{"action":"created","issue":{"url":"https://api.github.com/repos/km-git-acc/dbn_upper_bound/issues/126","repository_url":"https://api.github.com/repos/km-git-acc/dbn_upper_bound","labels_url":"https://api.github.com/repos/km-git-acc/dbn_upper_bound/issues/126/labels{/name}","comments_url":"https://api.github.com/repos/km-git-acc/dbn_upper_bound/issues/126/comments","events_url":"https://api.github.com/repos/km-git-acc/dbn_upper_bound/issues/126/events","html_url":"https://github.com/km-git-acc/dbn_upper_bound/issues/126","id":394518354,"node_id":"MDU6SXNzdWUzOTQ1MTgzNTQ=","number":126,"title":"Help need for evaluation of latest wiki page equations","user":{"login":"rudolph-git-acc","id":36714969,"node_id":"MDQ6VXNlcjM2NzE0OTY5","avatar_url":"https://avatars0.githubusercontent.com/u/36714969?v=4","gravatar_id":"","url":"https://api.github.com/users/rudolph-git-acc","html_url":"https://github.com/rudolph-git-acc","followers_url":"https://api.github.com/users/rudolph-git-acc/followers","following_url":"https://api.github.com/users/rudolph-git-acc/following{/other_user}","gists_url":"https://api.github.com/users/rudolph-git-acc/gists{/gist_id}","starred_url":"https://api.github.com/users/rudolph-git-acc/starred{/owner}{/repo}","subscriptions_url":"https://api.github.com/users/rudolph-git-acc/subscriptions","organizations_url":"https://api.github.com/users/rudolph-git-acc/orgs","repos_url":"https://api.github.com/users/rudolph-git-acc/repos","events_url":"https://api.github.com/users/rudolph-git-acc/events{/privacy}","received_events_url":"https://api.github.com/users/rudolph-git-acc/received_events","type":"User","site_admin":false},"labels":[],"state":"open","locked":false,"assignee":null,"assignees":[],"milestone":null,"comments":24,"created_at":"2018-12-28T00:41:30Z","updated_at":"2019-01-02T06:06:16Z","closed_at":null,"author_association":"COLLABORATOR","body":"Grateful for any help to properly evaluate the sequence of equations on this Wiki page: http://michaelnielsen.org/polymath1/index.php?title=Polymath15_test_problem#Large_negative_values_of_.5Bmath.5Dt.5B.2Fmath.5D\r\n\r\nBelow is the pari/gp code I have used so far. At the end of the code is a simple rootfinder that should (approximately) find this sequence of roots:\r\n\r\n```\r\n-10.00000000, 906.72649548\r\n-10.00000000, 931.47009390\r\n-10.00000000, 976.80075639\r\n-10.00000000, 990.45190417\r\n```\r\n\r\nhowever from eq 318 onwards it finds many more roots (also at different t, e.g. -100). Have experimented with the integral limits, the integral's fixed parameter, the real precision settings of pari and the size of the sums, but no success yet. \r\n\r\nHave re-injected some of the non-zero multiplication factors that can be switched on/off if needed.\r\n\r\nOne thing I noticed is that after eq 307 the function doesn't seem to be real anymore (see data at the end). Not sure if this should matter though, since I use the real value of Ht for root finding. \r\n\r\n```ruby\r\ndefault(realprecision, 40)\r\n\r\nXi(s)=return((s/2)*(s-1)*Pi^(-s/2)*gamma(s/2)*zeta(s));\r\nM0(s)=return(s*(s-1)/2*Pi^(-s/2)*sqrt(2*Pi)*exp((s/2-1/2)*log(s/2)-s/2));\r\n\r\nHt302(x,t) = {\r\n    A=intnum(v=-8,8,Xi((1+I*x)/2+I*sqrt(abs(t))*v)*exp(-v^2),1);\r\n    return(A);\r\n}\r\n\r\nHt303(x,t) = {\r\n    A=intnum(v=-8,8,Xi((1+I*x)/2+I*sqrt(abs(t))*v-Pi*I*abs(t)/8)*exp(-(v-Pi*sqrt(abs(t))/8)^2),1);\r\n    return(A);\r\n}\r\n\r\nHt304(x,t) = {\r\n    xt = x-Pi*abs(t)/4;\r\n    mulfact = exp(-Pi^2*abs(t)/64);\r\n    A=mulfact*intnum(v=-8,8,Xi((1+I*xt)/2+I*sqrt(abs(t))*v)*exp(-v^2+Pi*sqrt(abs(t))*v/4),1);\r\n    return(A);\r\n}\r\n\r\nHt307(x,t) = {\r\n    xt = x-Pi*abs(t)/4;\r\n    mulfact = exp(-Pi^2*abs(t)/64)*2;\r\n    A=mulfact*intnum(v=-8,8,real(M0((1+I*xt)/2+I*sqrt(abs(t))*v))*zeta((1+I*xt)/2+I*sqrt(abs(t))*v)*exp(-v^2+Pi*sqrt(abs(t))*v/4),1);\r\n    return(A);\r\n}\r\n\r\nHt312(x,t) = {\r\n    xt = x-Pi*abs(t)/4;\r\n    mulfact = exp(-Pi^2*abs(t)/64)*(M0((1+I*xt)/2));\r\n    A= mulfact*intnum(v=-8,8,exp((I*sqrt(abs(t))*v)/2*log(xt/(4*Pi))-Pi*sqrt(abs(t))*v/4+(I*abs(t)*v^2)/(2*xt))*zeta((1+I*xt)/2+I*sqrt(abs(t))*v)*exp(-v^2+Pi*sqrt(abs(t))*v/4),1);\r\n    return(A);\r\n}\r\n\r\nHt315(x,t) = {\r\n    xt = x-Pi*abs(t)/4;\r\n    N = sqrt(xt/(4*Pi));\r\n    u = 4*Pi*abs(t)/xt;\r\n    mulfact = exp(-Pi^2*abs(t)/64)*(M0((1+I*xt)/2));\r\n    A=mulfact*intnum(v=-8,8,exp(I*sqrt(abs(t))*v*log(N)+I*u*v^2/(8*Pi))*zeta((1+I*xt)/2+I*sqrt(abs(t))*v)*exp(-v^2),1);\r\n    return(A);\r\n}\r\n\r\nHt316(x,t) = {\r\n    xt = x-Pi*abs(t)/4;\r\n    N = sqrt(xt/(4*Pi));\r\n    u = 4*Pi*abs(t)/xt;\r\n    mulfact = exp(-Pi^2*abs(t)/64)*(M0((1+I*xt)/2));\r\n    A=mulfact*sum(n=1,1000, intnum(v=-8,8,exp(I*sqrt(abs(t))*v*log(N)+I*u*v^2/(8*Pi))*n^(-((1+I*xt)/2+I*sqrt(abs(t))*v))*exp(-v^2),1));\r\n    return(A);\r\n}\r\n\r\nHt317(x,t) = {\r\n    xt = x-Pi*abs(t)/4;\r\n    N = sqrt(xt/(4*Pi));\r\n    u = 4*Pi*abs(t)/xt;\r\n    mulfact = exp(-Pi^2*abs(t)/64)*(M0((1+I*xt)/2))*N^((1+I*xt)/2)/8;\r\n    A=mulfact*sum(n=1,1000, intnum(v=-8,8,exp(-I*sqrt(abs(t))*v*log(n/N)+I*u*v^2/(8*Pi)-((1+I*xt)/2)*log(n/N))*exp(-v^2),1));\r\n    return(A);\r\n}\r\n\r\nHt318(x,t) = {\r\n    xt = x-Pi*abs(t)/4;\r\n    N = sqrt(xt/(4*Pi));\r\n    u = 4*Pi*abs(t)/xt;\r\n    mulfact = exp(-Pi^2*abs(t)/64)*(M0((1+I*xt)/2))*N^((1+I*xt)/2)/8*sqrt(Pi);\r\n    A=mulfact*sum(n=1,1000,exp(-abs(t)*(log(n/N))^2/(4*(1-I*u/(8*Pi)))-(1+I*xt)/2*log(n/N)));\r\n    return(A);\r\n}\r\n\r\nHt320(x,t) = {\r\n    xt = x-Pi*abs(t)/4;\r\n    N = sqrt(xt/(4*Pi));\r\n    u = 4*Pi*abs(t)/xt;\r\n    mulfact = exp(-Pi^2*abs(t)/64)*(M0((1+I*xt)/2))*N^((1+I*xt)/2)/8*sqrt(Pi);\r\n    A = mulfact*sum(n=1,1000,exp(-(abs(t)*(n-N)^2)/(4*N^2*(1-I*u/(8*Pi)))-I*xt/2*(n-N)/N+I*xt*(n-N)^2/(4*N^2)));\r\n    return(A);\r\n}\r\n\r\nHt321(x,t) = {\r\n    xt = x-Pi*abs(t)/4;\r\n    N = sqrt(xt/(4*Pi));\r\n    u = 4*Pi*abs(t)/xt;\r\n    mulfact = exp(-Pi^2*abs(t)/64)*(M0((1+I*xt)/2))*N^((1+I*xt)/2)/8*sqrt(Pi);\r\n    A = mulfact*sum(n=1,1000,exp(-(2*Pi*u*(n-N)^2)/(8*Pi-I*u)-2*Pi*I*N*(n-N)+Pi*I*(n-N)^2));\r\n    return(A);\r\n}\r\n\r\nHt323(x,t) = {\r\n    xt = x-Pi*abs(t)/4;\r\n    N = sqrt(xt/(4*Pi));\r\n    u = 4*Pi*abs(t)/xt;\r\n    mulfact = exp(-Pi^2*abs(t)/64)*(M0((1+I*xt)/2))*N^((1+I*xt)/2)/8*sqrt(Pi);\r\n    A = mulfact*sum(n=1,1000,exp(-(2*Pi*u*(n-N)^2)/(8*Pi-I*u)-Pi*I*n^2+2*Pi*I*(n-N)^2));\r\n    return(A);\r\n}\r\n\r\nHt324(x,t) = {\r\n    xt = x-Pi*abs(t)/4;\r\n    N = sqrt(xt/(4*Pi));\r\n    u = 4*Pi*abs(t)/xt;\r\n    mulfact = exp(-Pi^2*abs(t)/64)*(M0((1+I*xt)/2))*N^((1+I*xt)/2)/8*sqrt(Pi);\r\n    A = mulfact*sum(n=1,1000,exp((16*Pi^2*I*(n-N)^2)/(8*Pi-I*u))*exp(Pi*I*n));\r\n    return(A);\r\n}\r\n\r\nHt328(x,t) = {\r\n    xt = x-Pi*abs(t)/4;\r\n    N = sqrt(xt/(4*Pi));\r\n    u = 4*Pi*abs(t)/xt;\r\n    mulfact = exp(-Pi^2*abs(t)/64)*(M0((1+I*xt)/2))*N^((1+I*xt)/2)/8*sqrt(Pi);\r\n    A = mulfact*sum(n=1,1000,exp(-Pi*I*n*(n+1)/2)*exp(2*Pi*I*(n+1/2)*N)*exp(-u*n*(n+1)/16));\r\n    return(A);\r\n}\r\n\r\nprint(\"eq.302 \",Ht302(1000, -10))\r\nprint(\"eq.303 \",Ht303(1000, -10))\r\nprint(\"eq.304 \",Ht304(1000, -10))\r\nprint(\"eq.307 \",Ht307(1000, -10))\r\nprint(\"eq.315 \",Ht315(1000, -10))\r\nprint(\"eq.316 \",Ht316(1000, -10))\r\nprint(\"eq.317 \",Ht317(1000, -10))\r\nprint(\"eq.318 \",Ht318(1000, -10))\r\nprint(\"eq.320 \",Ht320(1000, -10))\r\nprint(\"eq.321 \",Ht321(1000, -10))\r\nprint(\"eq.323 \",Ht323(1000, -10))\r\nprint(\"eq.324 \",Ht324(1000, -10))\r\nprint(\"eq.328 \",Ht324(1000, -10))\r\n\r\nxs = 900; xe = 1010;\r\nts = -10; te = -1;\r\nf=Ht318;\r\nforstep(t=ts, te, 10, {\r\n    xc = xs;\r\n    while(xc <= xe,\r\n        if(real(f(xc,t))*real(f(xc+1,t)) < 0, root=solve(a=xc, xc+1, real(f(a,t))); printf(\"%3.8f, %3.8f\\n\",t, root));\r\n        xc=xc+1);\r\n})\r\n```\r\n\r\n```\r\neq.302 1.201485926838305713580758240853488704456 E-165 + 3.514303326279329051 E-221*I\r\neq.303 1.201485926838305713582303927841938470920 E-165 + 3.522103826014908131 E-221*I\r\neq.304 1.201485926838305713582303927841938470920 E-165 + 3.522103826014922598 E-221*I\r\neq.307 1.201477416607903787215861199656638682224 E-165 + 3.992587328505261451855248837227037420869 E-169*I\r\neq.312 1.200242494404236645648308584282451391036 E-165 + 4.014384445003979486997306873566995218158 E-169*I\r\neq.315 1.200242494404236645648308584282451391036 E-165 + 4.014384445003979486997306873566995218158 E-169*I\r\neq.316 1.200242494404236645648308630967087228353 E-165 + 4.014384445003979487000178658198142256762 E-169*I\r\neq.317 1.210630457223022714918883693478442584040 E-165 - 5.581422172204223838770738013241198104262 E-166*I\r\neq.318 1.209227903617935860382758702133562335051 E-165 - 5.611945144292097211229689587084475461023 E-166*I\r\neq.320 2.791674127351571571455964690537038072397 E-167 - 2.126109533152847673586778474944806978351 E-167*I\r\neq.321 2.791674127351571571455964690537038072397 E-167 - 2.126109533152847673586778474944806978351 E-167*I\r\neq.323 -3.076893561828845392623560432368639943370 E-167 + 1.687160985046708488717820653125387727510 E-167*I\r\neq.324 -3.076893561828845392623560432368639943370 E-167 + 1.687160985046708488717820653125387727510 E-167*I\r\neq.328 -3.076893561828845392623560432368639943370 E-167 + 1.687160985046708488717820653125387727510 E-167*I\r\n```"},"comment":{"url":"https://api.github.com/repos/km-git-acc/dbn_upper_bound/issues/comments/450791959","html_url":"https://github.com/km-git-acc/dbn_upper_bound/issues/126#issuecomment-450791959","issue_url":"https://api.github.com/repos/km-git-acc/dbn_upper_bound/issues/126","id":450791959,"node_id":"MDEyOklzc3VlQ29tbWVudDQ1MDc5MTk1OQ==","user":{"login":"teorth","id":199308,"node_id":"MDQ6VXNlcjE5OTMwOA==","avatar_url":"https://avatars1.githubusercontent.com/u/199308?v=4","gravatar_id":"","url":"https://api.github.com/users/teorth","html_url":"https://github.com/teorth","followers_url":"https://api.github.com/users/teorth/followers","following_url":"https://api.github.com/users/teorth/following{/other_user}","gists_url":"https://api.github.com/users/teorth/gists{/gist_id}","starred_url":"https://api.github.com/users/teorth/starred{/owner}{/repo}","subscriptions_url":"https://api.github.com/users/teorth/subscriptions","organizations_url":"https://api.github.com/users/teorth/orgs","repos_url":"https://api.github.com/users/teorth/repos","events_url":"https://api.github.com/users/teorth/events{/privacy}","received_events_url":"https://api.github.com/users/teorth/received_events","type":"User","site_admin":false},"created_at":"2019-01-02T06:06:16Z","updated_at":"2019-01-02T06:06:16Z","author_association":"COLLABORATOR","body":"Thanks for this!  I guess the numerics are telling us that it's not a good idea to Taylor expand this particulr exponential factor.  (It's interesting though that the 1-periodicity survives even in these lousy approximations, though.  I'm also slightly curious to see how well each of these approximations maintain the real-valuedness of $latex H_t(x)$, though this is somewhat secondary.)\r\n\r\nI'll start writing a revised version of the wiki computations in which I keep this exponential phase factor.  The Theta function is now replaced by something a bit weirder - a series of Airy functions - but perhaps that's simply what the answer for what the asymptotic behaviour is."}}
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