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) |
| 1 | Here 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: |
| 8 | Example:: |
| 9 | {{{ |
| 10 | sage: 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 | }}} |
| 18 | Computing The first 'n' multiple zeta values using a fast algorithm using Initial, Middle and Final words |
| 19 | Example:: |
| 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 | |
| 34 | Computing Multiple Zeta values using Polylogarithm algorithm |
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? |
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: |