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Now we turn to an example of the usage of the Dragilev method with Maple. Let us consider the curve
from
http://www.mapleprimes.com/questions/143454-How-To-Produce-Such-Animation .

Its points can be found by the Dragilev method as follows.

 

 

> 

restart; with(LinearAlgebra); with(plots); N := 100; smin := 0.; smax := 0.46e-1; h := 0.1e-2; f1 := (x1-2)^2+(x2-2)^2+x3^2-9; f2 := x1^6+x2^6+x3^6-12; x01 := 3.9691163496*10^(-12); x02 := .3535805783; x03 := -1.5130442808; n := 2; x := seq(eval(cat('x', i)), i = 1 .. n+1); x0 := seq(eval(cat('x0', i)), i = 1 .. n+1); F := [seq(unapply(eval(cat('f', i)), [x]), i = 1 .. n)]; A := MTM:-jacobian(F(x), [x]); A := MTM:-subs(A, [x], [x(s)]); deqs := seq(diff(x[i](s), s) = simplify(Determinant(DeleteColumn(ColumnOperation(A, [i, n+1]), n+1))), i = 1 .. n), diff(x[n+1](s), s) = simplify(-Determinant(DeleteColumn(A, n+1))); ics := seq(x[i](0) = x0[i], i = 1 .. n+1); sln := dsolve([deqs, ics], numeric, method = rkf45, abserr = (1/100000)*h, maxfun = 100000, range = smin .. smax); a1 := sln(smin+smax*nn/N)[2]; a2 := sln(smin+smax*nn/N)[3]; a3 := sln(smin+smax*nn/N)[4]; LL := [seq([rhs(a1), rhs(a2), rhs(a3)], nn = 0 .. N)]

[[HFloat(3.96911635e-12), HFloat(0.3535805783), HFloat(-1.5130442808)], [HFloat(-0.07020454925501105), HFloat(0.44266112845900346), HFloat(-1.512927186499708)], [HFloat(-0.1365523096751349), HFloat(0.5346659092041354), HFloat(-1.51259384706809)], [HFloat(-0.1990356789337114), HFloat(0.6293154663866696), HFloat(-1.51177575875741)], [HFloat(-0.2577416479074561), HFloat(0.7262024700215282), HFloat(-1.5099809507788806)], [HFloat(-0.31289410917953364), HFloat(0.8246961946087024), HFloat(-1.5063803388153627)], [HFloat(-0.36490443012343554), HFloat(0.9238089112777212), HFloat(-1.4996798520674763)], [HFloat(-0.4144170105007935), HFloat(1.022036938154396), HFloat(-1.4880116198790554)], [HFloat(-0.4623180719047317), HFloat(1.117226615901122), HFloat(-1.468911388499481)], [HFloat(-0.5096485492984643), HFloat(1.2065657463441748), HFloat(-1.4394881381399327)], [HFloat(-0.55736954752636), HFloat(1.2868332259225372), HFloat(-1.3968729649312162)], [HFloat(-0.6060015789587125), HFloat(1.354990223950928), HFloat(-1.3389242495759375)], [HFloat(-0.6553014093659834), HFloat(1.4089861306540672), HFloat(-1.2649415008426375)], [HFloat(-0.7042180343442459), HFloat(1.4484078256378705), HFloat(-1.1759893021729204)], [HFloat(-0.7512174417121095), HFloat(1.4745825899025702), HFloat(-1.0745878704583272)], [HFloat(-0.7947819748668111), HFloat(1.4900727002426029), HFloat(-0.9639335242767559)], [HFloat(-0.833781952259619), HFloat(1.497883547083401), HFloat(-0.8470884941359643)], [HFloat(-0.8675792606788528), HFloat(1.5007869963080367), HFloat(-0.7264816857493535)], [HFloat(-0.8959296468872269), HFloat(1.5009581263139142), HFloat(-0.6037785854401849)], [HFloat(-0.9188194683454348), HFloat(1.499899650912559), HFloat(-0.4799919336697904)], [HFloat(-0.9363293875522539), HFloat(1.4985360604185338), HFloat(-0.35567362400238245)], [HFloat(-0.9485518434983851), HFloat(1.4973725215001998), HFloat(-0.23110084764560204)], [HFloat(-0.9555562902053749), HFloat(1.4966470555521214), HFloat(-0.10640901654497847)], [HFloat(-0.9573812746280482), HFloat(1.4964524707500564), HFloat(0.018328984371830188)], [HFloat(-0.954036428424389), HFloat(1.4968072760259679), HFloat(0.14305937049495035)], [HFloat(-0.9455039898135277), HFloat(1.497675978574668), HFloat(0.26772545371460366)], [HFloat(-0.9317362386255554), HFloat(1.498932183420108), HFloat(0.3922419831714783)], [HFloat(-0.912655492252804), HFloat(1.5002739352448655), HFloat(0.516441993606866)], [HFloat(-0.8881671097587187), HFloat(1.5010904052163099), HFloat(0.6399843831168849)], [HFloat(-0.8582073117066802), HFloat(1.5003015405271365), HFloat(0.7622025141065844)], [HFloat(-0.8228413104297032), HFloat(1.4962118712371768), HFloat(0.8819096972550735)], [HFloat(-0.7824123582363602), HFloat(1.4864521951859897), HFloat(0.9972212711917412)], [HFloat(-0.7376994579002023), HFloat(1.4681291973288075), HFloat(1.1054932706374472)], [HFloat(-0.6899691742413475), HFloat(1.4382865356600403), HFloat(1.2035547580845158)], [HFloat(-0.640793094886806), HFloat(1.3946468459636525), HFloat(1.2883165718012761)], [HFloat(-0.5916133985803501), HFloat(1.3363654708461847), HFloat(1.3576190236472425)], [HFloat(-0.5432622511235201), HFloat(1.2643769787034354), HFloat(1.4109132364999581)], [HFloat(-0.4957372555574966), HFloat(1.1811056107722093), HFloat(1.4493818684802757)], [HFloat(-0.44834855918973837), HFloat(1.0897229778443782), HFloat(1.47546104494275)], [HFloat(-0.40007801693974077), HFloat(0.9933761572714118), HFloat(1.492090530412655)], [HFloat(-0.3499131225687298), HFloat(0.894695802249036), HFloat(1.502068954641732)], [HFloat(-0.29703523677628824), HFloat(0.795635291681034), HFloat(1.507691911758092)], [HFloat(-0.2408717810027918), HFloat(0.6975301935918784), HFloat(1.510650903966005)], [HFloat(-0.18107200692358422), HFloat(0.6012475261975224), HFloat(1.512090130672608)], [HFloat(-0.11745719455376195), HFloat(0.5073401510433072), HFloat(1.5127266086383548)], [HFloat(-0.04997196405832138), HFloat(0.4161696292995799), HFloat(1.5129760383774014)], [HFloat(0.021354669525786785), HFloat(0.32799071164128085), HFloat(1.5130590813437668)], [HFloat(0.09643801964195603), HFloat(0.24300308723979325), HFloat(1.5130809048942586)], [HFloat(0.17515779330223594), HFloat(0.16137901275107697), HFloat(1.5130842575944754)], [HFloat(0.25737127682842803), HFloat(0.083274443621113), HFloat(1.5130791347588286)], [HFloat(0.3429200578880072), HFloat(0.00882881158017646), HFloat(1.513051075711483)], [HFloat(0.43162829363249994), HFloat(-0.061842569150896254), HFloat(1.5129493066626698)], [HFloat(0.5232893226524888), HFloat(-0.12866289707153564), HFloat(1.512653315590194)], [HFloat(0.6176345908667109), HFloat(-0.19161543821593474), HFloat(1.5119153034648503)], [HFloat(0.714276714653005), HFloat(-0.25077357488832147), HFloat(1.5102761911097695)], [HFloat(0.8126186873771224), HFloat(-0.30634194423967276), HFloat(1.5069547603323996)], [HFloat(0.9117245837944199), HFloat(-0.3587062010131653), HFloat(1.5007204446658222)], [HFloat(1.0101620022557067), HFloat(-0.40848060238894124), HFloat(1.4897791944962493)], [HFloat(1.105860930619992), HFloat(-0.4565241447108136), HFloat(1.4717352209620689)], [HFloat(1.19608265955441), HFloat(-0.5038680925129831), HFloat(1.4437318457504194)], [HFloat(1.2776309915167487), HFloat(-0.551501020289432), HFloat(1.4028634151375503)], [HFloat(1.3474032519856458), HFloat(-0.6000176353727356), HFloat(1.3468578510591316)], [HFloat(1.4031859901966037), HFloat(-0.6492777960331133), HFloat(1.2748099903564865)], [HFloat(1.4443474266977285), HFloat(-0.6983182957564676), HFloat(1.1875726736720857)], [HFloat(1.4720191947329169), HFloat(-0.7456377958104611), HFloat(1.0875245353923155)], [HFloat(1.4886537272379206), HFloat(-0.7896918655116054), HFloat(0.9778264454517267)], [HFloat(1.4972438276549478), HFloat(-0.8292925652585661), HFloat(0.8615916199954949)], [HFloat(1.5006160499887264), HFloat(-0.8637442991256964), HFloat(0.741339767242725)], [HFloat(1.5010271554263845), HFloat(-0.8927632645900738), HFloat(0.6188270083184392)], [HFloat(1.500058960407771), HFloat(-0.9163160885953419), HFloat(0.4951358051223638)], [HFloat(1.4986986823618642), HFloat(-0.9344774154656306), HFloat(0.3708634123602541)], [HFloat(1.4974939990538119), HFloat(-0.9473408386533276), HFloat(0.2463126385496126)], [HFloat(1.496708010050892), HFloat(-0.9549794374594281), HFloat(0.1216307619567666)], [HFloat(1.4964465594182643), HFloat(-0.957435484273183), HFloat(-0.003104623275179584)], [HFloat(1.4967350200030916), HFloat(-0.9547219916280205), HFloat(-0.12783865039638506)], [HFloat(1.4975453647794568), HFloat(-0.9468246088205576), HFloat(-0.2525162225822078)], [HFloat(1.4987657702497037), HFloat(-0.933699413200501), HFloat(-0.37705758357396724)], [HFloat(1.5001222705387016), HFloat(-0.9152721210702315), HFloat(-0.5013097059803103)], [HFloat(1.5010497537885552), HFloat(-0.8914484180251965), HFloat(-0.6249599896254425)], [HFloat(1.500534189379768), HFloat(-0.8621567966206845), HFloat(-0.7473908275135906)], [HFloat(1.4969611134309482), HFloat(-0.827439403373343), HFloat(-0.8674899701821565)], [HFloat(1.4880408584810398), HFloat(-0.7875965571243855), HFloat(-0.9834651826832739)], [HFloat(1.470926381444875), HFloat(-0.7433478348489277), HFloat(-1.0927599416200353)], [HFloat(1.4426331641903316), HFloat(-0.6959044753106292), HFloat(-1.1922423392912453)], [HFloat(1.400756989373389), HFloat(-0.6468200569757765), HFloat(-1.2787696806464606)], [HFloat(1.3442480434341646), HFloat(-0.5975803701366754), HFloat(-1.3500244652152593)], [HFloat(1.2738265441830154), HFloat(-0.5491115267150344), HFloat(-1.4052412735350015)], [HFloat(1.191769337237313), HFloat(-0.5015119490394248), HFloat(-1.445407111209715)], [HFloat(1.1012015382474256), HFloat(-0.45415822365192504), HFloat(-1.4728439369903699)], [HFloat(1.0053066320084918), HFloat(-0.40605209363976014), HFloat(-1.4904693623293839)], [HFloat(0.9067928452933773), HFloat(-0.35616707977118617), HFloat(-1.5011246711641846)], [HFloat(0.8076961093786404), HFloat(-0.3036558951686247), HFloat(-1.507176611517467)], [HFloat(0.7094201338189514), HFloat(-0.2479162817486269), HFloat(-1.5103894774382063)], [HFloat(0.6128804389159239), HFloat(-0.18857288073398004), HFloat(-1.5119684242856894)], [HFloat(0.5186609079333568), HFloat(-0.1254287185683981), HFloat(-1.5126756863301285)], [HFloat(0.42714118480309204), HFloat(-0.05841571443271286), HFloat(-1.5129574903052154)], [HFloat(0.33858570669550725), HFloat(0.012445687663004357), HFloat(-1.513053493520705)], [HFloat(0.25319898103191724), HFloat(0.08707638295347792), HFloat(-1.5130796115716185)], [HFloat(0.17115575687551046), HFloat(0.16535958625506061), HFloat(-1.5130841875616152)], [HFloat(0.09261379521910264), HFloat(0.2471548304523304), HFloat(-1.5130803537377422)], [HFloat(0.01771482145577431), HFloat(0.3323055134257799), HFloat(-1.5130568561204285)]]

 (1)
> 

NULL
> 

plots:-pointplot3d(LL)

 

Let us  compare the above with the output of

> 

DirectSearch:-SolveEquations([f1, f2], AllSolutions, solutions = 10)

Matrix(10, 4, {(1, 1) = 0.5879527055e-22, (1, 2) = Vector(2, {(1) = HFloat(3.0802027595200343e-12), (2) = HFloat(-7.02193858614919e-12)}), (1, 3) = [x1 = 1.49689379855953, x2 = -.953216966644196, x3 = .159354002146394], (1, 4) = 456, (2, 1) = 0.1925255113e-21, (2, 2) = Vector(2, {(1) = HFloat(-1.0190959187639237e-11), (2) = HFloat(-9.416467605660728e-12)}), (2, 3) = [x1 = -.553610165646297, x2 = 1.28096197032918, x3 = 1.40073531896101], (2, 4) = 401, (3, 1) = 0.6774810976e-21, (3, 2) = Vector(2, {(1) = HFloat(-4.304112621866807e-12), (2) = HFloat(-2.567013268617302e-11)}), (3, 3) = [x1 = .751705593125648, x2 = -.272419578356594, x3 = 1.50926151930788], (3, 4) = 415, (4, 1) = 0.7486975353e-21, (4, 2) = Vector(2, {(1) = HFloat(-2.6187052526438492e-11), (2) = HFloat(-7.933209644761519e-12)}), (4, 3) = [x1 = -0.191238253240398e-2, x2 = .355905238460614, x3 = 1.51304303564571], (4, 4) = 498, (5, 1) = 0.1320643164e-20, (5, 2) = Vector(2, {(1) = HFloat(-3.6415315207705135e-12), (2) = HFloat(-3.61577434659921e-11)}), (5, 3) = [x1 = -.956217277283830, x2 = 1.49657732918864, x3 = -0.857031154514981e-1], (5, 4) = 478, (6, 1) = 0.2489782556e-20, (6, 2) = Vector(2, {(1) = HFloat(1.1542766742422828e-11), (2) = HFloat(4.8544279707130045e-11)}), (6, 3) = [x1 = .293304904587475, x2 = 0.512045111444833e-1, x3 = 1.51307236903023], (6, 4) = 542, (7, 1) = 0.2675024253e-20, (7, 2) = Vector(2, {(1) = HFloat(-5.134559444286424e-11), (2) = HFloat(-6.217248937900877e-12)}), (7, 3) = [x1 = 1.49739611208694, x2 = -.948322160097376, x3 = -.234063606915019], (7, 4) = 464, (8, 1) = 0.8458109414e-20, (8, 2) = Vector(2, {(1) = HFloat(-8.924061489778978e-11), (2) = HFloat(-2.2231105845094135e-11)}), (8, 3) = [x1 = -.897343263267087, x2 = 1.50092120229124, x3 = .596927439802496], (8, 4) = 409, (9, 1) = 0.1119585997e-19, (9, 2) = Vector(2, {(1) = HFloat(-1.0581047149571532e-10), (2) = HFloat(-6.394884621840902e-14)}), (9, 3) = [x1 = -.502984528327884, x2 = 1.19446857392350, x3 = 1.44436407198760], (9, 4) = 489, (10, 1) = 0.1646898986e-19, (10, 2) = Vector(2, {(1) = HFloat(-1.2816414596272807e-10), (2) = HFloat(6.552980380547524e-12)}), (10, 3) = [x1 = .131278835637824, x2 = .205993893763216, x3 = 1.51308403617309], (10, 4) = 444})

 (2)
> 

NULL

 

Download dragilev1.mw

Another serious application of the Dragilev method can be seen in the recent article http://jap.aip.org/resource/1/japiau/v113/i8/p083103_s1?isAuthorized=no .
numerical-analysis

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