Changes between Version 3 and Version 6 of Ticket #18010


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Timestamp:
08/25/16 12:50:15 (5 years ago)
Author:
akhi
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  • Ticket #18010

    • Property Status changed from new to needs_review
    • Property Summary changed from This introducing function compute multiple zeta values and finding relations between them to This introduces function to compute multiple zeta values and finding relations between them
    • Property Milestone changed from sage-6.6 to sage-7.4
    • Property Commit changed from c40c4f19230485f7b02aef06f13f402b890bc66d to 43f19c7c1eaed13ff371773fdcb0377a5975383d

  • Ticket #18010 – Description

    v3 v6  
    1 This program computes multiple zeta values for any given precision (composition input) and also it helps to compute the  relation between multiple zeta values (composition input)
     1Here it is introducing to three functions that compute multiple zeta values,
     2\\The first one multizeta it computing multiple zeta values using Double Tails, this is fastest algorithm to compute one MZV
     3\\The second allmultizetaprint is returning the first n multiple zeta values using intial, Middle and final words, this algorith is very efficient to compute a plenty of MZV together
     4\\ The third one mzeta that compute multiple zeta values using polylogarithm
     5\\References: Double tails of multiple zeta values, P. Akhilesh, Journal of Number Theory, Volume 170, January 2017, Pages 228–249
     6\\http://www.sciencedirect.com/science/article/pii/S0022314X16301718
     7\\Multiple zeta values Computing using Double Tails:
     8Example::
     9{{{
     10sage: multizeta([2],170,100)
     11    1.6449340668482264364724151666460251892189499012067984377355582293700074704032008738336289006197587?
     12    sage: multizeta([2,3],170,100)
     13    0.7115661975505724320969738060864026120925612044383392364922224964576860857450582651154252344636008?
     14    sage: multizeta([2,1],170,100)
     15    1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093353?
     16    sage:
     17}}}
     18Computing The first 'n' multiple zeta values using a fast algorithm using Initial, Middle and Final words
     19Example::
     20{{{
     21    sage: allmultizetaprint(10,170,100)
     22    multizeta( [2] )= 1.644934066848226436472415166646025189218949901206798437735558229370007470403200873833628900619758706?
     23    multizeta( [3] )= 1.202056903159594285399738161511449990764986292340498881792271555341838205786313090186455873609335258?
     24    multizeta( [2, 1] )= 1.202056903159594285399738161511449990764986292340498881792271555341838205786313090186455873609335258?
     25    multizeta( [4] )= 1.082323233711138191516003696541167902774750951918726907682976215444120616186968846556909635941699917?
     26    multizeta( [3, 1] )= 0.270580808427784547879000924135291975693687737979681726920744053861030154046742211639227408985424980?
     27    multizeta( [2, 2] )= 0.811742425283353643637002772405875927081063213939045180762232161583090462140226634917682226956274938?
     28    multizeta( [2, 1, 1] )= 1.082323233711138191516003696541167902774750951918726907682976215444120616186968846556909635941699918?
     29    multizeta( [5] )= 1.036927755143369926331365486457034168057080919501912811974192677903803589786281484560043106557133337?
     30    multizeta( [4, 1] )= 0.0965511599894437344656455314289427640320103723436914152525630787528921454259587614177018405925170654?
     31
     32}}}
     33
     34Computing Multiple Zeta values using Polylogarithm algorithm
    235Example::
    336{{{
    437
    5 sage: mzeta([2,1])
    6 mpf('1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581461991577952607
    7 19418491995998673283213776396837207900161453941782949360066719191575522242494243961563909664103291159095780965514651279918
    8 40510571525598801543710978110203982753256678760352233698494166181105701471577863949973752378527793703095602570185318279000
    9 30765471075630488433208697115737423807934450316076253177145354444118311781822497185263570918244899879620350833575617202260
    10 339378587032813126780799005417734869115253706562370574409662217129026273207323614922429130405285553723410330775777980642420
    11 243048828152100091460265382206962715520208227433500101529480119869011762595167636699817183557523488070371955574234729408359
    12 5208861666202572853755813079282586487282173705566196898952662018776810629200817792338135876828426412432431480282173674506720
    13 69350762689530434593937503296636377575062473323992348288310773390527680200757984356793711505090050273660471140085335034364672
    14 24856531518117766181092227918431')
    15 sage: mzeta([2,5],100)
    16 mpf('0.6587533875711093581412522186346254271044356998380703541143384794612078112362164544354656566174209515505697938')
    17 sage: mzeta([2],100)
    18 mpf('1.644934066848226436472415166646025189218949901206798437735558229370007470403200873833628900619758705304004264')
    19 sage: zeta([3],100)-zeta([2,1],100)
    20 mpf('0.0')
    21 sage:
     38    sage: mzeta([2,1])
     39    1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581461991577952607194184919959986
     40    732832137763968372079001614539417829493600667191915755222424942439615639096641032911590957809655146512799184051057152559880154371097
     41    811020398275325667876035223369849416618110570147157786394997375237852779370309560257018531827900030765471075630488433208697115737423
     42    8079344503160762531771453544441183117818224971852635709182448998796203508335756172022603393785870328131267807990054177348691152537065
     43    6237057440966221712902627320732361492242913040528555372341033077577798064242024304882815210009146026538220696271552020822743350010152
     44    9480119869011762595167636699817183557523488070371955574234729408359520886166620257285375581307928258648728217370556619689895266201877
     45    68106292008177923381358768284264124324314802821736745067206935076268953043459393750329663637757506247332399234828831077339052768020075
     46    7984356793711505090050273660471140085335034364672248565315181177661811?
     47    sage: mzeta([2,1],100)
     48    1.20205690315959428539973816151144999076498629234049888179227155534183820578631309018645587360933526?
    2249}}}
    2350
    … …  
    2552Example ::
    2653{{{
    27 
    28 sage: Rmzeta([[2,1],[3]],100,100,1000)
    29 [1, -1]
    30 sage: Rmzeta([[4],[2,2]],100,100,1000)
    31 [-3, 4]
    32 sage:
     54    sage: Rmultizeta([[2,1],[3]])
     55    [1, -1]
     56    sage: Rmultizeta([[2,1],[3]],100,100,1000)
     57    [1, -1]
     58    sage:
    3359}}}