(*---------------- RKF45ext ----------------*)
c={0,1/4,3/8,12/13,1,1/2,1/3,3/5,4/5};
    
b={16/135,0,6656/12825,28561/56430,-9/50,2/55,0,0,0};
hatb={1/360,0,-128/4275,-2197/75240,1/50,2/55,0,0,0};

tildea={{0,0,0,0,0,0,0,0,0},
{1/4,0,0,0,0,0,0,0,0},
{3/32,9/32,0,0,0,0,0,0,0},
{1932/2197,-7200/2197,7296/2197,0,0,0,0,0,0},
{439/216,-8,3680/513,-845/4104,0,0,0,0,0},
{-8/27,2,-3544/2565,1859/4104,-11/40,0,0,0,0},
{1151813151244382707/45305133264090000000,2000248/3256875,-11508143414648933006/33624903594441796875,1585591870494041752121/9468772852194810000000,-736039189085276843/6292379620012500000,-12530720546283017/865202197751718750,0,0,0},
{-290003176149606751/1668777594545040000,2462810904/2326276339,-639617035530247258/1238545870951396875,12406535109573846761/31706774296355760000,-148962728455253787/695323997727100000,-33/98,910028196/2326276339,0,0},
{-130784667389/106832325600,11104/875,-55799122264/2003106105,-2250055900937/287625492000,109/24,824/183,193/33,2326276339/229864635,0}};

tildeb={9095900507/102618140625,0,-(8611500949504/29246170078125),-(3301514764249/46193950687500),1600439227/38006718750,-(30965859296/41807390625),3667/4004,2326276339/2758375620,1/60};
hattildeb={177633295016020969583/1955579105182614912000,0,1970464956543831633859/2721386938364478759375,342911949222073238078851/1100389319570063698560000,-88744154150740158761/452680348421901600000,-79040589217/107026920000,17312099983595/111772925536272,7769655827/17653603968,5/384};