Coefficients of a FOUR-stage tsrk method. The Four Nodes: .8000000000000000000000000000000000000000e-2 .4000000000000000000000000000000000000000 .6500000000000000000000000000000000000000 1. The Weights v: -.8127799435382267625137715943265067214953e-1 .7965686344432509846043680630146795560329 -1.668067448132098056391900698275678696500 .2600000000000000000000000000000000000000 The Weights w: 1.748749571114974324350032193755217180747 -1.331861744728542758099408345713764432976 1.267555648322904848454952613318863731513 .8333333333333333333333333333333333333333e-2 The Rows of Matrix A: -.1447570119911742739965980663060397104312 .6146793356760925300082406504439756054049 -.6435447014583161091437688814022684871291 .1816223777733978531321262972643325921554 -.6413476276208447393893685147331042395560 3.039486325669699982932027487335446878360 -3.606499853719834309385608359388653489646 .9583611556709790658429493867863108508427 -.4551429906477246353785282613110144113400 2.073098705296997351012914167851498591503 -2.309969469995246144890460133399085272894 .2417186004526748752546056798568685593814 -1.058546247221100238151304671694690735111 4.650288193412154230927416487097843999365 -4.758663811764654134122764845886130748662 .8000000000000000000000000000000000000000 The Rows of Matrix B: (lower triangular) 0. 0. 0. 0. .6500000000000000000000000000000000000000 0. 0. 0. .6250000000000000000000000000000000000000 .4752951548932985540014685470017325333494 0. 0. .1428571428571428571428571428571428571429 .8888888888888888888888888888888888888889 .3351758338275683953149069987369457383763 0. The Weights v of an embedded method -.2304838870457205917782498929835601511481 1.812611229621236926283687081230561811330 -3.201508918603364940866643395427129536799 .1000000000000000000000000000000000000000 The Weights w of an embedded method 2.715961394795365216379572730950212390662 -1.237791165410929896815059300408413255935 .9412113466434132867966927766383287418895 .1000000000000000000000000000000000000000 The coefficients of FOUR starting methods each having six stages. Starting method for c[1] = 1/125: Nodes 0. .1250000000000000000000000000000000000000 .3750000000000000000000000000000000000000 .1875000014699737012090742886454303899931 .7500000000000000000000000000000000000000 1. Weights -40997261017.11891262733406661190237695694 0. -130990505063.0832895007860645413788831078 134452175117.0500059725273276858183008600 54495738388.33600082027000622589190453149 -16960147424.18380466467720275842894532673 Rows of the Runge--Kutta Matrix 0. 0. 0. 0. 0. 0. .1250000000000000000000000000000000000000 0. 0. 0. 0. 0. -.1875000000000000000000000000000000000000 .5625000000000000000000000000000000000000 0. 0. 0. 0. -.4687500146997370409017115313237880336938e-1 .2812500033074408277204170816900313720052 -.4687500036749342242117163991222217864272e-1 0. 0. 0. -.1942785004345620360816308500944032835297e-8 1.124999991141503087340151526738890307898 1.125000006864172270240637046482226811393 -1.499999996062890353235168212404808618346 0. 0. 1.796127746098389062049694358442352199467 1.499999980320368048750010555605609539239 4.684848148747397839260472348875565020722 -6.740291311933061304459224352346311760391 -.2406845632330936456009529105772149990346 0. Starting Method for c[2] = 2/5 Nodes 0. .1250000000000000000000000000000000000000 .3750000000000000000000000000000000000000 .1901966296628804797526367937426058822713 .7500000000000000000000000000000000000000 1. Weights -716.2777731037872585099035318076597738925 0. -2252.714259164221813288540969448239592273 2350.153367068674395064719990767890328013 886.2482704803350785808129196295059937949 -266.4096052810004018470884091414969556423 Rows of Runge-Kutta Matrix A 0. 0. 0. 0. 0. 0. .1250000000000000000000000000000000000000 0. 0. 0. 0. 0. -.1875000000000000000000000000000000000000 .5625000000000000000000000000000000000000 0. 0. 0. 0. -.4958104652260040776615477351162686838470e-1 .2873169984079835176839492018823295595203 -.4753932222250263016515763462809680886430e-1 0. 0. 0. -.2354291954174406604127703643012097092780e-2 1.106502716363096489579194761687490743522 1.137939187981192122530286399702739314043 -1.492087612390114205505353457747217960472 0. 0. 1.823745706965917838436039285023302376039 1.459119158610240835283076341381368179584 4.855971948229336534639764698033244181104 -6.878751431280020195224763537953969893378 -.2600853825254750131341167864839448433519 0. Starting Method for c[3] = 13/20 Nodes 0. .1250000000000000000000000000000000000000 .3750000000000000000000000000000000000000 .2156019118630981486537424194081270001391 .7500000000000000000000000000000000000000 1. Weights -48.46610530662485924987008220043352608515 0. -197.6652837036865673731508539482569514743 197.7325906655141791601135171685073828665 69.62336616417214581762502298556991685044 -20.22456781937489835471760400538682215747 Rows of Runge-Kutta Matrix A 0. 0. 0. 0. 0. 0. .1250000000000000000000000000000000000000 0. 0. 0. 0. 0. -.1875000000000000000000000000000000000000 .5625000000000000000000000000000000000000 0. 0. 0. 0. -.7571424079870092808478314267782281551235e-1 .3440058601946523306966757430052667481389 -.5268970753285325395815018091931693248745e-1 0. 0. 0. -.4043259779847150758988407717588880218194e-2 .9270465670928767620634955996891656943925 1.271464578387244166249658829951051693690 -1.444467885700273777554166021922628507866 0. 0. 1.581631422048919882700270053535153783939 1.057058456821476890059278708871191103605 5.351665930786362503459991706139135184397 -6.743586674606121357683858155457645103205 -.2467691350506379185356823130878349687411 0. Starting Method for c[4] = 1 Nodes 0. .1250000000000000000000000000000000000000 .3750000000000000000000000000000000000000 .2868428481403036421228459367954546759117 .7500000000000000000000000000000000000000 1. Weights -11.42716902544568410270968975801101617537 0. -113.4748151258952160470980659744585711117 107.0964551990895310839155064938866829107 25.88930206031173549681514413512015984687 -7.083773108060366430922894896537255470504 Rows of Runge-Kutta Matrix A 0. 0. 0. 0. 0. 0. .1250000000000000000000000000000000000000 0. 0. 0. 0. 0. -.1875000000000000000000000000000000000000 .5625000000000000000000000000000000000000 0. 0. 0. 0. -.1454328408292962163533717437007129835337 .4838558943959171945097967192558353810149 -.5158020542631733603357903875966772156954e-1 0. 0. 0. .9013256168326704287758173025514935512402e-1 .4539666844257586583294212142093008005205 1.876682745261131132397862207880749973440 -1.670781991370156833604865152345200129084 0. 0. 1.384018537489696011175090603453801725672 -.3634387620612690907294653226916162122132e-1 7.969365089925599242745963985154798609761 -8.104831964999387183647633901374270655193 -.2122077862097811612004741549651680590186 0. Coefficients of a New 7-stage Runge-Kutta method of order 6 for computing y_1: Nodes 0. .1250000000000000000000000000000000000000 .5714285714285714285714285714285714285714 .4827586206896551724137931034482758620690 .4285714285714285714285714285714285714286 .7500000000000000000000000000000000000000 1. Weights .9788359788359788359788359788359788359788e-1 0. 2.519567901234567901234567901234567901235 -5.834655800193434602036752574386982989134 4.075771604938271604938271604938271604938 .4485862206292313819195539625647152528873e-1 .9657407407407407407407407407407407407407e-1 Rows of Runge-Kutta Matrix 0. 0. 0. 0. 0. 0. 0. .1250000000000000000000000000000000000000 0. 0. 0. 0. 0. 0. -.7346938775510204081632653061224489795918 1.306122448979591836734693877551020408163 0. 0. 0. 0. 0. -.6754889499364467587846980195989995489585 1.221534298249210709746197056049858542762 -.6328672762310877854770593300258313173450e-1 0. 0. 0. 0. -.5080174927113702623906705539358600582889 .9795918367346938775510204081632653060966 -.1111111111111111111111111111111111111042 .6810819565921606737933268545513443472487e-1 0. 0. 0. -1.186215883725649350649350649350649351498 -.4772727272727272727272727272727272725202 27.78955324731691919191919191919191920237 -86.06054617040945165945165945165945169192 60.68448153409090909090909090909090911353 0. 0. 1.361902479112450349267223667990686207980 -1.395973154362416107382550335570469799088 -.8612975391498881431767337807606264023348 5.850831672786022830868654021470029895620 -4.071588366890380313199105145413870255195 .1161249085042113836225115722842503530552 0.