2Descriptionofthenumericalintegrations
（本部分涉及比较复杂的积分计算，作者君就不贴上来了，贴上来了起点也不一定能成功显示。）
2.3Numericalmethod
Weutilizeasecond-orderWisdom–Holmansymplecticmapasourmainintegrationmethod(Wisdom&Kinoshita，Yoshida&&Nakai1991)withaspecialstart-upproceduretoreducethetruncationerrorofanglevariables，‘warmstart’(Saha&&Tremaine1992，1994).
Thestepsizeforthenumericalintegrationsis8dthroughoutallintegrationsofthenineplanets(N±1，2，3)，whichisabout111oftheorbitalperiodoftheinnermostplanet(Mercury).Asforthedeterminationofstepsize，wepartlyfollowthepreviousnumericalintegrationofallnineplanetsinSussman&&Wisdom(1988，7.2d)andSaha&&Tremaine(1994，22532d).Weroundedthedecimalpartofthetheirstepsizesto8tomakethestepsizeamultipleof2inordertoreducetheaccumulationofround-offerrorinthecomputationprocesses.Inrelationtothis，Wisdom&&Holman(1991)performednumericalintegrationsoftheouterfiveplanetaryorbitsusingthesymplecticmapwithastepsizeof400d，110.83oftheorbitalperiodofJupiter.Theirresultseemstobeaccurateenough，whichpartlyjustifiesourmethodofdeterminingthestepsize.However，sincetheeccentricityofJupiter(∼0.05)ismuchsmallerthanthatofMercury(∼0.2)，weneedsomecarewhenwecomparetheseintegrationssimplyintermsofstepsizes.
Intheintegrationoftheouterfiveplanets(F±)，wefixedthestepsizeat400d.
WeadoptGauss'fandgfunctionsinthesymplecticmaptogetherwiththethird-orderHalleymethod(Danby1992)asasolverforKeplerequations.ThenumberofmaximumiterationswesetinHalley'smethodis15，buttheyneverreachedthemaximuminanyofourintegrations.
Theintervalofthedataoutputis200000d(∼547yr)forthecalculationsofallnineplanets(N±1，2，3)，andabout8000000d(∼21903yr)fortheintegrationoftheouterfiveplanets(F±).
Althoughnooutputfilteringwasdonewhenthenumericalintegrationswereinprocess，weappliedalow-passfiltertotheraworbitaldataafterwehadcompletedallthecalculations.SeeSection4.1formoredetail.
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