01/17/2019, 06:44 PM
(This post was last modified: 01/18/2019, 11:01 PM by sheldonison.)
This is a second post analyzing Jay's slog vs Kneser. \( \text{JaySlog}(z)=\text{JayTaylor}(z)+\frac{\ln(z-L)}{L}+\frac{\ln(z-L^*)}{L^*};\;\;\;L\approx1.337i+0.3181 \) L is the fixed point. By judiciously choosing how to take the 1-cyclic Fourier series of JaySlog(SexpKneser(z))-z, I was able to get two additional accurate terms to the 1-cyclic Fourier series from earlier, for eight complex conjugate pairs of terms plus the constant; (see previous post: updated jaytheta function). Without the theta Fourier series, Jays slog matches Kneser to about 27 decimal digits at the real axis. With the theta mapping, the KneserSlog+theta(KneserSlog) approximation matches Jay's slog series to >76 decimal digits over most of a circle of radius 1.1, except for a byte taken out of the circle near both L and L*. This image is a color contour plot of \( \frac{10^{-72}}{\text{JaySlog}(z)-\text{KneserSlog}(z)-\theta(\text{KneserSlog})} \) The main region matches Jay's slog series to better than 76 decimal digits inside most of the circle! The results are so good that for all intents and purposes, the two functions are the same, except near the two singularities as can be seen from the image. Even near the singularity but before it turns black, at z=i, the two match each other to within 5E-74. The plot turns black when consistency is below ~70 decimal digits and consistency is better than 5E-64 everywhere in the circle of radius 1.1. As before, I am using the first 692 Taylor series coefficients of Jay's slog, where a_692~=2E-111.
So, is there any way to know that Jay's slog doesn't match Kneser's slog? How would one know if Jay's slog is correct and Sheldon's program gave the incorrect slog? It turns we can also compare any slog valid over enough of the complex plane to the inverse Schroder function, where the Schroder function is turned into a complex valued Abel function \( \alpha(z) \) for base(e) exponentiation. And its inverse \( \alpha^{-1}(z) \) is a superfunction for base e, which is also complex valued. In the plot below, we can see the inverse Schroder function extending out from the two complex valued fixed points. Superimposed is a circle of radius abs(L), which is the maximum possible radius of convergence for the slog since both Kneser's slog and Jay's slog functions have a singularity at the fixed point. The smaller circle has a radius of 1.1 within which Jay's slog was consistent to ~76 decimal digits with a theta mapping discussed earlier; except for near the singularity. The green dots represent a sequence of sample points where we take the 1-cyclic Fourier series of \( \text{JaySlog}(\alpha^{-1}(z))-z \). Here, \( \alpha^{-1}(z) \) is a complex valued superfunction for base(e) developed at the fixed point of L. This superfunction is the inverse of the complex valued Abel where S(z) is the Schroder function and \( \alpha(z)=\frac{\ln(S(z))}{L} \).
If we subtract out the constant term, the resulting 1-cyclic function has an amplitude of about 0.0009, and looks visually like a simple 1 term 1-cyclic exponential, but of course it is much more complicated than that. The details of the 1-cyclic function are critical to understanding the behavior of the slog in the complex plane. So let us analyze both the 1-cyclic theta Schroder function mapping for Jay's slog, and for Sheldon's slog, where Sheldon's slog is also 692 terms printed with 77 decimal digits of precision centered at 0 and using the same format as Jay's slog.
For simplicity I am using the substitution \( y=\exp(2\pi i (z-z_0)) \) where the 1-cyclic Fourier series was taken on 60 equally spaced samples between \( \alpha^{-1}(z_0)\;...\;\alpha^{-1}(z_0+1) \) My \( z_0 \) definition is somewhat random. Any number whose sexp is is in the well behaved range will do. \( z_0=\alpha(\text{sexp}(-1.2+\frac{-\ln(100)}{2\pi i}))\approx\alpha(\text{sexp}(-1.2+0.733i))\approx\alpha(-0.0161+0.75768i) \)
For brevity; I'm only printing the terms accurate to 25 decimal digits even though I calculated them to much higher precision. Remember that \( y=\exp(2\pi i)(z-z_0) \). The terms in y, y^2, y^n will decay to zero in the limit as \( \Im(z)\to\infty \). These are very similar to the well behaved terms we expect in Kneser's slog. But the terms in y^-1, y^-2, y^-n grow in amplitude as \( \Im(z)\to\infty \) and these are the terms that tell us that Jay's slog is not Kneser's slog. The error term is dominated by the y^-1 whose magnitude is approximately 10^-25. This term will improve to 10^-27 near the real axis, but will misbehave as we get closer to the singularity at L.
For comparison; here is Sheldon's version of Kneser's slog Fourier series, which behaves as expected. In doing numerical calculations, I truncate and ignore all of the terms with y^-n as numerical calculation noise (whose values are<1E-80) for Kneser's slog. As the slog approaches the fixed point, \( \Im(z)\to\infty;\;\;\;y=\exp(2\pi i z)\to 0 \) and the Fourier series goes to a constant and Kneser's slog approaches the Schroder function slog. This means, if we have a Kneser slog accurate ~77 decimal digits in the red circle radius=1.1 (as posted), then we can extend it to everywhere in the complex plane. When \( \Im(\alpha(z))>\Im(z_0) \) then \( \text{slog(z)}=\alpha(z)+\theta(\alpha(z)) \) will similarly be accurate to >76 decimal digits.
Why does Jay's slog choose the theta mapping that it chooses, rather than for example Kneser's slog solution? If we had a 4200 term matrix, the results would have been equally self consistent. Would they have matched Jay's result with a 4000 term matrix to 75 decimal digits? I imagine the theta mapping in Jay's slog gradually becomes more and more well behaved as the number of terms used in the Matrix calculation approaches infinity, but that's about all we can guess.
So, is there any way to know that Jay's slog doesn't match Kneser's slog? How would one know if Jay's slog is correct and Sheldon's program gave the incorrect slog? It turns we can also compare any slog valid over enough of the complex plane to the inverse Schroder function, where the Schroder function is turned into a complex valued Abel function \( \alpha(z) \) for base(e) exponentiation. And its inverse \( \alpha^{-1}(z) \) is a superfunction for base e, which is also complex valued. In the plot below, we can see the inverse Schroder function extending out from the two complex valued fixed points. Superimposed is a circle of radius abs(L), which is the maximum possible radius of convergence for the slog since both Kneser's slog and Jay's slog functions have a singularity at the fixed point. The smaller circle has a radius of 1.1 within which Jay's slog was consistent to ~76 decimal digits with a theta mapping discussed earlier; except for near the singularity. The green dots represent a sequence of sample points where we take the 1-cyclic Fourier series of \( \text{JaySlog}(\alpha^{-1}(z))-z \). Here, \( \alpha^{-1}(z) \) is a complex valued superfunction for base(e) developed at the fixed point of L. This superfunction is the inverse of the complex valued Abel where S(z) is the Schroder function and \( \alpha(z)=\frac{\ln(S(z))}{L} \).
If we subtract out the constant term, the resulting 1-cyclic function has an amplitude of about 0.0009, and looks visually like a simple 1 term 1-cyclic exponential, but of course it is much more complicated than that. The details of the 1-cyclic function are critical to understanding the behavior of the slog in the complex plane. So let us analyze both the 1-cyclic theta Schroder function mapping for Jay's slog, and for Sheldon's slog, where Sheldon's slog is also 692 terms printed with 77 decimal digits of precision centered at 0 and using the same format as Jay's slog.
For simplicity I am using the substitution \( y=\exp(2\pi i (z-z_0)) \) where the 1-cyclic Fourier series was taken on 60 equally spaced samples between \( \alpha^{-1}(z_0)\;...\;\alpha^{-1}(z_0+1) \) My \( z_0 \) definition is somewhat random. Any number whose sexp is is in the well behaved range will do. \( z_0=\alpha(\text{sexp}(-1.2+\frac{-\ln(100)}{2\pi i}))\approx\alpha(\text{sexp}(-1.2+0.733i))\approx\alpha(-0.0161+0.75768i) \)
For brevity; I'm only printing the terms accurate to 25 decimal digits even though I calculated them to much higher precision. Remember that \( y=\exp(2\pi i)(z-z_0) \). The terms in y, y^2, y^n will decay to zero in the limit as \( \Im(z)\to\infty \). These are very similar to the well behaved terms we expect in Kneser's slog. But the terms in y^-1, y^-2, y^-n grow in amplitude as \( \Im(z)\to\infty \) and these are the terms that tell us that Jay's slog is not Kneser's slog. The error term is dominated by the y^-1 whose magnitude is approximately 10^-25. This term will improve to 10^-27 near the real axis, but will misbehave as we get closer to the singularity at L.
Code:
{JaySchroderTheta=
0.4884150884437966033074146 + 0.9223068629260569360028236*I
+y^ 1* ( 0.0008527801667562909114972082 + 0.0003125556838491044418014920*I)
+y^ 2* (-9.708080486410223227916803 E-7 + 9.710323531077750244488204 E-7*I)
+y^ 3* (-5.878024574251115606103497 E-10 - 1.978287506966856063528219 E-9*I)
+y^ 4* ( 3.001780705849975137758605 E-12 + 7.045220893442265392671639 E-13*I)
+y^ 5* (-3.208477845886844079339461 E-15 + 3.530961767417970849545504 E-15*I)
+y^ 6* (-2.624381171326872906024063 E-18 - 7.335020072903456702338085 E-18*I)
+y^ 7* ( 1.321074202132256455724085 E-20 + 1.903016287529692366359941 E-21*I)
+y^ 8* (-1.402639507282101778108773 E-23 + 1.895166893801443575271719 E-23*I)
+y^ 9* (-1.772669741397395815497444 E-26 - 3.843052544312580188258504 E-26*I)
+y^10* ( 7.645694514460890241495210 E-29 + 5.552243517074882110745036 E-30*I)
+y^11* (-7.631015450702930275457768 E-32 + 1.172015128721544589139540 E-31*I)
+y^12* (-1.209607734584374483299469 E-34 - 2.267112332210102955224326 E-34*I)
+y^13* ( 4.753532085143246163488097 E-37 + 5.924245742645747236000230 E-39*I)
+y^14* (-4.356128499530166755058963 E-40 + 7.703675719233272983112047 E-40*I)
+y^15* (-8.714832983894441953540382 E-43 - 1.408639493017801276231409 E-42*I)
+y^16* ( 3.110276412626715226311838 E-45 - 1.613579774192629849189697 E-46*I)
+y^17* (-2.559407501923749769404915 E-48 + 5.292507245138397232431039 E-48*I)
+y^18* (-6.452498657290520087576675 E-51 - 9.075146965735464393804413 E-51*I)
+y^19* ( 2.101441013456621272587330 E-53 - 2.444372952775288835443245 E-54*I)
+y^20* (-1.519200972127019065550623 E-56 + 3.733376757028580035512240 E-56*I)
+y^21* (-4.853204347358686770736136 E-59 - 5.975957490470546490388492 E-59*I)
+y^22* ( 1.449959209297115275635694 E-61 - 2.636213849095307318093880 E-62*I)
+y^23* (-8.957079055517393516146350 E-65 + 2.681409149566454402820860 E-64*I)
+y^24* (-3.688379555089395707123036 E-67 - 3.989745448215004096507666 E-67*I)
+y^25* ( 1.015123824848710522724926 E-69 - 2.526500725632739298899299 E-70*I)
+y^26* (-5.152996566028510960117872 E-73 + 1.950331086549317835189971 E-72*I)
+y^27* (-2.822342478896528913521354 E-75 - 2.685431212030569314974869 E-75*I)
+y^28* ( 7.178220565561244820058272 E-78 - 2.275652974604442218826640 E-78*I)
+y^29* (-4.049403841842832213780212 E-81 + 2.266443088077442119730075 E-80*I)
+ ...
+y^-10*(-2.498280149110708915077648 E-81 + 1.652447658636857959861455 E-80*I)
+y^-9* (-2.756326912393882620142116 E-81 + 1.817695787154244726071448 E-80*I)
+y^-8* ( 3.516799343332071483243507 E-79 + 1.129899714092047963753501 E-79*I)
+y^-7* (-1.570613227969063404295141 E-72 + 3.924015122582248643571331 E-73*I)
+y^-6* ( 9.849786695584270675395796 E-66 - 6.670060862127669849687576 E-66*I)
+y^-5* (-1.140067387869215638195915 E-58 + 1.105707324092836584333242 E-58*I)
+y^-4* ( 3.239403384690151114449161 E-51 - 2.919774220191775456469721 E-51*I)
+y^-3* (-2.701367092076952153394282 E-43 + 1.243406536013408682927275 E-43*I)
+y^-2* ( 6.760727404214144941914633 E-35 + 1.571472569348109887119765 E-35*I)
+y^-1* (-1.427564270910666936171602 E-26 - 1.075015698770442630959924 E-25*I)For comparison; here is Sheldon's version of Kneser's slog Fourier series, which behaves as expected. In doing numerical calculations, I truncate and ignore all of the terms with y^-n as numerical calculation noise (whose values are<1E-80) for Kneser's slog. As the slog approaches the fixed point, \( \Im(z)\to\infty;\;\;\;y=\exp(2\pi i z)\to 0 \) and the Fourier series goes to a constant and Kneser's slog approaches the Schroder function slog. This means, if we have a Kneser slog accurate ~77 decimal digits in the red circle radius=1.1 (as posted), then we can extend it to everywhere in the complex plane. When \( \Im(\alpha(z))>\Im(z_0) \) then \( \text{slog(z)}=\alpha(z)+\theta(\alpha(z)) \) will similarly be accurate to >76 decimal digits.
Why does Jay's slog choose the theta mapping that it chooses, rather than for example Kneser's slog solution? If we had a 4200 term matrix, the results would have been equally self consistent. Would they have matched Jay's result with a 4000 term matrix to 75 decimal digits? I imagine the theta mapping in Jay's slog gradually becomes more and more well behaved as the number of terms used in the Matrix calculation approaches infinity, but that's about all we can guess.
Code:
{KneserTht=
0.4884150884437966033074146 + 0.9223068629260569360028236*I
+y^ 1* ( 0.0008527801667562909114972082 + 0.0003125556838491044418014920*I)
+y^ 2* (-9.708080486410223227916804 E-7 + 9.710323531077750244488204 E-7*I)
+y^ 3* (-5.878024574251115606103495 E-10 - 1.978287506966856063528219 E-9*I)
+y^ 4* ( 3.001780705849975137758605 E-12 + 7.045220893442265392671649 E-13*I)
+y^ 5* (-3.208477845886844079339465 E-15 + 3.530961767417970849545504 E-15*I)
+y^ 6* (-2.624381171326872906024058 E-18 - 7.335020072903456702338094 E-18*I)
+y^ 7* ( 1.321074202132256455724086 E-20 + 1.903016287529692366359967 E-21*I)
+y^ 8* (-1.402639507282101778108782 E-23 + 1.895166893801443575271721 E-23*I)
+y^ 9* (-1.772669741397395815497438 E-26 - 3.843052544312580188258528 E-26*I)
+y^10* ( 7.645694514460890241495263 E-29 + 5.552243517074882110745464 E-30*I)
+y^11* (-7.631015450702930275457929 E-32 + 1.172015128721544589139547 E-31*I)
+y^12* (-1.209607734584374483299469 E-34 - 2.267112332210102955224372 E-34*I)
+y^13* ( 4.753532085143246163488202 E-37 + 5.924245742645747236005303 E-39*I)
+y^14* (-4.356128499530166755059191 E-40 + 7.703675719233272983112227 E-40*I)
+y^15* (-8.714832983894441953540533 E-43 - 1.408639493017801276231479 E-42*I)
+y^16* ( 3.110276412626715226312007 E-45 - 1.613579774192629849189277 E-46*I)
+y^17* (-2.559407501923749769405186 E-48 + 5.292507245138397232431359 E-48*I)
+y^18* (-6.452498657290520087577073 E-51 - 9.075146965735464393805332 E-51*I)
+y^19* ( 2.101441013456621272587567 E-53 - 2.444372952775288835443119 E-54*I)
+y^20* (-1.519200972127019065550898 E-56 + 3.733376757028580035512723 E-56*I)
+y^21* (-4.853204347358686770736722 E-59 - 5.975957490470546490390507 E-59*I)
+y^22* ( 1.449959209297115275649282 E-61 - 2.636213849095307319005724 E-62*I)
+y^23* (-8.957079055517393386050392 E-65 + 2.681409149566454313657751 E-64*I)
+y^24* (-3.688379555089382930300230 E-67 - 3.989745448215091595910955 E-67*I)
+y^25* ( 1.015123824849969112322006 E-69 - 2.526500725718867043243022 E-70*I)
+y^26* (-5.152996553594352134719516 E-73 + 1.950331078046532757613616 E-72*I)
+y^27* (-2.822341246957428036475944 E-75 - 2.685439630379945122000459 E-75*I)
+y^28* ( 7.179444561035288527697040 E-78 - 2.284011229705630628906773 E-78*I)
+y^29* (-2.829930076922271673723029 E-81 + 1.434277383231040198217646 E-80*I)
+y^30* (-3.588756110041497691714038 E-83 + 1.032843579771145257285948 E-83*I)
+...
+y^-4* (-6.954413041155221807894541 E-83 + 1.315319285416430611784253 E-82*I)
+y^-3* (-9.285463472073180073827514 E-83 + 1.723366120220390227761641 E-82*I)
+y^-2* (-1.398954602589024216439116 E-82 + 2.505664252483493625419186 E-82*I)
+y^-1* (-2.804523765697162374597977 E-82 + 4.582257135192752691376739 E-82*I)}
- Sheldon