.)
The nonsensical continued fraction I got for 1/(1-Tan[x]) should be 0 for x→π/2. Instead I got (4 + Pi^2)/(2*Pi + 2*Pi^2) + (Pi*ContinuedFractionK[(-(-1 - 2*k + Pi))*(3 - 2*k + Pi), (-2 + 8*k^2 - 4*k*Pi)/Pi, {k, 1, Infinity}])/(2*(-1 + Pi)*(1 + Pi)), which Plouffe's new Inverter (and ries) positively identify as (3 - 2/π)/(π^2 - 1). It seems to blow up obediently at x→π/4. But Plouffe & ries boggle for other values of x. I have a general formula for ContinuedFractionK[quadratic,linear,...], but not ...[quadratic,quadratic]. This may wind up requiring Mozart, a fish, a shower, and massive amounts of Stash Double Bergamot. --rwg On Fri, Jun 8, 2018 at 3:01 PM Bill Gosper <billgosper@gmail.com> wrote:
On Mon, Jun 4, 2018 at 12:44 PM Bill Gosper <billgosper@gmail.com> wrote:
https://books.google.com/books?id=oYGz3S_WTHUC&pg=PA261&lpg=PA261&dq=NYU+Com... pp 267-268. This is the formula which I've used regularly to convert an arbitrary composition of homographic functions (e.g., a sum) to a (non regular) continued fraction. To my bewilderment and distress, it just stopped working, even after I rederived it. The rarely used terminating case went undetected. Empirically, the final term should be x = c[m+1](y + d[m]/c[m]), not c[m+1](y + d[m]/d[m+1]). --rwg
It converted the sum 1 0 1 0 ( ) Product[ , {n, ∞}] == x x E 0 - 1 n to a CF: 1 + x + ContinuedFractionK[x (1 - n), n + x, {n, 2, ∞}] ==
x 1 + x - --------------------------------------------- 2 x 2 + x - ------------------------------------- 3 x 3 + x - ----------------------------- 4 x 4 + x - --------------------- 5 x 5 + x - ------------- 6 x 6 + x - ----- . . . == x E^x/(E^x - 1) I.e., the factorial generating function for BernoulliB[k,1], as the Bernoulli numbers *should* have been defined. (Wikipedia notates Subsuperscript[B,k,+].)
But given a 2x2 product for Tan x,
Product[ -1 + 2 k , {k, ∞}] -1 + -------- z 1
-1 + 2 k -2 + -------- z 1
1 == ---------- 1 - Tan[x]
In[281]:= Dot @@ Table[%215[[1]], {k, 5}] // FullSimplify
Out[281]= {{(945 - z (105 + z (420 + z (-45 + (-15 + z) z))))/z^5, 1 + 105/z^4 - 45/z^2}, {(945 + z (-1050 + z (-315 + z (150 + (5 - 2 z) z))))/z^5, (105 + z (-105 + z (-45 + z (10 + z))))/z^4}}
In[283]:= Series[Divide @@ %% - 1/(1 - Tan@z), {z, 0, 5}]
Out[283]= {O[z]^6,O[z]^6}
it gives a nonsensical continued fraction. --rwg This problem arose trying to work around Mathematica's K notation, which lacks provision for terms with an alternating formula. I recently saw the notation
a(k) --------- K b(k) + c(k) k ----- d(k)
for a period 2, e.g.. Maybe it's time to just deprecate nonregular CFs in favor of ∏ Matrix[k], with some way to specify which elements to divide as k→∞, when the product doesn't converge elementwise. Julian & I once sought unsuccessfully an f[k]^-1 M[k] f[k+1] transformation that would assure elementwise convergence. (Which would do something weird to regular CFs.)
Taking ries right to the end of its -l7 rope, the exact value of the nonsensical continued fraction: ContinuedFractionK[(-(-3 + 2*k - 2*z))*(1 + 2*k - 2*z)*z^2, -1 + 4*k^2 - 4*k*z, {k, 0, ∞}] == (z^2*(3 + 2*z)*(-1 + Tan[z]))/(1 + z + z*Tan[z]) Five cf terms give 13 good Taylor terms: In[489]:= Series[cfCF[%/.\[Infinity]->5],{z,0,14}] Out[489]= -3 z^2+4 z^3+z^4-4 z^5+(14 z^6)/3-(8 z^7)/5-(41 z^8)/15+(556 z^9)/105-(418 z^10)/105-(376 z^11)/945+(2678 z^12)/567-(11896 z^13)/2079+(88638397 z^14)/36018675+O[z]^15== -3 z^2+4 z^3+z^4-4 z^5+(14 z^6)/3-(8 z^7)/5-(41 z^8)/15+(556 z^9)/105-(418 z^10)/105-(376 z^11)/945+(2678 z^12)/567-(11896 z^13)/2079+(383716 z^14)/155925+O[z]^15 Which is what my cf canonicalizer turned 1/(1-Tan[z]) into. I'd rather it hadn't. --rwg On Tue, Jun 12, 2018 at 6:11 AM Bill Gosper <billgosper@gmail.com> wrote:
The nonsensical continued fraction I got for 1/(1-Tan[x]) should be 0 for x→π/2. Instead I got (4 + Pi^2)/(2*Pi + 2*Pi^2) + (Pi*ContinuedFractionK[(-(-1 - 2*k + Pi))*(3 - 2*k + Pi), (-2 + 8*k^2 - 4*k*Pi)/Pi, {k, 1, Infinity}])/(2*(-1 + Pi)*(1 + Pi)),
which Plouffe's new Inverter (and ries) positively identify as (3 - 2/π)/(π^2 - 1). It seems to blow up obediently at x→π/4. But Plouffe & ries boggle for other values of x. I have a general formula for ContinuedFractionK[quadratic,linear,...], but not ...[quadratic,quadratic]. This may wind up requiring Mozart, a fish, a shower, and massive amounts of Stash Double Bergamot. --rwg
On Fri, Jun 8, 2018 at 3:01 PM Bill Gosper <billgosper@gmail.com> wrote:
On Mon, Jun 4, 2018 at 12:44 PM Bill Gosper <billgosper@gmail.com> wrote:
https://books.google.com/books?id=oYGz3S_WTHUC&pg=PA261&lpg=PA261&dq=NYU+Com... pp 267-268. This is the formula which I've used regularly to convert an arbitrary composition of homographic functions (e.g., a sum) to a (non regular) continued fraction. To my bewilderment and distress, it just stopped working, even after I rederived it. The rarely used terminating case went undetected. Empirically, the final term should be x = c[m+1](y + d[m]/c[m]), not c[m+1](y + d[m]/d[m+1]). --rwg
It converted the sum 1 0 1 0 ( ) Product[ , {n, ∞}] == x x E 0 - 1 n to a CF: 1 + x + ContinuedFractionK[x (1 - n), n + x, {n, 2, ∞}] ==
x 1 + x - --------------------------------------------- 2 x 2 + x - ------------------------------------- 3 x 3 + x - ----------------------------- 4 x 4 + x - --------------------- 5 x 5 + x - ------------- 6 x 6 + x - ----- . . . == x E^x/(E^x - 1) I.e., the factorial generating function for BernoulliB[k,1], as the Bernoulli numbers *should* have been defined. (Wikipedia notates Subsuperscript[B,k,+].)
But given a 2x2 product for Tan x,
Product[ -1 + 2 k , {k, ∞}] -1 + -------- z 1
-1 + 2 k -2 + -------- z 1
1 == ---------- 1 - Tan[x]
In[281]:= Dot @@ Table[%215[[1]], {k, 5}] // FullSimplify
Out[281]= {{(945 - z (105 + z (420 + z (-45 + (-15 + z) z))))/z^5, 1 + 105/z^4 - 45/z^2}, {(945 + z (-1050 + z (-315 + z (150 + (5 - 2 z) z))))/z^5, (105 + z (-105 + z (-45 + z (10 + z))))/z^4}}
In[283]:= Series[Divide @@ %% - 1/(1 - Tan@z), {z, 0, 5}]
Out[283]= {O[z]^6,O[z]^6}
it gives a nonsensical continued fraction. --rwg This problem arose trying to work around Mathematica's K notation, which lacks provision for terms with an alternating formula. I recently saw the notation
a(k) --------- K b(k) + c(k) k ----- d(k)
for a period 2, e.g.. Maybe it's time to just deprecate nonregular CFs in favor of ∏ Matrix[k], with some way to specify which elements to divide as k→∞, when the product doesn't converge elementwise. Julian & I once sought unsuccessfully an f[k]^-1 M[k] f[k+1] transformation that would assure elementwise convergence. (Which would do something weird to regular CFs.)
participants (6)
-
Bill Gosper -
David Wilson -
Fred Lunnon -
James Propp -
Mike Speciner -
Tomas Rokicki