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tomleslie

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tomleslie 13876
November 22 2018
1 3

solve explicitly, using dsolve() then plot the solution, as in the attached.Periodic behaviour is exhibite when b=3, but not when b=0.5


 

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with(plots):
with(DEtools):
pendsys:={diff(x(t),t)= 1 - x(t) - b*x(t) + (x(t)^2)*y(t),diff(y(t),t)=b*x(t) - (x(t)^2)*y(t)};
animate(phaseportrait,[pendsys,{x(t),y(t)},t=0..40, [[x(0)=0,y(0)=0],[x(0)=2,y(0)=3]],x = 0 .. 5, y = 0 .. 5,linecolor = [sin(t),tan(t)], numpoints = 75, dirgrid = [30, 30]], b=0.5..3);

{diff(x(t), t) = 1-x(t)-b*x(t)+x(t)^2*y(t), diff(y(t), t) = b*x(t)-x(t)^2*y(t)}

  

 
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b:=3:
sol:=dsolve(`union`(pendsys, {x(0)=0, y(0)=0}), numeric);
plots:-odeplot(sol, [[t,x(t)], [t,y(t)]], 0..50, numpoints=5000);

proc (x_rkf45) local _res, _dat, _vars, _solnproc, _xout, _ndsol, _pars, _n, _i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; if 1 < nargs then error "invalid input: too many arguments" end if; _EnvDSNumericSaveDigits := Digits; Digits := 15; if _EnvInFsolve = true then _xout := evalf[_EnvDSNumericSaveDigits](x_rkf45) else _xout := evalf(x_rkf45) end if; _dat := Array(1..4, {(1) = proc (_xin) local _xout, _dtbl, _dat, _vmap, _x0, _y0, _val, _dig, _n, _ne, _nd, _nv, _pars, _ini, _par, _i, _j, _k, _src; option `Copyright (c) 2002 by Waterloo Maple Inc. All rights reserved.`; table( [( "complex" ) = false ] ) _xout := _xin; _pars := []; _dtbl := array( 1 .. 4, [( 1 ) = (array( 1 .. 26, [( 1 ) = (datatype = float[8], order = C_order, storage = rectangular), ( 2 ) = (datatype = float[8], order = C_order, storage = rectangular), ( 3 ) = ([0, 0, 0, Array(1..0, 1..2, {}, datatype = float[8], order = C_order)]), ( 4 ) = (Array(1..63, {(1) = 2, (2) = 2, (3) = 0, (4) = 0, (5) = 0, (6) = 0, (7) = 1, (8) = 0, (9) = 0, (10) = 0, (11) = 0, (12) = 0, (13) = 0, (14) = 0, (15) = 0, (16) = 0, (17) = 0, (18) = 1, (19) = 30000, (20) = 0, (21) = 0, (22) = 1, (23) = 4, (24) = 0, (25) = 1, (26) = 15, (27) = 1, (28) = 0, (29) = 1, (30) = 3, (31) = 3, (32) = 0, (33) = 1, (34) = 0, (35) = 0, (36) = 0, (37) = 0, (38) = 0, (39) = 0, (40) = 0, (41) = 0, (42) = 0, (43) = 1, (44) = 0, (45) = 0, (46) = 0, (47) = 0, (48) = 0, (49) = 0, (50) = 50, (51) = 1, (52) = 0, (53) = 0, (54) = 0, (55) = 0, (56) = 0, (57) = 0, (58) = 0, (59) = 10000, (60) = 0, (61) = 1000, (62) = 0, (63) = 0}, datatype = integer[8])), ( 5 ) = (Array(1..28, {(1) = .0, (2) = 0.10e-5, (3) = .0, (4) = 0.500001e-14, (5) = .0, (6) = 0.5047658755841546e-2, (7) = .0, (8) = 0.10e-5, (9) = .0, (10) = .0, (11) = .0, (12) = .0, (13) = 1.0, (14) = .0, (15) = .49999999999999, (16) = .0, (17) = 1.0, (18) = 1.0, (19) = .0, (20) = .0, (21) = 1.0, (22) = 1.0, (23) = .0, (24) = .0, (25) = 0.10e-14, (26) = .0, (27) = .0, (28) = .0}, datatype = float[8], order = C_order)), ( 6 ) = (Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order)), ( 7 ) = ([Array(1..4, 1..7, {(1, 1) = .0, (1, 2) = .203125, (1, 3) = .3046875, (1, 4) = .75, (1, 5) = .8125, (1, 6) = .40625, (1, 7) = .8125, (2, 1) = 0.6378173828125e-1, (2, 2) = .0, (2, 3) = .279296875, (2, 4) = .27237892150878906, (2, 5) = -0.9686851501464844e-1, (2, 6) = 0.1956939697265625e-1, (2, 7) = .5381584167480469, (3, 1) = 0.31890869140625e-1, (3, 2) = .0, (3, 3) = -.34375, (3, 4) = -.335235595703125, (3, 5) = .2296142578125, (3, 6) = .41748046875, (3, 7) = 11.480712890625, (4, 1) = 0.9710520505905151e-1, (4, 2) = .0, (4, 3) = .40350341796875, (4, 4) = 0.20297467708587646e-1, (4, 5) = -0.6054282188415527e-2, (4, 6) = -0.4770040512084961e-1, (4, 7) = .77858567237854}, datatype = float[8], order = C_order), Array(1..6, 1..6, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (1, 6) = 1.0, (2, 1) = .25, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (2, 6) = 1.0, (3, 1) = .1875, (3, 2) = .5625, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (3, 6) = 2.0, (4, 1) = .23583984375, (4, 2) = -.87890625, (4, 3) = .890625, (4, 4) = .0, (4, 5) = .0, (4, 6) = .2681884765625, (5, 1) = .1272735595703125, (5, 2) = -.5009765625, (5, 3) = .44921875, (5, 4) = -0.128936767578125e-1, (5, 5) = .0, (5, 6) = 0.626220703125e-1, (6, 1) = -0.927734375e-1, (6, 2) = .626220703125, (6, 3) = -.4326171875, (6, 4) = .1418304443359375, (6, 5) = -0.861053466796875e-1, (6, 6) = .3131103515625}, datatype = float[8], order = C_order), Array(1..6, {(1) = .0, (2) = .386, (3) = .21, (4) = .63, (5) = 1.0, (6) = 1.0}, datatype = float[8], order = C_order), Array(1..6, {(1) = .25, (2) = -.1043, (3) = .1035, (4) = -0.362e-1, (5) = .0, (6) = .0}, datatype = float[8], order = C_order), Array(1..6, 1..5, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (2, 1) = 1.544, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (3, 1) = .9466785280815533, (3, 2) = .25570116989825814, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (4, 1) = 3.3148251870684886, (4, 2) = 2.896124015972123, (4, 3) = .9986419139977808, (4, 4) = .0, (4, 5) = .0, (5, 1) = 1.2212245092262748, (5, 2) = 6.019134481287752, (5, 3) = 12.537083329320874, (5, 4) = -.687886036105895, (5, 5) = .0, (6, 1) = 1.2212245092262748, (6, 2) = 6.019134481287752, (6, 3) = 12.537083329320874, (6, 4) = -.687886036105895, (6, 5) = 1.0}, datatype = float[8], order = C_order), Array(1..6, 1..5, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (2, 1) = -5.6688, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (3, 1) = -2.4300933568337584, (3, 2) = -.20635991570891224, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (4, 1) = -.10735290581452621, (4, 2) = -9.594562251021896, (4, 3) = -20.470286148096154, (4, 4) = .0, (4, 5) = .0, (5, 1) = 7.496443313968615, (5, 2) = -10.246804314641219, (5, 3) = -33.99990352819906, (5, 4) = 11.708908932061595, (5, 5) = .0, (6, 1) = 8.083246795922411, (6, 2) = -7.981132988062785, (6, 3) = -31.52159432874373, (6, 4) = 16.319305431231363, (6, 5) = -6.0588182388340535}, datatype = float[8], order = C_order), Array(1..3, 1..5, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (2, 1) = 10.126235083446911, (2, 2) = -7.487995877607633, (2, 3) = -34.800918615557414, (2, 4) = -7.9927717075687275, (2, 5) = 1.0251377232956207, (3, 1) = -.6762803392806898, (3, 2) = 6.087714651678606, (3, 3) = 16.43084320892463, (3, 4) = 24.767225114183653, (3, 5) = -6.5943891257167815}, datatype = float[8], order = C_order)]), ( 9 ) = ([Array(1..2, {(1) = .1, (2) = .1}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, 1..2, {(1, 1) = .0, (1, 2) = .0, (2, 1) = .0, (2, 2) = .0}, datatype = float[8], order = C_order), Array(1..2, 1..2, {(1, 1) = .0, (1, 2) = .0, (2, 1) = .0, (2, 2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, 1..2, {(1, 1) = .0, (1, 2) = .0, (2, 1) = .0, (2, 2) = .0}, datatype = float[8], order = C_order), Array(1..2, 1..6, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (1, 6) = .0, (2, 1) = .0, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (2, 6) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = 0, (2) = 0}, datatype = integer[8]), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..4, {(1) = .0, (2) = .0, (3) = .0, (4) = .0}, datatype = float[8], order = C_order)]), ( 8 ) = ([Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = 1.0, (2) = .0}, datatype = float[8], order = C_order), 0, 0]), ( 11 ) = (Array(1..6, 0..2, {(1, 1) = .0, (1, 2) = .0, (2, 0) = .0, (2, 1) = .0, (2, 2) = .0, (3, 0) = .0, (3, 1) = .0, (3, 2) = .0, (4, 0) = .0, (4, 1) = .0, (4, 2) = .0, (5, 0) = .0, (5, 1) = .0, (5, 2) = .0, (6, 0) = .0, (6, 1) = .0, (6, 2) = .0}, datatype = float[8], order = C_order)), ( 10 ) = ([proc (N, X, Y, YP) option `[Y[1] = x(t), Y[2] = y(t)]`; YP[1] := Y[1]^2*Y[2]-4*Y[1]+1; YP[2] := -Y[1]^2*Y[2]+3*Y[1]; 0 end proc, -1, 0, 0, 0, 0, 0, 0]), ( 13 ) = (), ( 12 ) = (), ( 15 ) = ("rkf45"), ( 14 ) = ([0, 0]), ( 18 ) = ([]), ( 19 ) = (0), ( 16 ) = ([0, 0, 0, []]), ( 17 ) = ([proc (N, X, Y, YP) option `[Y[1] = x(t), Y[2] = y(t)]`; YP[1] := Y[1]^2*Y[2]-4*Y[1]+1; YP[2] := -Y[1]^2*Y[2]+3*Y[1]; 0 end proc, -1, 0, 0, 0, 0, 0, 0]), ( 22 ) = (0), ( 23 ) = (0), ( 20 ) = ([]), ( 21 ) = (0), ( 26 ) = (Array(1..0, {})), ( 25 ) = (Array(1..0, {})), ( 24 ) = (0) ] )) ] ); _y0 := Array(0..2, {(1) = 0., (2) = 0.}); _vmap := array( 1 .. 2, [( 1 ) = (1), ( 2 ) = (2) ] ); _x0 := _dtbl[1][5][5]; _n := _dtbl[1][4][1]; _ne := _dtbl[1][4][3]; _nd := _dtbl[1][4][4]; _nv := _dtbl[1][4][16]; if not type(_xout, 'numeric') then if member(_xout, ["start", "left", "right"]) then if _Env_smart_dsolve_numeric = true or _dtbl[1][4][10] = 1 then if _xout = "left" then if type(_dtbl[2], 'table') then return _dtbl[2][5][1] end if elif _xout = "right" then if type(_dtbl[3], 'table') then return _dtbl[3][5][1] end if end if end if; return _dtbl[1][5][5] elif _xout = "method" then return _dtbl[1][15] elif _xout = "storage" then return evalb(_dtbl[1][4][10] = 1) elif _xout = "leftdata" then if not type(_dtbl[2], 'array') then return NULL else return eval(_dtbl[2]) end if elif _xout = "rightdata" then if not type(_dtbl[3], 'array') then return NULL else return eval(_dtbl[3]) end if elif _xout = "enginedata" then return eval(_dtbl[1]) elif _xout = "enginereset" then _dtbl[2] := evaln(_dtbl[2]); _dtbl[3] := evaln(_dtbl[3]); return NULL elif _xout = "initial" then return procname(_y0[0]) elif _xout = "laxtol" then return _dtbl[`if`(member(_dtbl[4], {2, 3}), _dtbl[4], 1)][5][18] elif _xout = "numfun" then return `if`(member(_dtbl[4], {2, 3}), _dtbl[_dtbl[4]][4][18], 0) elif _xout = "parameters" then return [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] elif _xout = "initial_and_parameters" then return procname(_y0[0]), [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] elif _xout = "last" then if _dtbl[4] <> 2 and _dtbl[4] <> 3 or _x0-_dtbl[_dtbl[4]][5][1] = 0. then error "no information is available on last computed point" else _xout := _dtbl[_dtbl[4]][5][1] end if elif _xout = "function" then if _dtbl[1][4][33]-2. = 0 then return eval(_dtbl[1][10], 1) else return eval(_dtbl[1][10][1], 1) end if elif _xout = "map" then return copy(_vmap) elif type(_xin, `=`) and type(rhs(_xin), 'list') and member(lhs(_xin), {"initial", "parameters", "initial_and_parameters"}) then _ini, _par := [], []; if lhs(_xin) = "initial" then _ini := rhs(_xin) elif lhs(_xin) = "parameters" then _par := rhs(_xin) elif select(type, rhs(_xin), `=`) <> [] then _par, _ini := selectremove(type, rhs(_xin), `=`) elif nops(rhs(_xin)) < nops(_pars)+1 then error "insufficient data for specification of initial and parameters" else _par := rhs(_xin)[-nops(_pars) .. -1]; _ini := rhs(_xin)[1 .. -nops(_pars)-1] end if; _xout := lhs(_xout); if _par <> [] then `dsolve/numeric/process_parameters`(_n, _pars, _par, _y0) end if; if _ini <> [] then `dsolve/numeric/process_initial`(_n-_ne, _ini, _y0, _pars, _vmap) end if; `dsolve/numeric/SC/reinitialize`(_dtbl, _y0, _n, procname, _pars); if _Env_smart_dsolve_numeric = true and type(_y0[0], 'numeric') and _dtbl[1][4][10] <> 1 then procname("right") := _y0[0]; procname("left") := _y0[0] end if; if _xout = "initial" then return [_y0[0], seq(_y0[_vmap[_i]], _i = 1 .. _n-_ne)] elif _xout = "parameters" then return [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] else return [_y0[0], seq(_y0[_vmap[_i]], _i = 1 .. _n-_ne)], [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] end if elif _xin = "eventstop" then if _nv = 0 then error "this solution has no events" end if; _i := _dtbl[4]; if _i <> 2 and _i <> 3 then return 0 end if; if _dtbl[_i][4][10] = 1 and assigned(_dtbl[5-_i]) and _dtbl[_i][4][9] < 100 and 100 <= _dtbl[5-_i][4][9] then _i := 5-_i; _dtbl[4] := _i; _j := round(_dtbl[_i][4][17]); return round(_dtbl[_i][3][1][_j, 1]) elif 100 <= _dtbl[_i][4][9] then _j := round(_dtbl[_i][4][17]); return round(_dtbl[_i][3][1][_j, 1]) else return 0 end if elif _xin = "eventstatus" then if _nv = 0 then error "this solution has no events" end if; _i := [selectremove(proc (a) options operator, arrow; _dtbl[1][3][1][a, 7] = 1 end proc, {seq(_j, _j = 1 .. round(_dtbl[1][3][1][_nv+1, 1]))})]; return ':-enabled' = _i[1], ':-disabled' = _i[2] elif _xin = "eventclear" then if _nv = 0 then error "this solution has no events" end if; _i := _dtbl[4]; if _i <> 2 and _i <> 3 then error "no events to clear" end if; if _dtbl[_i][4][10] = 1 and assigned(_dtbl[5-_i]) and _dtbl[_i][4][9] < 100 and 100 < _dtbl[5-_i][4][9] then _dtbl[4] := 5-_i; _i := 5-_i end if; if _dtbl[_i][4][9] < 100 then error "no events to clear" elif _nv < _dtbl[_i][4][9]-100 then error "event error condition cannot be cleared" else _j := _dtbl[_i][4][9]-100; if irem(round(_dtbl[_i][3][1][_j, 4]), 2) = 1 then error "retriggerable events cannot be cleared" end if; _j := round(_dtbl[_i][3][1][_j, 1]); for _k to _nv do if _dtbl[_i][3][1][_k, 1] = _j then if _dtbl[_i][3][1][_k, 2] = 3 then error "range events cannot be cleared" end if; _dtbl[_i][3][1][_k, 8] := _dtbl[_i][3][1][_nv+1, 8] end if end do; _dtbl[_i][4][17] := 0; _dtbl[_i][4][9] := 0; if _dtbl[1][4][10] = 1 then if _i = 2 then try procname(procname("left")) catch: end try else try procname(procname("right")) catch: end try end if end if end if; return elif type(_xin, `=`) and member(lhs(_xin), {"eventdisable", "eventenable"}) then if _nv = 0 then error "this solution has no events" end if; if type(rhs(_xin), {('list')('posint'), ('set')('posint')}) then _i := {op(rhs(_xin))} elif type(rhs(_xin), 'posint') then _i := {rhs(_xin)} else error "event identifiers must be integers in the range 1..%1", round(_dtbl[1][3][1][_nv+1, 1]) end if; if select(proc (a) options operator, arrow; _nv < a end proc, _i) <> {} then error "event identifiers must be integers in the range 1..%1", round(_dtbl[1][3][1][_nv+1, 1]) end if; _k := {}; for _j to _nv do if member(round(_dtbl[1][3][1][_j, 1]), _i) then _k := `union`(_k, {_j}) end if end do; _i := _k; if lhs(_xin) = "eventdisable" then _dtbl[4] := 0; _j := [evalb(assigned(_dtbl[2]) and member(_dtbl[2][4][17], _i)), evalb(assigned(_dtbl[3]) and member(_dtbl[3][4][17], _i))]; for _k in _i do _dtbl[1][3][1][_k, 7] := 0; if assigned(_dtbl[2]) then _dtbl[2][3][1][_k, 7] := 0 end if; if assigned(_dtbl[3]) then _dtbl[3][3][1][_k, 7] := 0 end if end do; if _j[1] then for _k to _nv+1 do if _k <= _nv and not type(_dtbl[2][3][4][_k, 1], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to defined init `, _dtbl[2][3][4][_k, 1]); _dtbl[2][3][1][_k, 8] := _dtbl[2][3][4][_k, 1] elif _dtbl[2][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[2][3][1][_k, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to rate hysteresis init `, _dtbl[2][5][24]); _dtbl[2][3][1][_k, 8] := _dtbl[2][5][24] elif _dtbl[2][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[2][3][1][_k, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to initial init `, _x0); _dtbl[2][3][1][_k, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to fireinitial init `, _x0-1); _dtbl[2][3][1][_k, 8] := _x0-1 end if end do; _dtbl[2][4][17] := 0; _dtbl[2][4][9] := 0; if _dtbl[1][4][10] = 1 then procname(procname("left")) end if end if; if _j[2] then for _k to _nv+1 do if _k <= _nv and not type(_dtbl[3][3][4][_k, 2], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to defined init `, _dtbl[3][3][4][_k, 2]); _dtbl[3][3][1][_k, 8] := _dtbl[3][3][4][_k, 2] elif _dtbl[3][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[3][3][1][_k, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to rate hysteresis init `, _dtbl[3][5][24]); _dtbl[3][3][1][_k, 8] := _dtbl[3][5][24] elif _dtbl[3][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[3][3][1][_k, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to initial init `, _x0); _dtbl[3][3][1][_k, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to fireinitial init `, _x0+1); _dtbl[3][3][1][_k, 8] := _x0+1 end if end do; _dtbl[3][4][17] := 0; _dtbl[3][4][9] := 0; if _dtbl[1][4][10] = 1 then procname(procname("right")) end if end if else for _k in _i do _dtbl[1][3][1][_k, 7] := 1 end do; _dtbl[2] := evaln(_dtbl[2]); _dtbl[3] := evaln(_dtbl[3]); _dtbl[4] := 0; if _dtbl[1][4][10] = 1 then if _x0 <= procname("right") then try procname(procname("right")) catch: end try end if; if procname("left") <= _x0 then try procname(procname("left")) catch: end try end if end if end if; return elif type(_xin, `=`) and lhs(_xin) = "eventfired" then if not type(rhs(_xin), 'list') then error "'eventfired' must be specified as a list" end if; if _nv = 0 then error "this solution has no events" end if; if _dtbl[4] <> 2 and _dtbl[4] <> 3 then error "'direction' must be set prior to calling/setting 'eventfired'" end if; _i := _dtbl[4]; _val := NULL; if not assigned(_EnvEventRetriggerWarned) then _EnvEventRetriggerWarned := false end if; for _k in rhs(_xin) do if type(_k, 'integer') then _src := _k elif type(_k, 'integer' = 'anything') and type(evalf(rhs(_k)), 'numeric') then _k := lhs(_k) = evalf[max(Digits, 18)](rhs(_k)); _src := lhs(_k) else error "'eventfired' entry is not valid: %1", _k end if; if _src < 1 or round(_dtbl[1][3][1][_nv+1, 1]) < _src then error "event identifiers must be integers in the range 1..%1", round(_dtbl[1][3][1][_nv+1, 1]) end if; _src := {seq(`if`(_dtbl[1][3][1][_j, 1]-_src = 0., _j, NULL), _j = 1 .. _nv)}; if nops(_src) <> 1 then error "'eventfired' can only be set/queried for root-finding events and time/interval events" end if; _src := _src[1]; if _dtbl[1][3][1][_src, 2] <> 0. and _dtbl[1][3][1][_src, 2]-2. <> 0. then error "'eventfired' can only be set/queried for root-finding events and time/interval events" elif irem(round(_dtbl[1][3][1][_src, 4]), 2) = 1 then if _EnvEventRetriggerWarned = false then WARNING(`'eventfired' has no effect on events that retrigger`) end if; _EnvEventRetriggerWarned := true end if; if _dtbl[_i][3][1][_src, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_src, 4]), 32), 2) = 1 then _val := _val, undefined elif type(_dtbl[_i][3][4][_src, _i-1], 'undefined') or _i = 2 and _dtbl[2][3][1][_src, 8] < _dtbl[2][3][4][_src, 1] or _i = 3 and _dtbl[3][3][4][_src, 2] < _dtbl[3][3][1][_src, 8] then _val := _val, _dtbl[_i][3][1][_src, 8] else _val := _val, _dtbl[_i][3][4][_src, _i-1] end if; if type(_k, `=`) then if _dtbl[_i][3][1][_src, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_src, 4]), 32), 2) = 1 then error "cannot set event code for a rate hysteresis event" end if; userinfo(3, {'events', 'eventreset'}, `manual set event code `, _src, ` to value `, rhs(_k)); _dtbl[_i][3][1][_src, 8] := rhs(_k); _dtbl[_i][3][4][_src, _i-1] := rhs(_k) end if end do; return [_val] elif type(_xin, `=`) and lhs(_xin) = "direction" then if not member(rhs(_xin), {-1, 1, ':-left', ':-right'}) then error "'direction' must be specified as either '1' or 'right' (positive) or '-1' or 'left' (negative)" end if; _src := `if`(_dtbl[4] = 2, -1, `if`(_dtbl[4] = 3, 1, undefined)); _i := `if`(member(rhs(_xin), {1, ':-right'}), 3, 2); _dtbl[4] := _i; _dtbl[_i] := `dsolve/numeric/SC/IVPdcopy`(_dtbl[1], `if`(assigned(_dtbl[_i]), _dtbl[_i], NULL)); if 0 < _nv then for _j to _nv+1 do if _j <= _nv and not type(_dtbl[_i][3][4][_j, _i-1], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to defined init `, _dtbl[_i][3][4][_j, _i-1]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][3][4][_j, _i-1] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to rate hysteresis init `, _dtbl[_i][5][24]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][5][24] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to initial init `, _x0); _dtbl[_i][3][1][_j, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to fireinitial init `, _x0-2*_i+5.0); _dtbl[_i][3][1][_j, 8] := _x0-2*_i+5.0 end if end do end if; return _src elif _xin = "eventcount" then if _dtbl[1][3][1] = 0 or _dtbl[4] <> 2 and _dtbl[4] <> 3 then return 0 else return round(_dtbl[_dtbl[4]][3][1][_nv+1, 12]) end if else return "procname" end if end if; if _xout = _x0 then return [_x0, seq(evalf(_dtbl[1][6][_vmap[_i]]), _i = 1 .. _n-_ne)] end if; _i := `if`(_x0 <= _xout, 3, 2); if _xin = "last" and 0 < _dtbl[_i][4][9] and _dtbl[_i][4][9] < 100 then _dat := eval(_dtbl[_i], 2); _j := _dat[4][20]; return [_dat[11][_j, 0], seq(_dat[11][_j, _vmap[_i]], _i = 1 .. _n-_ne-_nd), seq(_dat[8][1][_vmap[_i]], _i = _n-_ne-_nd+1 .. _n-_ne)] end if; if not type(_dtbl[_i], 'array') then _dtbl[_i] := `dsolve/numeric/SC/IVPdcopy`(_dtbl[1], `if`(assigned(_dtbl[_i]), _dtbl[_i], NULL)); if 0 < _nv then for _j to _nv+1 do if _j <= _nv and not type(_dtbl[_i][3][4][_j, _i-1], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to defined init `, _dtbl[_i][3][4][_j, _i-1]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][3][4][_j, _i-1] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to rate hysteresis init `, _dtbl[_i][5][24]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][5][24] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to initial init `, _x0); _dtbl[_i][3][1][_j, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to fireinitial init `, _x0-2*_i+5.0); _dtbl[_i][3][1][_j, 8] := _x0-2*_i+5.0 end if end do end if end if; if _xin <> "last" then if 0 < 0 then if `dsolve/numeric/checkglobals`(op(_dtbl[1][14]), _pars, _n, _y0) then `dsolve/numeric/SC/reinitialize`(_dtbl, _y0, _n, procname, _pars, _i) end if end if; if _dtbl[1][4][7] = 0 then error "parameters must be initialized before solution can be computed" end if end if; _dat := eval(_dtbl[_i], 2); _dtbl[4] := _i; try _src := `dsolve/numeric/SC/IVPrun`(_dat, _xout) catch: userinfo(2, `dsolve/debug`, print(`Exception in solnproc:`, [lastexception][2 .. -1])); error end try; if _dat[17] <> _dtbl[1][17] then _dtbl[1][17] := _dat[17]; _dtbl[1][10] := _dat[10] end if; if _src = 0 and 100 < _dat[4][9] then _val := _dat[3][1][_nv+1, 8] else _val := _dat[11][_dat[4][20], 0] end if; if _src <> 0 or _dat[4][9] <= 0 then _dtbl[1][5][1] := _xout else _dtbl[1][5][1] := _val end if; if _i = 3 and _val < _xout then Rounding := -infinity; if _dat[4][9] = 1 then error "cannot evaluate the solution further right of %1, probably a singularity", evalf[8](_val) elif _dat[4][9] = 2 then error "cannot evaluate the solution further right of %1, maxfun limit exceeded (see ?dsolve,maxfun for details)", evalf[8](_val) elif _dat[4][9] = 3 then if _dat[4][25] = 3 then error "cannot evaluate the solution past the initial point, problem may be initially singular or improperly set up" else error "cannot evaluate the solution past the initial point, problem may be complex, initially singular or improperly set up" end if elif _dat[4][9] = 4 then error "cannot evaluate the solution further right of %1, accuracy goal cannot be achieved with specified 'minstep'", evalf[8](_val) elif _dat[4][9] = 5 then error "cannot evaluate the solution further right of %1, too many step failures, tolerances may be too loose for problem", evalf[8](_val) elif _dat[4][9] = 6 then error "cannot evaluate the solution further right of %1, cannot downgrade delay storage for problems with delay derivative order > 1, try increasing delaypts", evalf[8](_val) elif _dat[4][9] = 10 then error "cannot evaluate the solution further right of %1, interrupt requested", evalf[8](_val) elif 100 < _dat[4][9] then if _dat[4][9]-100 = _nv+1 then error "constraint projection failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-100 = _nv+2 then error "index-1 and derivative evaluation failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-100 = _nv+3 then error "maximum number of event iterations reached (%1) at t=%2", round(_dat[3][1][_nv+1, 3]), evalf[8](_val) else if _Env_dsolve_nowarnstop <> true then `dsolve/numeric/warning`(StringTools:-FormatMessage("cannot evaluate the solution further right of %1, event #%2 triggered a halt", evalf[8](_val), round(_dat[3][1][_dat[4][9]-100, 1]))) end if; Rounding := 'nearest'; _xout := _val end if else error "cannot evaluate the solution further right of %1", evalf[8](_val) end if elif _i = 2 and _xout < _val then Rounding := infinity; if _dat[4][9] = 1 then error "cannot evaluate the solution further left of %1, probably a singularity", evalf[8](_val) elif _dat[4][9] = 2 then error "cannot evaluate the solution further left of %1, maxfun limit exceeded (see ?dsolve,maxfun for details)", evalf[8](_val) elif _dat[4][9] = 3 then if _dat[4][25] = 3 then error "cannot evaluate the solution past the initial point, problem may be initially singular or improperly set up" else error "cannot evaluate the solution past the initial point, problem may be complex, initially singular or improperly set up" end if elif _dat[4][9] = 4 then error "cannot evaluate the solution further left of %1, accuracy goal cannot be achieved with specified 'minstep'", evalf[8](_val) elif _dat[4][9] = 5 then error "cannot evaluate the solution further left of %1, too many step failures, tolerances may be too loose for problem", evalf[8](_val) elif _dat[4][9] = 6 then error "cannot evaluate the solution further left of %1, cannot downgrade delay storage for problems with delay derivative order > 1, try increasing delaypts", evalf[8](_val) elif _dat[4][9] = 10 then error "cannot evaluate the solution further right of %1, interrupt requested", evalf[8](_val) elif 100 < _dat[4][9] then if _dat[4][9]-100 = _nv+1 then error "constraint projection failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-100 = _nv+2 then error "index-1 and derivative evaluation failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-100 = _nv+3 then error "maximum number of event iterations reached (%1) at t=%2", round(_dat[3][1][_nv+1, 3]), evalf[8](_val) else if _Env_dsolve_nowarnstop <> true then `dsolve/numeric/warning`(StringTools:-FormatMessage("cannot evaluate the solution further left of %1, event #%2 triggered a halt", evalf[8](_val), round(_dat[3][1][_dat[4][9]-100, 1]))) end if; Rounding := 'nearest'; _xout := _val end if else error "cannot evaluate the solution further left of %1", evalf[8](_val) end if end if; if _EnvInFsolve = true then _dig := _dat[4][26]; if type(_EnvDSNumericSaveDigits, 'posint') then _dat[4][26] := _EnvDSNumericSaveDigits else _dat[4][26] := Digits end if; _Env_dsolve_SC_native := true; if _dat[4][25] = 1 then _i := 1; _dat[4][25] := 2 else _i := _dat[4][25] end if; _val := `dsolve/numeric/SC/IVPval`(_dat, _xout, _src); _dat[4][25] := _i; _dat[4][26] := _dig; [_xout, seq(_val[_vmap[_i]], _i = 1 .. _n-_ne)] else Digits := _dat[4][26]; _val := `dsolve/numeric/SC/IVPval`(eval(_dat, 2), _xout, _src); [_xout, seq(_val[_vmap[_i]], _i = 1 .. _n-_ne)] end if end proc, (2) = Array(0..0, {}), (3) = [t, x(t), y(t)], (4) = []}); _vars := _dat[3]; _pars := map(rhs, _dat[4]); _n := nops(_vars)-1; _solnproc := _dat[1]; if not type(_xout, 'numeric') then if member(x_rkf45, ["start", 'start', "method", 'method', "left", 'left', "right", 'right', "leftdata", "rightdata", "enginedata", "eventstop", 'eventstop', "eventclear", 'eventclear', "eventstatus", 'eventstatus', "eventcount", 'eventcount', "laxtol", 'laxtol', "numfun", 'numfun', NULL]) then _res := _solnproc(convert(x_rkf45, 'string')); if 1 < nops([_res]) then return _res elif type(_res, 'array') then return eval(_res, 1) elif _res <> "procname" then return _res end if elif member(x_rkf45, ["last", 'last', "initial", 'initial', "parameters", 'parameters', "initial_and_parameters", 'initial_and_parameters', NULL]) then _xout := convert(x_rkf45, 'string'); _res := _solnproc(_xout); if _xout = "parameters" then return [seq(_pars[_i] = _res[_i], _i = 1 .. nops(_pars))] elif _xout = "initial_and_parameters" then return [seq(_vars[_i+1] = [_res][1][_i+1], _i = 0 .. _n), seq(_pars[_i] = [_res][2][_i], _i = 1 .. nops(_pars))] else return [seq(_vars[_i+1] = _res[_i+1], _i = 0 .. _n)] end if elif type(_xout, `=`) and member(lhs(_xout), ["initial", 'initial', "parameters", 'parameters', "initial_and_parameters", 'initial_and_parameters', NULL]) then _xout := convert(lhs(x_rkf45), 'string') = rhs(x_rkf45); if type(rhs(_xout), 'list') then _res := _solnproc(_xout) else error "initial and/or parameter values must be specified in a list" end if; if lhs(_xout) = "initial" then return [seq(_vars[_i+1] = _res[_i+1], _i = 0 .. _n)] elif lhs(_xout) = "parameters" then return [seq(_pars[_i] = _res[_i], _i = 1 .. nops(_pars))] else return [seq(_vars[_i+1] = [_res][1][_i+1], _i = 0 .. _n), seq(_pars[_i] = [_res][2][_i], _i = 1 .. nops(_pars))] end if elif type(_xout, `=`) and member(lhs(_xout), ["eventdisable", 'eventdisable', "eventenable", 'eventenable', "eventfired", 'eventfired', "direction", 'direction', NULL]) then return _solnproc(convert(lhs(x_rkf45), 'string') = rhs(x_rkf45)) elif _xout = "solnprocedure" then return eval(_solnproc) elif _xout = "sysvars" then return _vars end if; if procname <> unknown then return ('procname')(x_rkf45) else _ndsol := 1; _ndsol := _ndsol; _ndsol := pointto(_dat[2][0]); return ('_ndsol')(x_rkf45) end if end if; try _res := _solnproc(_xout); [seq(_vars[_i+1] = _res[_i+1], _i = 0 .. _n)] catch: error end try end proc

  

 
> 

 


 

Download ODEperiod.mw

 

Define the plots...

tomleslie 13876
November 22 2018
1 2

as entries in an Array(), then use the plots:-display() command. As shown in the following

ArrPlot.mw

Serious! numerical problems...

tomleslie 13876
November 18 2018
0 0

In the attached, I have used the 'parfrac' approach as well as your solution process: both will a Digits setting of 50 (display precision is restricted to 10 just to keep the output readable).

The parfrac approach produces an answer, but your method does not, because you generate nine equations in eight unknowns. Toggling the Digits setting at the top of the worksheet to 10 (ie the default), then both the 'parfrac' approach and your solution method will produce 'answers' - because the latter now produces eight equations in eight unknowns.

The discrepancy in you solution method is with the coefficient of s^8. With Digits=50, one has A+C=0 (whihc is the ninth equation), whereas with Digits=10, your solution process generates 0=0, which the solve() command will ignore

Check the output of the last four commands in the attached whilst toggling the Digits setting at the top of the worksheet between 10 and 50


 

> 

  restart;
> 

#
# Perform all subsequent calculations to 50 digits
# but restrict the display to 10 digits (just because
# *displaying* 50 digits everywhere gets a bit lengthy
#
  Digits := 50:
  interface(displayprecision = 10):
  EQ1 := Psi2(s) = -Fnk*(92.28065040*s^5+669248.354777463*s^4+1.64532969786998*10^8*s^3+1.08423033340076*10^10*s^2+1.24776819370483*10^11*s+6.63421455605816*10^11)/(-6.17394404982986*10^15+s^8+308.467148218142*s^7+406813.735483657*s^6+9.51175570810348*10^7*s^5+8.24976363159764*10^7*s^4-1.45856901564772*10^12*s^3-1.00514781049853*10^14*s^2-1.16120016128822*10^15*s);
  EQ2 := EQ1/Fnk;
#
# So now generate the partial fractions to 50 digits
# of precision, while restricting the display to 10
# digits
#
  convert(EQ2, parfrac, s);

Psi2(s) = -Fnk*(92.28065040*s^5+669248.354777463*s^4+164532969.78699800000000*s^3+10842303334.00760000000000*s^2+124776819370.48300000000000*s+663421455605.81600000000000)/(-6173944049829860.00000000000000+s^8+308.467148218142*s^7+406813.735483657*s^6+95117557.08103480000000*s^5+82497636.31597640000000*s^4-1458569015647.72000000000000*s^3-100514781049853.00000000000000*s^2-1161200161288220.00000000000000*s)

  

Psi2(s)/Fnk = -(92.28065040*s^5+669248.354777463*s^4+164532969.78699800000000*s^3+10842303334.00760000000000*s^2+124776819370.48300000000000*s+663421455605.81600000000000)/(-6173944049829860.00000000000000+s^8+308.467148218142*s^7+406813.735483657*s^6+95117557.08103480000000*s^5+82497636.31597640000000*s^4-1458569015647.72000000000000*s^3-100514781049853.00000000000000*s^2-1161200161288220.00000000000000*s)

  

Psi2(s)/Fnk = 0.64042295579977213788046916059066589107062170760700e-2/(s+123.95936359750265297185596051550647036659011563743)+(0.7643159897602552756027104517654469081066598585e-5*s+1.5890423004588031903113205817619392444069997838271)/(s^2+56.142264645708354225809475665278814478118570596152*s+391252.79097567023079015071909098045312869106244420)+0.21385070248775650918926525929539038300482000000000e-10/(s+92.660187384035638553231027927412989704630476880467)-0.64118727739976523124277853283336615030487620393305e-2/(s-123.94958488594237203382816480277908306679338388130)+(-0.37113307487564011080664244084110529526709533182466e-12*s-0.53763095780522081670273019775061001527205e-11)/(s^2+12.861160988207490696337262517730984452695598400261*s+75.505757318432963550499582955755873806102180182642)+0.35088391206967055809789464634563545359460595331825e-10/(s+146.79375648863023558659443817684982406475862236699)

 (1)
> 

Denom1 := factor(rhs(EQ2));
Denom2 := denom(Denom1)

-92.280650400000000000000000000000000000000000000000*(s+7000.0000000000027316207043682331242314501606180021)*(s+146.79375634986232257346558402415105608383294099801)*(s+92.660187473798661650246473701129380609422508207435)*(s^2+12.861160984732823155269688087449521024717494191815*s+75.505757268373322849603714155278816756606438453469)/((s+146.79375648863023558659443817684982406475862236699)*(s+123.95936359750265297185596051550647036659011563743)*(s+92.660187384035638553231027927412989704630476880467)*(s-123.94958488594237203382816480277908306679338388130)*(s^2+56.142264645708354225809475665278814478118570596152*s+391252.79097567023079015071909098045312869106244420)*(s^2+12.861160988207490696337262517730984452695598400261*s+75.505757318432963550499582955755873806102180182642))

  

(s+146.79375648863023558659443817684982406475862236699)*(s+123.95936359750265297185596051550647036659011563743)*(s+92.660187384035638553231027927412989704630476880467)*(s-123.94958488594237203382816480277908306679338388130)*(s^2+56.142264645708354225809475665278814478118570596152*s+391252.79097567023079015071909098045312869106244420)*(s^2+12.861160988207490696337262517730984452695598400261*s+75.505757318432963550499582955755873806102180182642)

 (2)

 

 

 

 

 

 

 

Automatical getting Simply1

``

Denom1 := factor(rhs(EQ2)); Denom2 := denom(Denom1)

(s+146.79375648863023558659443817684982406475862236699)*(s+123.95936359750265297185596051550647036659011563743)*(s+92.660187384035638553231027927412989704630476880467)*(s-123.94958488594237203382816480277908306679338388130)*(s^2+56.142264645708354225809475665278814478118570596152*s+391252.79097567023079015071909098045312869106244420)*(s^2+12.861160988207490696337262517730984452695598400261*s+75.505757318432963550499582955755873806102180182642)

 (3)
> 

 local D;
 Denoms := [op(Denom2)];
 S := [[A, B], [C, D], E, F, G, H];
 Simply1 := `+`(seq(`if`(nops(S[i]) = 2, (s*S[i, 1]+S[i, 2])/Denoms[i], S[i]/Denoms[i]), i = 1 .. nops(S)));
 Simply2 := normal(Simply1);
 Numer := numer(Simply2);

Warning, A new binding for the name `D` has been created. The global instance of this name is still accessible using the :- prefix, :-`D`.  See ?protect for details.

  

[s+146.79375648863023558659443817684982406475862236699, s+123.95936359750265297185596051550647036659011563743, s+92.660187384035638553231027927412989704630476880467, s-123.94958488594237203382816480277908306679338388130, s^2+56.142264645708354225809475665278814478118570596152*s+391252.79097567023079015071909098045312869106244420, s^2+12.861160988207490696337262517730984452695598400261*s+75.505757318432963550499582955755873806102180182642]

  

[[A, B], [C, D], E, F, G, H]

  

(A*s+B)/(s+146.79375648863023558659443817684982406475862236699)+(C*s+D)/(s+123.95936359750265297185596051550647036659011563743)+E/(s+92.660187384035638553231027927412989704630476880467)+F/(s-123.94958488594237203382816480277908306679338388130)+G/(s^2+56.142264645708354225809475665278814478118570596152*s+391252.79097567023079015071909098045312869106244420)+H/(s^2+12.861160988207490696337262517730984452695598400261*s+75.505757318432963550499582955755873806102180182642)

  

(-66629954289230.853351964793698355440910791176473210*E-15779936123.736896752352323444813538247190635638921*G-81767858095804.252692630340308479057393577857396095*H-42058628360723.616307294413258076907769937601261231*B+49810122845599.559373052446569045921832073977569162*F+383942.26792329745908486350776331130623428840884256*C*s^6+47524317.891080990440120410672548754935735096209837*C*s^5+383942.26792329745908486350776331130623428840884256*s^5*D+47524317.891080990440120410672548754935735096209837*s^4*D+C*s^8+184.50778462063934702814403948449352963340988436257*C*s^7+s^7*D+184.50778462063934702814403948449352963340988436257*s^6*D+E*s^7+215.80696083410636144676897207258701029536952311953*E*s^6+386817.02205398946456229464104774416710710212015297*E*s^5+59275019.334177490049864812212209564510234527679724*E*s^4-5409936762.3812452194670802975059988184512985605757*E*s^3-957283261509.69073359764351612211142859298462693447*E*s^2-11812734658764.265702481442194215870629708925210542*E*s+F*s^7+432.41673310408437203382816480277908306679338388130*F*s^6+460411.61004964359284784553091349240166684402248386*F*s^5+152185385.02335649660105099107708122967023007133954*F*s^4+18945812935.668325433871566592281474322316488242538*F*s^3+889756633055.08615184018191100410310211280663669879*F*s^2+9770184266838.6796264068483962798183615978889977543*F*s+G*s^6+252.32488357243364577419052433472118552188142940385*G*s^5+1394.8541177656510445370732616558211257728483534122*G*s^4-3683568.1183126636160798257645837738641321950415443*G*s^3-256439074.12493727370906890754263425670134219029679*G*s^2-2965638243.8698618355031711444968889917370787314637*G*s+H*s^6+295.60598722993450930366273748226901554730440159974*H*s^5+402936.39353549637164746650712934864411382455075645*H*s^4+89913007.301833393890135453305394454749793734226506*H*s^3-1104312043.0718342886981361979321082809364175920587*H*s^2-1451155230389.6595379602185696933802326999232777717*H*s-7623904174794.9162815415989571038814356493076426121*A*s^2-7623904174794.9162815415989571038814356493076426121*s*B-632798554223.61137138093817629704816089972257384343*A*s^3-632798554223.61137138093817629704816089972257384343*s^2*B-5625378634.4657503088637321277824131160302176852356*A*s^4-5625378634.4657503088637321277824131160302176852356*s^3*B+383081.09098742412456669395855838386875466521403514*A*s^6+38883644.695228060251888451945070801510724697210725*A*s^5+383081.09098742412456669395855838386875466521403514*s^5*B+38883644.695228060251888451945070801510724697210725*s^4*B+A*s^8+161.67339172951176441340556182315017593524137763301*A*s^7+s^7*B+161.67339172951176441340556182315017593524137763301*s^6*B-8965792785078.0004116646708411617062572964824490740*C*s^2-8965792785078.0004116646708411617062572964824490740*s*D-738540321665.69941204354252236784632028838867030614*C*s^3-738540321665.69941204354252236784632028838867030614*s^2*D-5808586564.8678325769651462969069225989988084232500*C*s^4-5808586564.8678325769651462969069225989988084232500*s^3*D-42058628360723.616307294413258076907769937601261231*A*s-49806193502870.188576279201173210192418660124732386*C*s-49806193502870.188576279201173210192418660124732386*D)/((s+146.79375648863023558659443817684982406475862236699)*(s+123.95936359750265297185596051550647036659011563743)*(s+92.660187384035638553231027927412989704630476880467)*(s-123.94958488594237203382816480277908306679338388130)*(s^2+56.142264645708354225809475665278814478118570596152*s+391252.79097567023079015071909098045312869106244420)*(s^2+12.861160988207490696337262517730984452695598400261*s+75.505757318432963550499582955755873806102180182642))

  

-66629954289230.853351964793698355440910791176473210*E-15779936123.736896752352323444813538247190635638921*G-81767858095804.252692630340308479057393577857396095*H-42058628360723.616307294413258076907769937601261231*B+49810122845599.559373052446569045921832073977569162*F+383942.26792329745908486350776331130623428840884256*C*s^6+47524317.891080990440120410672548754935735096209837*C*s^5+383942.26792329745908486350776331130623428840884256*s^5*D+47524317.891080990440120410672548754935735096209837*s^4*D+C*s^8+184.50778462063934702814403948449352963340988436257*C*s^7+s^7*D+184.50778462063934702814403948449352963340988436257*s^6*D+E*s^7+215.80696083410636144676897207258701029536952311953*E*s^6+386817.02205398946456229464104774416710710212015297*E*s^5+59275019.334177490049864812212209564510234527679724*E*s^4-5409936762.3812452194670802975059988184512985605757*E*s^3-957283261509.69073359764351612211142859298462693447*E*s^2-11812734658764.265702481442194215870629708925210542*E*s+F*s^7+432.41673310408437203382816480277908306679338388130*F*s^6+460411.61004964359284784553091349240166684402248386*F*s^5+152185385.02335649660105099107708122967023007133954*F*s^4+18945812935.668325433871566592281474322316488242538*F*s^3+889756633055.08615184018191100410310211280663669879*F*s^2+9770184266838.6796264068483962798183615978889977543*F*s+G*s^6+252.32488357243364577419052433472118552188142940385*G*s^5+1394.8541177656510445370732616558211257728483534122*G*s^4-3683568.1183126636160798257645837738641321950415443*G*s^3-256439074.12493727370906890754263425670134219029679*G*s^2-2965638243.8698618355031711444968889917370787314637*G*s+H*s^6+295.60598722993450930366273748226901554730440159974*H*s^5+402936.39353549637164746650712934864411382455075645*H*s^4+89913007.301833393890135453305394454749793734226506*H*s^3-1104312043.0718342886981361979321082809364175920587*H*s^2-1451155230389.6595379602185696933802326999232777717*H*s-7623904174794.9162815415989571038814356493076426121*A*s^2-7623904174794.9162815415989571038814356493076426121*s*B-632798554223.61137138093817629704816089972257384343*A*s^3-632798554223.61137138093817629704816089972257384343*s^2*B-5625378634.4657503088637321277824131160302176852356*A*s^4-5625378634.4657503088637321277824131160302176852356*s^3*B+383081.09098742412456669395855838386875466521403514*A*s^6+38883644.695228060251888451945070801510724697210725*A*s^5+383081.09098742412456669395855838386875466521403514*s^5*B+38883644.695228060251888451945070801510724697210725*s^4*B+A*s^8+161.67339172951176441340556182315017593524137763301*A*s^7+s^7*B+161.67339172951176441340556182315017593524137763301*s^6*B-8965792785078.0004116646708411617062572964824490740*C*s^2-8965792785078.0004116646708411617062572964824490740*s*D-738540321665.69941204354252236784632028838867030614*C*s^3-738540321665.69941204354252236784632028838867030614*s^2*D-5808586564.8678325769651462969069225989988084232500*C*s^4-5808586564.8678325769651462969069225989988084232500*s^3*D-42058628360723.616307294413258076907769937601261231*A*s-49806193502870.188576279201173210192418660124732386*C*s-49806193502870.188576279201173210192418660124732386*D

 (4)
> 

#
# Solve generates no solutions with Digits set to 50,
# but setting Digits=10 at the top of this worksheet
# produce a 'solution' here - although I wouldn't
# believe it for a bet!
#
 solve({seq(coeff(Numer, s, k) = coeff(numer(Denom1), s, k), k = 0 .. degree(Numer, s))});
#
# Notice that with Digiits=50 at teh top  of this
# worksheet, there are nine equations with eight
# unknowns. Changing the Digits setting to 10 at
# the top of this worksheet will result in eight
# equations with eight unknowns, because one of
# nine becomes 0=0 which is ignored. Given eight
# equations in eight unknowns, a solution can be
# generated, although as I said above, I wouldn't
# believe it
#
 numelems({seq(coeff(Numer, s, k) = coeff(numer(Denom1), s, k), k = 0 .. degree(Numer, s))});
 indets({seq(coeff(Numer, s, k) = coeff(numer(Denom1), s, k), k = 0 .. degree(Numer, s))});
 coeff(Numer, s, 8);
 coeff(numer(Denom1), s, 8)

9

  

{A, B, C, D, E, F, G, H}

  

A+C

  

0

 (5)
> 

``

Download partfrac.mw

Like this...

tomleslie 13876
November 16 2018
1 0

The attached code will export the data to the specified fileName. You will have to change the fileName (variable fname in the attached) to something appropriate for your machine.

Many formats are possible for this data - I have used the "simplest"

> 

  restart;
  with(DynamicSystems):
  with(ExcelTools):
#
# OP will have to change output path/file
# name to something appropriate for his/her machine
#
  fname:="C:/Users/TomLeslie/Desktop/BodeData.xls":
  T2 := TransferFunction((1.224867518*10^(-7)*s+2.259345677*10^(-6))/(s^2+11.83692270*s+238.3951363)):
  BP2 := BodePlot(T2, output = [magnitudedata, phasedata]):
#
# Construct output matrix from Bode plot data
# (Many options and manipulations possible:
# eg linear frequency, dBs, degrees/radians,
# whatever
#
  M:=Matrix( op([1,1,1],BP2),
             3,
            (i,j) -> `if`( j<3,
                           BP2[1][i,j],
                           BP2[2][i,2])
                         ):
  Export(M, fname, "Sheet1", "B2");

``


 

Download Bode.mw

BodeData.xls

 

Something like...

tomleslie 13876
November 14 2018
1 2

the solution shown in the attached - perhaps?

> 

restart;
with(LinearAlgebra):
k:=2:
M:=3:
F:= Matrix(M,M, (i,j)-> `if`(`and`(i=1,j=1),2,0));
L:= BandMatrix( [ [ seq(-sqrt(2*i-1)/(2*i-1)*sqrt(2*i-3), i=2..M)],
                  [ 1, seq(0*i,i=1..M-1) ],
                  [ seq(sqrt(2*i-3)/((2*i-3)*sqrt(2*i-1)),i=2..M)]
                ]
              );
P:= Matrix( 2^(k-1),
            2^(k-1),
            (i,j)-> `if`( i=j,
                        (1/2^k)*~F,
                        `if`( j>i,
                              (1/2^k)*~L,
                              Matrix( M, M, fill=0 )
                            )
                       )
          );

Matrix(3, 3, {(1, 1) = 2, (1, 2) = 0, (1, 3) = 0, (2, 1) = 0, (2, 2) = 0, (2, 3) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0})

  

Matrix(3, 3, {(1, 1) = 1, (1, 2) = (1/3)*sqrt(3), (1, 3) = 0, (2, 1) = -(1/3)*sqrt(3), (2, 2) = 0, (2, 3) = (1/15)*sqrt(5)*sqrt(3), (3, 1) = 0, (3, 2) = -(1/5)*sqrt(5)*sqrt(3), (3, 3) = 0})

  

Matrix(%id = 18446744074368910390)

 (1)
> 

 

> 

 

Download doMat.mw

 

 

When...

tomleslie 13876
November 13 2018
1 1

computing finite sums and products of numeric terms, you should be using add() and mul(), not sum() and product().

Assuming I haven't made a typo somewhere (always a possibility, so you should check!), then answer you want is shown in the attached

> 

  restart;
  k:=3;
  j:=0;
  expr:= add
         ( add
           ( `if`
             ( l=m, 0, mul( `if`(`or`(q=m, q=l),1,(r-q+1) ),
                            q=0..k
                          )
             ),
             l=0..k
           )
           /
           mul( `if`(l=m, 1, m-l),
                l=0..k
              ),
           m=j+1..k
         );
  eval(expr, r=1);

3

  

0

  

(1/2)*(r-1)*(r-2)+(2/3)*(r+1)*(r-1)-(1/2)*r*(r-2)-(1/3)*(r+1)*r+(1/6)*r*(r-1)

  

-1/6

 (1)
> 

 

Download sumProd.mw

More of an observation really...

tomleslie 13876
November 12 2018
0 1

Bear in mind I know very little about Latex, and the following may not be what you want

Looking at the output of the Latex() command, it appears that you are getting a 'picture' of a graph, rather than the graph itself. If a 'picture' suffices, you can create one by 'right-clicking' on the graph and selecting 'Export as'. Then pick a format which can be imported into your Latex document.

Please post worksheet code...

tomleslie 13876
November 12 2018
0 0

using the big green up-arrow in the Mapleprimes toolbar;

The code

restart;
pde:=d*ep*diff(T(t,z),t)-k*diff(T(t,z),z,z)-Q;
pdsolve(pde, build);

will return an explicit solution in terms of four constants: three 'constants of integration' and one 'variable separation constant' - so you *might* need one more BC/IC. Alternatively the BCs/ICs you supply *might* be inconsistent with the PDE

This one hurt...

tomleslie 13876
November 12 2018
1 3

All arithmetic operations have to be replaced byt their GI[` `] equivalents!

I too got the

rror, (in GI[`*`]) invalid subscript selector

but that is because I forgot to fix the `<>` in the 'while' loop!

The attached now executes - it may even provide the correct answer!

> 

EEA:=proc(ED,a,b)
    description
    "Extended Euclidean Algorithm"
    "INPUT: an Euclidean Domain ED and two elements from said domain"
    "OUTPUT: r,s,t such that r = gcd(a,b) = s*a + t*b ";
    local r_0, r_1, r_aux, s_0, s_1, s_aux, t_0, t_1, t_aux, q;
    # Domain checks
    # TODO: check that ED is an euclidean domain
    if not ED[Type](a) then error "1st argument must be of type ED" end if;
    if not ED[Type](b) then error "2nd argument must be of type ED" end if;
    
    # Initialization
    r_0 := a; r_1 := b; # gcd series
    s_0 := GI[Input](1); s_1 := GI[Input](0); # 1st cofactor series
    t_0 := GI[Input](0); t_1 := GI[Input](1); # 2nd cofactor series

    # Loop
    while GI[`<>`](r_1,GI[Input](0)) do;
        print("All is fine before the Quo");
        print(r_0); print(r_1);
        q := ED[Quo](r_0, r_1);
        print("All is fine after the Quo");
        
     #   r_aux := r_0 - q * r_1;
         r_aux:= GI[`-`](r_0, GI[`*`](q, r_1));
         r_0 := r_1; r_1 := r_aux;
        
     #   s_aux := s_0 - q * s_1;
         s_aux:= GI[`-`](s_0, GI[`*`](q, s_1));
         s_0 := s_1; s_1 := s_aux;

     #   t_aux := t_0 - q * t_1;
         t_aux:= GI[`-`](t_0, GI[`*`](q, t_1));
         t_0 := t_1; t_1 := t_aux;
    od;

    # Result
    return r_0, s_0, t_0;
    end proc:
> 

with(Domains):
GI := Gaussian(Z):
a := GI[Input](-87+47*I):
b := GI[Input](-90+43*I):
r, s, t := EEA(GI, a, b);

---------------------- Domains version 1.0 ---------------------

Initially defined domains are Z and Q the integers and rationals
Abbreviations, e.g. DUP for DenseUnivariatePolynomial, also made

  

"All is fine before the Quo"

  

`domains/Gaussian/badge0`(-87, 47)

  

`domains/Gaussian/badge0`(-90, 43)

  

"All is fine after the Quo"

  

"All is fine before the Quo"

  

`domains/Gaussian/badge0`(-90, 43)

  

`domains/Gaussian/badge0`(3, 4)

  

"All is fine after the Quo"

  

"All is fine before the Quo"

  

`domains/Gaussian/badge0`(3, 4)

  

`domains/Gaussian/badge0`(2, -1)

  

"All is fine after the Quo"

  

"All is fine before the Quo"

  

`domains/Gaussian/badge0`(2, -1)

  

`domains/Gaussian/badge0`(1, 0)

  

"All is fine after the Quo"

  

`domains/Gaussian/badge0`(1, 0), `domains/Gaussian/badge0`(-39, -8), `domains/Gaussian/badge0`(39, 6)

 (1)
> 

 

Download doms.mw

Syntax and other errors...

tomleslie 13876
November 12 2018
2 0

There are some simple syntax errors (easy to fix).

More significantly, Maple's numeric solvers can only handle systems with Initial/boundary conditions containing derivatives if the latter are 'normal' to the boundary: some of yours are not! So just for the purpose of getting some kind of solution, I "made up" some BCs  which work. This almost certainly solves a different problem from the one you want!!

See the attached

> 

restart; with(PDEtools):
> 

PDE := (diff(y(x,t), x,x,x,x))+m*(diff(y(x, t), t,t))=m*g+M*g*Dirac(x-v*t)-M*(diff(y(x,t), x,x)*v^2+2*v*diff(y(x,t), x,y)+a*diff(y(x,t), x)+diff(y(x, t), t,t))*Dirac(x-v*t);

diff(diff(diff(diff(y(x, t), x), x), x), x)+m*(diff(diff(y(x, t), t), t)) = m*g+M*g*Dirac(t*v-x)-M*((diff(diff(y(x, t), x), x))*v^2+a*(diff(y(x, t), x))+diff(diff(y(x, t), t), t))*Dirac(t*v-x)

 (1)
> 

#Analytical solution
anlyt_sol:=pdsolve(PDE);

(2)
> 

 

> 

#Numerical solution
  a:=1:m:=1:g:=1:M:=1:v:=1:
#
# Initial/boundary conditions can only contain
# derivatives "normal" to the boundary -
# Arbitrarily changed these (almost
# certainly giving invalid conditions!)
#
# BCs:=y(0,t) = 0, y(1,t) = 0,D[2](y)(0,t)=0,D[2](y)(1,t)=0;
#
  BCs:=y(0,t) = 0, y(1,t) = 0,D[1](y)(0,t)=0,D[1](y)(1,t)=0;
#
# Incorrect syntax for IC: so I guessed something
# "appropriate" - possibly invalid
#
# ICs:=y(x,0) =0,Dy(x,0)=0 ;
#
  ICs:=y(x,0) =0, D[2](y)(x,0)=0 ;
#
# sys:={PDE,BCs,ICs}:
#
# Incorrect syntax for pdsolve: - fixed
#
# num_sol := pdsolve(sys, numeric);
#
  num_sol := pdsolve(PDE, {BCs,ICs}, numeric);
#
# Produce general 3D-plot: many other plot
# options are available!
#
  num_sol:-plot3d(x=0..1, t=0..10);

y(0, t) = 0, y(1, t) = 0, (D[1](y))(0, t) = 0, (D[1](y))(1, t) = 0

  

y(x, 0) = 0, (D[2](y))(x, 0) = 0

  

_m700614144

  

 
> 

 
> 

 

> 

 

Download pdeProb.mw

Numerically, yes: analytically, well.......

tomleslie 13876
November 11 2018
1 0

the solution starts heading for +Infinity, very quickly. Is this expected? See the attached

> 

  restart:
  n:=1: N:=1: i:=1: j:=1: E:=1: M:=2:
  m:=2: II:=2: l:=1: a:=1: v_0:=10: mu:=1:
  gamma_0:=M/(m*l):
  a_0:=Pi^2*a/w1^2:

  M:= tau -> 1+gamma_0*sin(i*Pi*v_0*tau)*sin(j*Pi*v_0*tau):

  C:=tau -> 4*gamma_0*(i*Pi*v_0*tau)*sin(i*Pi*v_0*tau)*cos(j*Pi*v_0*tau):
  K:=tau -> n^4*Pi^2-2*mu*(i*Pi*v_0)^2*sin(i*Pi*v_0*tau)*sin(j*Pi*v_0*tau)+2*mu*(i*Pi*a_0)*sin(i*Pi*v_0*tau)*cos(j*Pi*v_0*tau):
  f:=tau -> gamma_0*Pi^2*(Pi/2+v_0^2*sum(((1-(-1)^k)/k^3)*sin(k*Pi*v_0*tau),k=1..N)-a_0*sum(((1-(-1)^k)/k^4)*cos(k*Pi*v_0*tau),k=1..N))*sin(i*Pi*v_0*tau):
   X:=x(t):   
 
#X:= Vector(n, i-> x[i](tau)):
sys:= M(t)*diff~(X, t$2)+C(t)*diff~(X, t)+K(t)*X=f(t);

(1+sin(10*Pi*t)^2)*(diff(diff(x(t), t), t))+40*Pi*t*sin(10*Pi*t)*cos(10*Pi*t)*(diff(x(t), t))+(Pi^2-200*Pi^2*sin(10*Pi*t)^2+2*Pi^3*sin(10*Pi*t)*cos(10*Pi*t)/w1^2)*x(t) = Pi^2*((1/2)*Pi+200*sin(10*Pi*t)-2*Pi^2*cos(10*Pi*t)/w1^2)*sin(10*Pi*t)

 (1)
> 

#numerical solution
  IC:=x(0)=0,D(x)(0)=0; # or you can define any Initial conditions
  sys1:={sys,IC};
# indets(sys);
#
# In order to achieve a numerical solution, it is necessary
# for w1 to be defined. Hence the following.
#
  w1:=1.0:
#
# Now solve it
#
  numSol:=dsolve(sys1,numeric,method=rkf45, range=0..1);
#
# Plot the solution - for tau-values much greater
# than 0.1, the solution "grows" at a ridiculous rate!
# Consider the following
#
  plots:-odeplot(numSol, [t, x(t)], t=0..0.1);
  plots:-odeplot(numSol, [t, x(t)], t=0..1.0);

x(0) = 0, (D(x))(0) = 0

  

{(1+sin(10*Pi*t)^2)*(diff(diff(x(t), t), t))+40*Pi*t*sin(10*Pi*t)*cos(10*Pi*t)*(diff(x(t), t))+(Pi^2-200*Pi^2*sin(10*Pi*t)^2+2*Pi^3*sin(10*Pi*t)*cos(10*Pi*t)/w1^2)*x(t) = Pi^2*((1/2)*Pi+200*sin(10*Pi*t)-2*Pi^2*cos(10*Pi*t)/w1^2)*sin(10*Pi*t), x(0) = 0, (D(x))(0) = 0}

  

proc (x_rkf45) local _res, _dat, _vars, _solnproc, _xout, _ndsol, _pars, _n, _i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; if 1 < nargs then error "invalid input: too many arguments" end if; _EnvDSNumericSaveDigits := Digits; Digits := 15; if _EnvInFsolve = true then _xout := evalf[_EnvDSNumericSaveDigits](x_rkf45) else _xout := evalf(x_rkf45) end if; _dat := Array(1..4, {(1) = proc (_xin) local _xout, _dtbl, _dat, _vmap, _x0, _y0, _val, _dig, _n, _ne, _nd, _nv, _pars, _ini, _par, _i, _j, _k, _src; option `Copyright (c) 2002 by Waterloo Maple Inc. All rights reserved.`; table( [( "complex" ) = false, ( "right" ) = 1., ( "left" ) = 0. ] ) _xout := _xin; _pars := []; _dtbl := array( 1 .. 4, [( 1 ) = (array( 1 .. 26, [( 1 ) = (datatype = float[8], order = C_order, storage = rectangular), ( 2 ) = (datatype = float[8], order = C_order, storage = rectangular), ( 3 ) = ([0, 0, 0, Array(1..0, 1..2, {}, datatype = float[8], order = C_order)]), ( 4 ) = (Array(1..63, {(1) = 2, (2) = 2, (3) = 0, (4) = 0, (5) = 0, (6) = 0, (7) = 1, (8) = 0, (9) = 0, (10) = 1, (11) = 0, (12) = 0, (13) = 0, (14) = 0, (15) = 0, (16) = 0, (17) = 0, (18) = 21, (19) = 30000, (20) = 0, (21) = 0, (22) = 1, (23) = 4, (24) = 0, (25) = 1, (26) = 15, (27) = 1, (28) = 0, (29) = 1, (30) = 3, (31) = 3, (32) = 0, (33) = 1, (34) = 0, (35) = 0, (36) = 0, (37) = 0, (38) = 0, (39) = 0, (40) = 0, (41) = 0, (42) = 0, (43) = 1, (44) = 0, (45) = 0, (46) = 0, (47) = 0, (48) = 0, (49) = 0, (50) = 50, (51) = 1, (52) = 0, (53) = 0, (54) = 0, (55) = 0, (56) = 0, (57) = 0, (58) = 0, (59) = 10000, (60) = 0, (61) = 1000, (62) = 0, (63) = 0}, datatype = integer[8])), ( 5 ) = (Array(1..28, {(1) = 1.0, (2) = 0.10e-5, (3) = .0, (4) = 0.500001e-14, (5) = .0, (6) = 0.6049812287291186e-3, (7) = .0, (8) = 0.10e-5, (9) = .0, (10) = .0, (11) = .0, (12) = .0, (13) = 1.0, (14) = .0, (15) = .49999999999999, (16) = .0, (17) = 1.0, (18) = 1.0, (19) = .0, (20) = .0, (21) = 1.0, (22) = 1.0, (23) = .0, (24) = .0, (25) = 0.10e-14, (26) = .0, (27) = .0, (28) = .0}, datatype = float[8], order = C_order)), ( 6 ) = (Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order)), ( 7 ) = ([Array(1..4, 1..7, {(1, 1) = .0, (1, 2) = .203125, (1, 3) = .3046875, (1, 4) = .75, (1, 5) = .8125, (1, 6) = .40625, (1, 7) = .8125, (2, 1) = 0.6378173828125e-1, (2, 2) = .0, (2, 3) = .279296875, (2, 4) = .27237892150878906, (2, 5) = -0.9686851501464844e-1, (2, 6) = 0.1956939697265625e-1, (2, 7) = .5381584167480469, (3, 1) = 0.31890869140625e-1, (3, 2) = .0, (3, 3) = -.34375, (3, 4) = -.335235595703125, (3, 5) = .2296142578125, (3, 6) = .41748046875, (3, 7) = 11.480712890625, (4, 1) = 0.9710520505905151e-1, (4, 2) = .0, (4, 3) = .40350341796875, (4, 4) = 0.20297467708587646e-1, (4, 5) = -0.6054282188415527e-2, (4, 6) = -0.4770040512084961e-1, (4, 7) = .77858567237854}, datatype = float[8], order = C_order), Array(1..6, 1..6, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (1, 6) = 1.0, (2, 1) = .25, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (2, 6) = 1.0, (3, 1) = .1875, (3, 2) = .5625, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (3, 6) = 2.0, (4, 1) = .23583984375, (4, 2) = -.87890625, (4, 3) = .890625, (4, 4) = .0, (4, 5) = .0, (4, 6) = .2681884765625, (5, 1) = .1272735595703125, (5, 2) = -.5009765625, (5, 3) = .44921875, (5, 4) = -0.128936767578125e-1, (5, 5) = .0, (5, 6) = 0.626220703125e-1, (6, 1) = -0.927734375e-1, (6, 2) = .626220703125, (6, 3) = -.4326171875, (6, 4) = .1418304443359375, (6, 5) = -0.861053466796875e-1, (6, 6) = .3131103515625}, datatype = float[8], order = C_order), Array(1..6, {(1) = .0, (2) = .386, (3) = .21, (4) = .63, (5) = 1.0, (6) = 1.0}, datatype = float[8], order = C_order), Array(1..6, {(1) = .25, (2) = -.1043, (3) = .1035, (4) = -0.362e-1, (5) = .0, (6) = .0}, datatype = float[8], order = C_order), Array(1..6, 1..5, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (2, 1) = 1.544, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (3, 1) = .9466785280815533, (3, 2) = .25570116989825814, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (4, 1) = 3.3148251870684886, (4, 2) = 2.896124015972123, (4, 3) = .9986419139977808, (4, 4) = .0, (4, 5) = .0, (5, 1) = 1.2212245092262748, (5, 2) = 6.019134481287752, (5, 3) = 12.537083329320874, (5, 4) = -.687886036105895, (5, 5) = .0, (6, 1) = 1.2212245092262748, (6, 2) = 6.019134481287752, (6, 3) = 12.537083329320874, (6, 4) = -.687886036105895, (6, 5) = 1.0}, datatype = float[8], order = C_order), Array(1..6, 1..5, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (2, 1) = -5.6688, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (3, 1) = -2.4300933568337584, (3, 2) = -.20635991570891224, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (4, 1) = -.10735290581452621, (4, 2) = -9.594562251021896, (4, 3) = -20.470286148096154, (4, 4) = .0, (4, 5) = .0, (5, 1) = 7.496443313968615, (5, 2) = -10.246804314641219, (5, 3) = -33.99990352819906, (5, 4) = 11.708908932061595, (5, 5) = .0, (6, 1) = 8.083246795922411, (6, 2) = -7.981132988062785, (6, 3) = -31.52159432874373, (6, 4) = 16.319305431231363, (6, 5) = -6.0588182388340535}, datatype = float[8], order = C_order), Array(1..3, 1..5, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (2, 1) = 10.126235083446911, (2, 2) = -7.487995877607633, (2, 3) = -34.800918615557414, (2, 4) = -7.9927717075687275, (2, 5) = 1.0251377232956207, (3, 1) = -.6762803392806898, (3, 2) = 6.087714651678606, (3, 3) = 16.43084320892463, (3, 4) = 24.767225114183653, (3, 5) = -6.5943891257167815}, datatype = float[8], order = C_order)]), ( 9 ) = ([Array(1..2, {(1) = .1, (2) = .1}, datatype = float[8], order = C_order), Array(1..2, {(1) = 46013014117.14629, (2) = 1614166122272.2595}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = 1606480840439.0854, (2) = -10361588507740.117}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, 1..2, {(1, 1) = .0, (1, 2) = .0, (2, 1) = .0, (2, 2) = .0}, datatype = float[8], order = C_order), Array(1..2, 1..2, {(1, 1) = .0, (1, 2) = .0, (2, 1) = .0, (2, 2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, 1..2, {(1, 1) = .0, (1, 2) = .0, (2, 1) = .0, (2, 2) = .0}, datatype = float[8], order = C_order), Array(1..2, 1..6, {(1, 1) = 1606480840439.0854, (1, 2) = 1616084072802.6145, (1, 3) = 1615091119436.1099, (1, 4) = 1608428564526.9526, (1, 5) = 1606166866407.9324, (1, 6) = 1614937932474.9717, (2, 1) = -10361588507740.117, (2, 2) = 2780032942948.5684, (2, 3) = 524538970763.5669, (2, 4) = -9070259691978.488, (2, 5) = -10359358761437.225, (2, 6) = -1711579351060.1375}, datatype = float[8], order = C_order), Array(1..2, {(1) = 0, (2) = 0}, datatype = integer[8]), Array(1..2, {(1) = 45250973444.09453, (2) = 1615095261461.6143}, datatype = float[8], order = C_order), Array(1..2, {(1) = 47793021367.78502, (2) = 1606480840439.0854}, datatype = float[8], order = C_order), Array(1..2, {(1) = 653.284309387207, (2) = 172220.03051757812}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = 1613910321089.7012, (2) = -3924465926183.4907}, datatype = float[8], order = C_order), Array(1..4, {(1) = .0, (2) = .0, (3) = .0, (4) = .0}, datatype = float[8], order = C_order)]), ( 8 ) = ([Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = -.0}, datatype = float[8], order = C_order), 0, 0]), ( 11 ) = (Array(1..6, 0..2, {(1, 1) = .0, (1, 2) = .0, (2, 0) = .0, (2, 1) = .0, (2, 2) = .0, (3, 0) = .0, (3, 1) = .0, (3, 2) = .0, (4, 0) = .0, (4, 1) = .0, (4, 2) = .0, (5, 0) = .0, (5, 1) = .0, (5, 2) = .0, (6, 0) = .0, (6, 1) = .0, (6, 2) = .0}, datatype = float[8], order = C_order)), ( 10 ) = ([proc (N, X, Y, YP) option `[Y[1] = x(t), Y[2] = diff(x(t),t)]`; YP[2] := (9.86960440108934*(1.57079632679490+200*sin(31.4159265358979*X)-19.73920881*cos(31.4159265358979*X))*sin(31.4159265358979*X)-125.663706143592*X*sin(31.4159265358979*X)*cos(31.4159265358979*X)*Y[2]-(9.86960440108934-1973.92088021787*sin(31.4159265358979*X)^2+62.01255338*sin(31.4159265358979*X)*cos(31.4159265358979*X))*Y[1])/(1+sin(31.4159265358979*X)^2); YP[1] := Y[2]; 0 end proc, -1, 0, 0, 0, 0, 0, 0]), ( 13 ) = (), ( 12 ) = (), ( 15 ) = ("rkf45"), ( 14 ) = ([0, 0]), ( 18 ) = ([]), ( 19 ) = (0), ( 16 ) = ([0, 0, 0, []]), ( 17 ) = ([proc (N, X, Y, YP) option `[Y[1] = x(t), Y[2] = diff(x(t),t)]`; YP[2] := (9.86960440108934*(1.57079632679490+200*sin(31.4159265358979*X)-19.73920881*cos(31.4159265358979*X))*sin(31.4159265358979*X)-125.663706143592*X*sin(31.4159265358979*X)*cos(31.4159265358979*X)*Y[2]-(9.86960440108934-1973.92088021787*sin(31.4159265358979*X)^2+62.01255338*sin(31.4159265358979*X)*cos(31.4159265358979*X))*Y[1])/(1+sin(31.4159265358979*X)^2); YP[1] := Y[2]; 0 end proc, -1, 0, 0, 0, 0, 0, 0]), ( 22 ) = (0), ( 23 ) = (0), ( 20 ) = ([]), ( 21 ) = (0), ( 26 ) = (Array(1..0, {})), ( 25 ) = (Array(1..0, {})), ( 24 ) = (0) ] )), ( 3 ) = (array( 1 .. 26, [( 1 ) = (datatype = float[8], order = C_order, storage = rectangular), ( 2 ) = (datatype = float[8], order = C_order, storage = rectangular), ( 3 ) = ([0, 0, 0, Array(1..0, 1..2, {}, datatype = float[8], order = C_order)]), ( 4 ) = (Array(1..63, {(1) = 2, (2) = 2, (3) = 0, (4) = 0, (5) = 0, (6) = 0, (7) = 1, (8) = 1, (9) = 0, (10) = 1, (11) = 1073, (12) = 1073, (13) = 0, (14) = 0, (15) = 0, (16) = 0, (17) = 0, (18) = 1982, (19) = 30000, (20) = 5, (21) = 0, (22) = 1, (23) = 4, (24) = 0, (25) = 1, (26) = 15, (27) = 1, (28) = 0, (29) = 1, (30) = 3, (31) = 3, (32) = 0, (33) = 1, (34) = 0, (35) = 0, (36) = 0, (37) = 0, (38) = 0, (39) = 0, (40) = 0, (41) = 0, (42) = 0, (43) = 1, (44) = 0, (45) = 0, (46) = 0, (47) = 0, (48) = 0, (49) = 0, (50) = 50, (51) = 1, (52) = 0, (53) = 0, (54) = 0, (55) = 0, (56) = 0, (57) = 0, (58) = 0, (59) = 10000, (60) = 0, (61) = 1000, (62) = 0, (63) = 0}, datatype = integer[8])), ( 5 ) = (Array(1..28, {(1) = 1.0, (2) = 0.10e-5, (3) = 0.3360296953602959e-2, (4) = 0.500001e-14, (5) = .0, (6) = 0.6049812287291186e-3, (7) = .0, (8) = 0.10e-5, (9) = .0, (10) = .0, (11) = .0, (12) = .0, (13) = 1.0, (14) = .0, (15) = .49999999999999, (16) = .0, (17) = 1.0, (18) = 1.0, (19) = .0, (20) = .0, (21) = 1.0, (22) = 1.0, (23) = .0, (24) = .0, (25) = 0.10e-14, (26) = .0, (27) = .0, (28) = .0}, datatype = float[8], order = C_order)), ( 6 ) = (Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order)), ( 7 ) = ([Array(1..4, 1..7, {(1, 1) = .0, (1, 2) = .203125, (1, 3) = .3046875, (1, 4) = .75, (1, 5) = .8125, (1, 6) = .40625, (1, 7) = .8125, (2, 1) = 0.6378173828125e-1, (2, 2) = .0, (2, 3) = .279296875, (2, 4) = .27237892150878906, (2, 5) = -0.9686851501464844e-1, (2, 6) = 0.1956939697265625e-1, (2, 7) = .5381584167480469, (3, 1) = 0.31890869140625e-1, (3, 2) = .0, (3, 3) = -.34375, (3, 4) = -.335235595703125, (3, 5) = .2296142578125, (3, 6) = .41748046875, (3, 7) = 11.480712890625, (4, 1) = 0.9710520505905151e-1, (4, 2) = .0, (4, 3) = .40350341796875, (4, 4) = 0.20297467708587646e-1, (4, 5) = -0.6054282188415527e-2, (4, 6) = -0.4770040512084961e-1, (4, 7) = .77858567237854}, datatype = float[8], order = C_order), Array(1..6, 1..6, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (1, 6) = 1.0, (2, 1) = .25, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (2, 6) = 1.0, (3, 1) = .1875, (3, 2) = .5625, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (3, 6) = 2.0, (4, 1) = .23583984375, (4, 2) = -.87890625, (4, 3) = .890625, (4, 4) = .0, (4, 5) = .0, (4, 6) = .2681884765625, (5, 1) = .1272735595703125, (5, 2) = -.5009765625, (5, 3) = .44921875, (5, 4) = -0.128936767578125e-1, (5, 5) = .0, (5, 6) = 0.626220703125e-1, (6, 1) = -0.927734375e-1, (6, 2) = .626220703125, (6, 3) = -.4326171875, (6, 4) = .1418304443359375, (6, 5) = -0.861053466796875e-1, (6, 6) = .3131103515625}, datatype = float[8], order = C_order), Array(1..6, {(1) = .0, (2) = .386, (3) = .21, (4) = .63, (5) = 1.0, (6) = 1.0}, datatype = float[8], order = C_order), Array(1..6, {(1) = .25, (2) = -.1043, (3) = .1035, (4) = -0.362e-1, (5) = .0, (6) = .0}, datatype = float[8], order = C_order), Array(1..6, 1..5, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (2, 1) = 1.544, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (3, 1) = .9466785280815533, (3, 2) = .25570116989825814, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (4, 1) = 3.3148251870684886, (4, 2) = 2.896124015972123, (4, 3) = .9986419139977808, (4, 4) = .0, (4, 5) = .0, (5, 1) = 1.2212245092262748, (5, 2) = 6.019134481287752, (5, 3) = 12.537083329320874, (5, 4) = -.687886036105895, (5, 5) = .0, (6, 1) = 1.2212245092262748, (6, 2) = 6.019134481287752, (6, 3) = 12.537083329320874, (6, 4) = -.687886036105895, (6, 5) = 1.0}, datatype = float[8], order = C_order), Array(1..6, 1..5, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (2, 1) = -5.6688, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (3, 1) = -2.4300933568337584, (3, 2) = -.20635991570891224, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (4, 1) = -.10735290581452621, (4, 2) = -9.594562251021896, (4, 3) = -20.470286148096154, (4, 4) = .0, (4, 5) = .0, (5, 1) = 7.496443313968615, (5, 2) = -10.246804314641219, (5, 3) = -33.99990352819906, (5, 4) = 11.708908932061595, (5, 5) = .0, (6, 1) = 8.083246795922411, (6, 2) = -7.981132988062785, (6, 3) = -31.52159432874373, (6, 4) = 16.319305431231363, (6, 5) = -6.0588182388340535}, datatype = float[8], order = C_order), Array(1..3, 1..5, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (2, 1) = 10.126235083446911, (2, 2) = -7.487995877607633, (2, 3) = -34.800918615557414, (2, 4) = -7.9927717075687275, (2, 5) = 1.0251377232956207, (3, 1) = -.6762803392806898, (3, 2) = 6.087714651678606, (3, 3) = 16.43084320892463, (3, 4) = 24.767225114183653, (3, 5) = -6.5943891257167815}, datatype = float[8], order = C_order)]), ( 9 ) = ([Array(1..2, {(1) = .1, (2) = .1}, datatype = float[8], order = C_order), Array(1..2, {(1) = 46013014117.14629, (2) = 1614166122272.2595}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = 1606480840439.0854, (2) = -10361588507740.117}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, 1..2, {(1, 1) = .0, (1, 2) = .0, (2, 1) = .0, (2, 2) = .0}, datatype = float[8], order = C_order), Array(1..2, 1..2, {(1, 1) = .0, (1, 2) = .0, (2, 1) = .0, (2, 2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, 1..2, {(1, 1) = .0, (1, 2) = .0, (2, 1) = .0, (2, 2) = .0}, datatype = float[8], order = C_order), Array(1..2, 1..6, {(1, 1) = 1606480840439.0854, (1, 2) = 1616084072802.6145, (1, 3) = 1615091119436.1099, (1, 4) = 1608428564526.9526, (1, 5) = 1606166866407.9324, (1, 6) = 1614937932474.9717, (2, 1) = -10361588507740.117, (2, 2) = 2780032942948.5684, (2, 3) = 524538970763.5669, (2, 4) = -9070259691978.488, (2, 5) = -10359358761437.225, (2, 6) = -1711579351060.1375}, datatype = float[8], order = C_order), Array(1..2, {(1) = 0, (2) = 0}, datatype = integer[8]), Array(1..2, {(1) = 45250973444.09453, (2) = 1615095261461.6143}, datatype = float[8], order = C_order), Array(1..2, {(1) = 47793021367.78502, (2) = 1606480840439.0854}, datatype = float[8], order = C_order), Array(1..2, {(1) = 653.284309387207, (2) = 172220.03051757812}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = 1613910321089.7012, (2) = -3924465926183.4907}, datatype = float[8], order = C_order), Array(1..4, {(1) = .0, (2) = .0, (3) = .0, (4) = .0}, datatype = float[8], order = C_order)]), ( 8 ) = ([Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = 1606480840439.0854, (2) = -10361588507740.117}, datatype = float[8], order = C_order), 0, 0]), ( 11 ) = (Array(1..6, 0..2, {(1, 1) = .9988145253544011, (1, 2) = 43337819073.835175, (2, 0) = 43337819073.835175, (2, 1) = 1611039626162.232, (2, 2) = .9995051089483713, (3, 0) = .9995051089483713, (3, 1) = 44451756999.93105, (3, 2) = 1614519865823.2795, (4, 0) = 1614519865823.2795, (4, 1) = 1.0001956925423416, (4, 2) = 45567017799.27907, (5, 0) = 45567017799.27907, (5, 1) = 1614884022583.912, (5, 2) = 1.000886276136312, (6, 0) = 1.000886276136312, (6, 1) = 46681471009.20963, (6, 2) = 1612177361785.0664}, datatype = float[8], order = C_order)), ( 10 ) = ([proc (N, X, Y, YP) option `[Y[1] = x(t), Y[2] = diff(x(t),t)]`; YP[2] := (9.86960440108934*(1.57079632679490+200*sin(31.4159265358979*X)-19.73920881*cos(31.4159265358979*X))*sin(31.4159265358979*X)-125.663706143592*X*sin(31.4159265358979*X)*cos(31.4159265358979*X)*Y[2]-(9.86960440108934-1973.92088021787*sin(31.4159265358979*X)^2+62.01255338*sin(31.4159265358979*X)*cos(31.4159265358979*X))*Y[1])/(1+sin(31.4159265358979*X)^2); YP[1] := Y[2]; 0 end proc, -1, 0, 0, 0, 0, 0, 0]), ( 13 ) = (), ( 12 ) = (Array(1..1073, 0..2, {(1, 1) = .0, (1, 2) = .0, (2, 0) = .0, (2, 1) = .0, (2, 2) = 0.15124530718227966e-3, (3, 0) = 0.15124530718227966e-3, (3, 1) = -0.3163558863216358e-8, (3, 2) = -0.621846418644414e-4, (4, 0) = -0.621846418644414e-4, (4, 1) = 0.3024906143645593e-3, (4, 2) = -0.24627177128764016e-7, (5, 0) = -0.24627177128764016e-7, (5, 1) = -0.23973541782304153e-3, (5, 2) = 0.453735921546839e-3, (6, 0) = 0.453735921546839e-3, (6, 1) = -0.8081640516889977e-7, (6, 2) = -0.5191333083014517e-3, (7, 0) = -0.5191333083014517e-3, (7, 1) = 0.6049812287291186e-3, (7, 2) = -0.18611164383375898e-6, (8, 0) = -0.18611164383375898e-6, (8, 1) = -0.8868402335623069e-3, (8, 2) = 0.9064516850113113e-3, (9, 0) = 0.9064516850113113e-3, (9, 1) = -0.589430282624001e-6, (9, 2) = -0.18294029386485293e-2, (10, 0) = -0.18294029386485293e-2, (10, 1) = 0.1207922141293504e-2, (10, 2) = -0.1308167846881165e-5, (11, 0) = -0.1308167846881165e-5, (11, 1) = -0.29614942464736543e-2, (11, 2) = 0.15093925975756964e-2, (12, 0) = 0.15093925975756964e-2, (12, 1) = -0.2383279001708325e-5, (12, 2) = -0.4175674477280768e-2, (13, 0) = -0.4175674477280768e-2, (13, 1) = 0.1810863053857889e-2, (13, 2) = -0.3823336817127368e-5, (14, 0) = -0.3823336817127368e-5, (14, 1) = -0.5364683017025421e-2, (14, 2) = 0.2113941436287329e-2, (15, 0) = 0.2113941436287329e-2, (15, 1) = -0.5614911606198617e-5, (15, 2) = -0.6426613508381817e-2, (16, 0) = -0.6426613508381817e-2, (16, 1) = 0.2417019818716769e-2, (16, 2) = -0.7694377763339832e-5, (17, 0) = -0.7694377763339832e-5, (17, 1) = -0.7246647450477593e-2, (17, 2) = 0.27200982011462093e-2, (18, 0) = 0.27200982011462093e-2, (18, 1) = -0.9972166462454376e-5, (18, 2) = -0.7717141203111435e-2, (19, 0) = -0.7717141203111435e-2, (19, 1) = 0.30231765835756494e-2, (19, 2) = -0.12326122391941498e-4, (20, 0) = -0.12326122391941498e-4, (20, 1) = -0.7731249148535493e-2, (20, 2) = 0.33320112681146513e-2, (21, 0) = 0.33320112681146513e-2, (21, 1) = -0.14643025768738502e-4, (21, 2) = -0.7166464990495078e-2, (22, 0) = -0.7166464990495078e-2, (22, 1) = 0.3640845952653653e-2, (22, 2) = -0.16680978413860794e-4, (23, 0) = -0.16680978413860794e-4, (23, 1) = -0.5906726261466463e-2, (23, 2) = 0.3949680637192655e-2, (24, 0) = 0.3949680637192655e-2, (24, 1) = -0.18208524042542287e-4, (24, 2) = -0.3842341820982854e-2, (25, 0) = -0.3842341820982854e-2, (25, 1) = 0.4258515321731657e-2, (25, 2) = -0.18960401393344728e-4, (26, 0) = -0.18960401393344728e-4, (26, 1) = -0.8650956663570604e-3, (26, 2) = 0.4574502559324291e-2, (27, 0) = 0.4574502559324291e-2, (27, 1) = -0.18615186267424916e-4, (27, 2) = 0.3237162269377877e-2, (28, 0) = 0.3237162269377877e-2, (28, 1) = 0.4890489796916926e-2, (28, 2) = -0.16790172994111325e-4, (29, 0) = -0.16790172994111325e-4, (29, 1) = 0.8519135423186448e-2, (29, 2) = 0.520647703450956e-2, (30, 0) = 0.520647703450956e-2, (30, 1) = -0.1309531930812665e-4, (30, 2) = 0.15091163143538542e-1, (31, 0) = 0.15091163143538542e-1, (31, 1) = 0.55224642721021944e-2, (31, 2) = -0.7105825622136676e-5, (32, 0) = -0.7105825622136676e-5, (32, 1) = 0.23061383537097146e-1, (32, 2) = 0.5858190406972379e-2, (33, 0) = 0.5858190406972379e-2, (33, 1) = 0.2285800567827309e-5, (33, 2) = 0.33179782479509645e-1, (34, 0) = 0.33179782479509645e-1, (34, 1) = 0.6193916541842564e-2, (34, 2) = 0.1537696786135307e-4, (35, 0) = 0.1537696786135307e-4, (35, 1) = 0.4512000618974452e-1, (35, 2) = 0.6529642676712749e-2, (36, 0) = 0.6529642676712749e-2, (36, 1) = 0.3279930278119319e-4, (36, 2) = 0.5900278139610222e-1, (37, 0) = 0.5900278139610222e-1, (37, 1) = 0.6865368811582933e-2, (37, 2) = 0.5522478646116198e-4, (38, 0) = 0.5522478646116198e-4, (38, 1) = 0.7494543263305684e-1, (38, 2) = 0.7249746536306753e-2, (39, 0) = 0.7249746536306753e-2, (39, 1) = 0.879605827977988e-4, (39, 2) = 0.9587405509480346e-1, (40, 0) = 0.9587405509480346e-1, (40, 1) = 0.7634124261030573e-2, (40, 2) = 0.12931532215242276e-3, (41, 0) = 0.12931532215242276e-3, (41, 1) = .1198170451788002, (41, 2) = 0.8018501985754392e-2, (42, 0) = 0.8018501985754392e-2, (42, 1) = 0.18047754409459768e-3, (42, 2) = .14693305759519557, (43, 0) = .14693305759519557, (43, 1) = 0.8402879710478213e-2, (43, 2) = 0.2426963874347988e-3, (44, 0) = 0.2426963874347988e-3, (44, 1) = .1773741792709786, (44, 2) = 0.8879080143840423e-2, (45, 0) = 0.8879080143840423e-2, (45, 1) = 0.3370750131336099e-3, (45, 2) = .21991595426359736, (46, 0) = .21991595426359736, (46, 1) = 0.9355280577202632e-2, (46, 2) = 0.45303351835175153e-3, (47, 0) = 0.45303351835175153e-3, (47, 1) = .2680462076584402, (47, 2) = 0.983148101056484e-2, (48, 0) = 0.983148101056484e-2, (48, 1) = 0.5932898479173473e-3, (48, 2) = .322010979098003, (49, 0) = .322010979098003, (49, 1) = 0.1030768144392705e-1, (49, 2) = 0.7606779727081789e-3, (50, 0) = 0.7606779727081789e-3, (50, 1) = .38204015185501805, (50, 2) = 0.1095091293708549e-1, (51, 0) = 0.1095091293708549e-1, (51, 1) = 0.10350888578351128e-2, (51, 2) = .47312997922371214, (52, 0) = .47312997922371214, (52, 1) = 0.11594144430243928e-1, (52, 2) = 0.13719090205366264e-2, (53, 0) = 0.13719090205366264e-2, (53, 1) = .5761526004183525, (53, 2) = 0.12237375923402367e-1, (54, 0) = 0.12237375923402367e-1, (54, 1) = 0.17789435845996689e-2, (54, 2) = .6915360083421236, (55, 0) = .6915360083421236, (55, 1) = 0.12880607416560805e-1, (55, 2) = 0.22642684615889765e-2, (56, 0) = 0.22642684615889765e-2, (56, 1) = .8196577849906359, (56, 2) = 0.13423520709433145e-1, (57, 0) = 0.13423520709433145e-1, (57, 1) = 0.2740959521941423e-2, (57, 2) = .9379544346871757, (58, 0) = .9379544346871757, (58, 1) = 0.13966434002305485e-1, (58, 2) = 0.32844430498925503e-2, (59, 0) = 0.32844430498925503e-2, (59, 1) = 1.0657312055947692, (59, 2) = 0.14509347295177824e-1, (60, 0) = 0.14509347295177824e-1, (60, 1) = 0.3899903254586565e-2, (60, 2) = 1.2031365339671365, (61, 0) = 1.2031365339671365, (61, 1) = 0.15052260588050164e-1, (61, 2) = 0.4592603501928192e-2, (62, 0) = 0.4592603501928192e-2, (62, 1) = 1.3502975940755144, (62, 2) = 0.15610920749335589e-1, (63, 0) = 0.15610920749335589e-1, (63, 1) = 0.5391648122915063e-2, (63, 2) = 1.5120234519722362, (64, 0) = 1.5120234519722362, (64, 1) = 0.16169580910621012e-1, (64, 2) = 0.6283983129200919e-2, (65, 0) = 0.6283983129200919e-2, (65, 1) = 1.68428628636815, (65, 2) = 0.16728241071906437e-1, (66, 0) = 0.16728241071906437e-1, (66, 1) = 0.72755133412007395e-2, (66, 2) = 1.8671611177081684, (67, 0) = 1.8671611177081684, (67, 1) = 0.17286901233191862e-1, (67, 2) = 0.8372184499807455e-2, (68, 0) = 0.8372184499807455e-2, (68, 1) = 2.060704978518161, (68, 2) = 0.17883531896730483e-1, (69, 0) = 0.17883531896730483e-1, (69, 1) = 0.9666246281097544e-2, (69, 2) = 2.2792295962302402, (70, 0) = 2.2792295962302402, (70, 1) = 0.18480162560269103e-1, (70, 2) = 0.1109433875232118e-1, (71, 0) = 0.1109433875232118e-1, (71, 1) = 2.5099965282526657, (71, 2) = 0.19076793223807724e-1, (72, 0) = 0.19076793223807724e-1, (72, 1) = 0.12663766702872729e-1, (72, 2) = 2.75301711209446, (73, 0) = 2.75301711209446, (73, 1) = 0.19673423887346345e-1, (73, 2) = 0.14381841271869786e-1, (74, 0) = 0.14381841271869786e-1, (74, 1) = 3.008287407452325, (74, 2) = 0.20323135188130496e-1, (75, 0) = 0.20323135188130496e-1, (75, 1) = 0.16430399226728648e-1, (75, 2) = 3.300179783556811, (76, 0) = 3.300179783556811, (76, 1) = 0.20972846488914645e-1, (76, 2) = 0.1867330503798365e-1, (77, 0) = 0.1867330503798365e-1, (77, 1) = 3.606536127811025, (77, 2) = 0.21622557789698793e-1, (78, 0) = 0.21622557789698793e-1, (78, 1) = 0.211199360543433e-1, (78, 2) = 3.9273028539159713, (79, 0) = 3.9273028539159713, (79, 1) = 0.22272269090482945e-1, (79, 2) = 0.23779635564256878e-1, (80, 0) = 0.23779635564256878e-1, (80, 1) = 4.262415734136786, (80, 2) = 0.22992113493543874e-1, (81, 0) = 0.22992113493543874e-1, (81, 1) = 0.2698649724111607e-1, (81, 2) = 4.650365782145515, (82, 0) = 4.650365782145515, (82, 1) = 0.23711957896604803e-1, (82, 2) = 0.30478895457745632e-1, (83, 0) = 0.30478895457745632e-1, (83, 1) = 5.0557250258308315, (83, 2) = 0.24431802299665732e-1, (84, 0) = 0.24431802299665732e-1, (84, 1) = 0.3426931814588043e-1, (84, 2) = 5.4783775093722875, (85, 0) = 5.4783775093722875, (85, 1) = 0.2515164670272666e-1, (85, 2) = 0.38370173001849105e-1, (86, 0) = 0.38370173001849105e-1, (86, 1) = 5.918205047778416, (86, 2) = 0.2596499148480011e-1, (87, 0) = 0.2596499148480011e-1, (87, 1) = 0.43392688570393495e-1, (87, 2) = 6.43567861498111, (88, 0) = 6.43567861498111, (88, 1) = 0.26778336266873558e-1, (88, 2) = 0.48844884383816047e-1, (89, 0) = 0.48844884383816047e-1, (89, 1) = 6.974757330349238, (89, 2) = 0.27591681048947003e-1, (90, 0) = 0.27591681048947003e-1, (90, 1) = 0.5474426657391406e-1, (90, 2) = 7.535281136046706, (91, 0) = 7.535281136046706, (91, 1) = 0.2840502583102045e-1, (91, 2) = 0.61108219786563986e-1, (92, 0) = 0.61108219786563986e-1, (92, 1) = 8.117102832545454, (92, 2) = 0.2935001181749971e-1, (93, 0) = 0.2935001181749971e-1, (93, 1) = 0.6910848602627714e-1, (93, 2) = 8.819665700600305, (94, 0) = 8.819665700600305, (94, 1) = 0.30294997803978967e-1, (94, 2) = 0.7778608564212239e-1, (95, 0) = 0.7778608564212239e-1, (95, 1) = 9.55061882548943, (95, 2) = 0.31239983790458226e-1, (96, 0) = 0.31239983790458226e-1, (96, 1) = 0.8716778015274916e-1, (96, 2) = 10.309818728938088, (97, 0) = 10.309818728938088, (97, 1) = 0.3218496977693748e-1, (97, 2) = 0.9728021759799521e-1, (98, 0) = 0.9728021759799521e-1, (98, 1) = 11.097162680548227, (98, 2) = 0.3325952508096116e-1, (99, 0) = 0.3325952508096116e-1, (99, 1) = .10970084720798659, (99, 2) = 12.026581827773606, (100, 0) = 12.026581827773606, (100, 1) = 0.3433408038498484e-1, (100, 2) = .12313967279105634, (101, 0) = .12313967279105634, (101, 1) = 12.99229005359974, (101, 2) = 0.3540863568900852e-1, (102, 0) = 0.3540863568900852e-1, (102, 1) = .1376357189084802, (102, 2) = 13.99433770430688, (103, 0) = 13.99433770430688, (103, 1) = 0.364831909930322e-1, (103, 2) = .15322810929172334, (104, 0) = .15322810929172334, (104, 1) = 15.032849860807085, (104, 2) = 0.37591563429203334e-1, (105, 0) = 0.37591563429203334e-1, (105, 1) = .17050143098697276, (105, 2) = 16.142458465197507, (106, 0) = 16.142458465197507, (106, 1) = 0.3869993586537447e-1, (106, 2) = .1890263484154588, (107, 0) = .1890263484154588, (107, 1) = 17.291368422554786, (107, 2) = 0.3980830830154561e-1, (108, 0) = 0.3980830830154561e-1, (108, 1) = .20884662499954695, (108, 2) = 18.479948095052066, (109, 0) = 18.479948095052066, (109, 1) = 0.40916680737716746e-1, (109, 2) = .2300064850146772, (110, 0) = .2300064850146772, (110, 1) = 19.70863552374894, (110, 2) = 0.4206988566651697e-1, (111, 0) = 0.4206988566651697e-1, (111, 1) = .253492323964606, (111, 2) = 21.030137402899697, (112, 0) = 21.030137402899697, (112, 1) = 0.432230905953172e-1, (112, 2) = .2785277635917081, (113, 0) = .2785277635917081, (113, 1) = 22.396234681397083, (113, 2) = 0.44376295524117426e-1, (114, 0) = 0.44376295524117426e-1, (114, 1) = .3051646127867727, (114, 2) = 23.807611856851565, (115, 0) = 23.807611856851565, (115, 1) = 0.4552950045291765e-1, (115, 2) = .3334555309033527, (116, 0) = .3334555309033527, (116, 1) = 25.26499634474468, (116, 2) = 0.4673193229066352e-1, (117, 0) = 0.4673193229066352e-1, (117, 1) = .3647734100631351, (117, 2) = 26.83440963415839, (118, 0) = 26.83440963415839, (118, 1) = 0.4793436412840939e-1, (118, 2) = .39800941082023716, (119, 0) = .39800941082023716, (119, 1) = 28.455558153530134, (119, 2) = 0.49136795966155264e-1, (120, 0) = 0.49136795966155264e-1, (120, 1) = .4332262320482749, (120, 2) = 30.12932412151795, (121, 0) = 30.12932412151795, (121, 1) = 0.5033922780390113e-1, (121, 2) = .47048770062169115, (122, 0) = .47048770062169115, (122, 1) = 31.856566558430245, (122, 2) = 0.51595088991325394e-1, (123, 0) = 0.51595088991325394e-1, (123, 1) = .5116581129257987, (123, 2) = 33.71853834635488, (124, 0) = 33.71853834635488, (124, 1) = 0.5285095017874965e-1, (124, 2) = .5552045641361357, (125, 0) = .5552045641361357, (125, 1) = 35.640602197596294, (125, 2) = 0.541068113661739e-1, (126, 0) = 0.541068113661739e-1, (126, 1) = .6012028880252012, (126, 2) = 37.62348901461209, (127, 0) = 37.62348901461209, (127, 1) = 0.5536267255359816e-1, (127, 2) = .64972989944481, (128, 0) = .64972989944481, (128, 1) = 39.66776416387905, (128, 2) = 0.5658704357734733e-1, (129, 0) = 0.5658704357734733e-1, (129, 1) = .6995485035359791, (129, 2) = 41.720234713225004, (130, 0) = 41.720234713225004, (130, 1) = 0.578114146010965e-1, (130, 2) = .7519161464363067, (131, 0) = .7519161464363067, (131, 1) = 43.83153752177719, (131, 2) = 0.5903578562484567e-1, (132, 0) = 0.5903578562484567e-1, (132, 1) = .8069046230437601, (132, 2) = 46.001507869043884, (133, 0) = 46.001507869043884, (133, 1) = 0.6026015664859484e-1, (133, 2) = .8645855494566124, (134, 0) = .8645855494566124, (134, 1) = 48.229644040059505, (134, 2) = 0.6145638392268217e-1, (135, 0) = 0.6145638392268217e-1, (135, 1) = .9236089695851015, (135, 2) = 50.46190345441642, (136, 0) = 50.46190345441642, (136, 1) = 0.626526111967695e-1, (136, 2) = .9853348764285976, (137, 0) = .9853348764285976, (137, 1) = 52.74764286780915, (137, 2) = 0.6384883847085683e-1, (138, 0) = 0.6384883847085683e-1, (138, 1) = 1.0498260393836105, (138, 2) = 55.08516073823658, (139, 0) = 55.08516073823658, (139, 1) = 0.6504506574494415e-1, (139, 2) = 1.117143164978303, (140, 0) = 1.117143164978303, (140, 1) = 57.47221133984476, (140, 2) = 0.6623037772343757e-1, (141, 0) = 0.6623037772343757e-1, (141, 1) = 1.1866904582551383, (141, 2) = 59.883554501856935, (142, 0) = 59.883554501856935, (142, 1) = 0.6741568970193099e-1, (142, 2) = 1.2591215868447028, (143, 0) = 1.2591215868447028, (143, 1) = 62.33736112259146, (143, 2) = 0.686010016804244e-1, (144, 0) = 0.686010016804244e-1, (144, 1) = 1.3344841775663525, (144, 2) = 64.82949169101188, (145, 0) = 64.82949169101188, (145, 1) = 0.6978631365891783e-1, (145, 2) = 1.41282086002064, (146, 0) = 1.41282086002064, (146, 1) = 67.35500048201015, (146, 2) = 0.7100245812631775e-1, (147, 0) = 0.7100245812631775e-1, (147, 1) = 1.496324804127228, (147, 2) = 69.97481619123346, (148, 0) = 69.97481619123346, (148, 1) = 0.7221860259371769e-1, (148, 2) = 1.5830290098403168, (149, 0) = 1.5830290098403168, (149, 1) = 72.61638498495442, (149, 2) = 0.7343474706111763e-1, (150, 0) = 0.7343474706111763e-1, (150, 1) = 1.6729544554249776, (150, 2) = 75.27128894247686, (151, 0) = 75.27128894247686, (151, 1) = 0.7465089152851756e-1, (151, 2) = 1.7661116986280567, (152, 0) = 1.7661116986280567, (152, 1) = 77.92993239882547, (152, 2) = 0.7608558437817659e-1, (153, 0) = 0.7608558437817659e-1, (153, 1) = 1.8801624704413276, (153, 2) = 81.05640539731884, (154, 0) = 81.05640539731884, (154, 1) = 0.7752027722783561e-1, (154, 2) = 1.9986802387674878, (155, 0) = 1.9986802387674878, (155, 1) = 84.15297228724793, (155, 2) = 0.7895497007749464e-1, (156, 0) = 0.7895497007749464e-1, (156, 1) = 2.121605310334176, (156, 2) = 87.19726162392942, (157, 0) = 87.19726162392942, (157, 1) = 0.8038966292715366e-1, (157, 2) = 2.2488454279184906, (158, 0) = 2.2488454279184906, (158, 1) = 90.16476748218658, (158, 2) = 0.8133888453160523e-1, (159, 0) = 0.8133888453160523e-1, (159, 1) = 2.3353413629087894, (159, 2) = 92.07306508579853, (160, 0) = 92.07306508579853, (160, 1) = 0.8228810613605682e-1, (160, 2) = 2.4236240687486226, (161, 0) = 2.4236240687486226, (161, 1) = 93.92808087216304, (161, 2) = 0.832373277405084e-1, (162, 0) = 0.832373277405084e-1, (162, 1) = 2.513639038290123, (162, 2) = 95.72155799928035, (163, 0) = 95.72155799928035, (163, 1) = 0.8418654934495998e-1, (163, 2) = 2.6053239054326127, (164, 0) = 2.6053239054326127, (164, 1) = 97.44517199226271, (164, 2) = 0.8513577094941155e-1, (165, 0) = 0.8513577094941155e-1, (165, 1) = 2.6986083904325504, (165, 2) = 99.09060766321062, (166, 0) = 99.09060766321062, (166, 1) = 0.8608499255386313e-1, (166, 2) = 2.793414329658189, (167, 0) = 2.793414329658189, (167, 1) = 100.64963281500431, (167, 2) = 0.8703421415831472e-1, (168, 0) = 0.8703421415831472e-1, (168, 1) = 2.889656113256554, (168, 2) = 102.11441622745653, (169, 0) = 102.11441622745653, (169, 1) = 0.879834357627663e-1, (169, 2) = 2.987240766434155, (170, 0) = 2.987240766434155, (170, 1) = 103.47768600464273, (170, 2) = 0.8887257904235711e-1, (171, 0) = 0.8887257904235711e-1, (171, 1) = 3.0797785817321657, (171, 2) = 104.65671218068343, (172, 0) = 104.65671218068343, (172, 1) = 0.8976172232194793e-1, (172, 2) = 3.1733206376889904, (173, 0) = 3.1733206376889904, (173, 1) = 105.73610195901804, (173, 2) = 0.9065086560153875e-1, (174, 0) = 0.9065086560153875e-1, (174, 1) = 3.267776863887372, (174, 2) = 106.7120208062077, (175, 0) = 106.7120208062077, (175, 1) = 0.9154000888112956e-1, (175, 2) = 3.3630539195002447, (176, 0) = 3.3630539195002447, (176, 1) = 107.5816927421004, (176, 2) = 0.9242441511319593e-1, (177, 0) = 0.9242441511319593e-1, (177, 1) = 3.458542858776289, (177, 2) = 108.33970241853716, (178, 0) = 108.33970241853716, (178, 1) = 0.933088213452623e-1, (178, 2) = 3.554654641757956, (179, 0) = 3.554654641757956, (179, 1) = 108.99061655016267, (179, 2) = 0.9419322757732866e-1, (180, 0) = 0.9419322757732866e-1, (180, 1) = 3.6512953613878967, (180, 2) = 109.53553521059094, (181, 0) = 109.53553521059094, (181, 1) = 0.9507763380939503e-1, (181, 2) = 3.7483722928224514, (182, 0) = 3.7483722928224514, (182, 1) = 109.97709093683262, (182, 2) = 0.9608235667240475e-1, (183, 0) = 0.9608235667240475e-1, (183, 1) = 3.859070733457438, (183, 2) = 110.35867604433525, (184, 0) = 110.35867604433525, (184, 1) = 0.9708707953541448e-1, (184, 2) = 3.970091512199092, (185, 0) = 3.970091512199092, (185, 1) = 110.6210635373472, (185, 2) = 0.980918023984242e-1, (186, 0) = 0.980918023984242e-1, (186, 1) = 4.081321244616854, (186, 2) = 110.7757007991817, (187, 0) = 110.7757007991817, (187, 1) = 0.9909652526143392e-1, (187, 2) = 4.192658462679894, (188, 0) = 4.192658462679894, (188, 1) = 110.83665098959017, (188, 2) = .10014199249128533, (189, 0) = .10014199249128533, (189, 1) = 4.308531312675858, (189, 2) = 110.81843355974247, (190, 0) = 110.81843355974247, (190, 1) = .10118745972113674, (190, 2) = 4.424350471182151, (191, 0) = 4.424350471182151, (191, 1) = 110.73814436247946, (191, 2) = .10223292695098816, (192, 0) = .10223292695098816, (192, 1) = 4.540063771579773, (192, 2) = 110.61931262382137, (193, 0) = 110.61931262382137, (193, 1) = .10327839418083958, (193, 2) = 4.65564392881655, (194, 0) = 4.65564392881655, (194, 1) = 110.48716567336754, (194, 2) = .10411988169561276, (195, 0) = .10411988169561276, (195, 1) = 4.748575187874751, (195, 2) = 110.38922513721197, (196, 0) = 110.38922513721197, (196, 1) = .10496136921038593, (196, 2) = 4.841432697893804, (197, 0) = 4.841432697893804, (197, 1) = 110.3142826238991, (197, 2) = .10580285672515911, (198, 0) = .10580285672515911, (198, 1) = 4.934241872222081, (198, 2) = 110.27677064508264, (199, 0) = 110.27677064508264, (199, 1) = .1066443442399323, (199, 2) = 5.027040261156286, (200, 0) = 5.027040261156286, (200, 1) = 110.29105674233996, (200, 2) = .10748823343266461, (201, 0) = .10748823343266461, (201, 1) = 5.1201425809378005, (201, 2) = 110.37170038361644, (202, 0) = 110.37170038361644, (202, 1) = .10833212262539695, (202, 2) = 5.213345981080991, (203, 0) = 5.213345981080991, (203, 1) = 110.5327708365935, (203, 2) = .10917601181812928, (204, 0) = .10917601181812928, (204, 1) = 5.306723907689875, (204, 2) = 110.78777668022703, (205, 0) = 110.78777668022703, (205, 1) = .11001990101086159, (205, 2) = 5.400361130050291, (206, 0) = 5.400361130050291, (206, 1) = 111.14961332370335, (206, 2) = .11097704955613187, (207, 0) = .11097704955613187, (207, 1) = 5.5070006844235335, (207, 2) = 111.70476380917276, (208, 0) = 111.70476380917276, (208, 1) = .11193419810140215, (208, 2) = 5.614251764599648, (209, 0) = 5.614251764599648, (209, 1) = 112.42975931791496, (209, 2) = .11289134664667244, (210, 0) = .11289134664667244, (210, 1) = 5.72228395141693, (210, 2) = 113.33994437713048, (211, 0) = 113.33994437713048, (211, 1) = .11384849519194272, (211, 2) = 5.831281336584734, (212, 0) = 5.831281336584734, (212, 1) = 114.4492947361824, (212, 2) = .11484759532103979, (213, 0) = .11484759532103979, (213, 1) = 5.946299397294085, (213, 2) = 115.83336800307299, (214, 0) = 115.83336800307299, (214, 1) = .11584669545013686, (214, 2) = 6.0628210976078325, (215, 0) = 6.0628210976078325, (215, 1) = 117.46094168419745, (215, 2) = .11684579557923393, (216, 0) = .11684579557923393, (216, 1) = 6.181094979875501, (216, 2) = 119.34329502221658, (217, 0) = 119.34329502221658, (217, 1) = .117844895708331, (217, 2) = 6.301380719290646, (218, 0) = 6.301380719290646, (218, 1) = 121.49033521538948, (218, 2) = .11884399583742807, (219, 0) = .11884399583742807, (219, 1) = 6.423947824886709, (219, 2) = 123.91067198926574, (220, 0) = 123.91067198926574, (220, 1) = .11984309596652513, (220, 2) = 6.549072838939048, (221, 0) = 6.549072838939048, (221, 1) = 126.61171582796682, (221, 2) = .1208421960956222, (222, 0) = .1208421960956222, (222, 1) = 6.677039193035116, (222, 2) = 129.59994866056087, (223, 0) = 129.59994866056087, (223, 1) = .12184129622471927, (223, 2) = 6.808136794626377, (224, 0) = 6.808136794626377, (224, 1) = 132.88105591778663, (224, 2) = .1228258871303874, (225, 0) = .1228258871303874, (225, 1) = 6.940681689157146, (225, 2) = 136.40587371716686, (226, 0) = 136.40587371716686, (226, 1) = .12381047803605555, (226, 2) = 7.0768412444672535, (227, 0) = 7.0768412444672535, (227, 1) = 140.22413151213277, (227, 2) = .12479506894172368, (228, 0) = .12479506894172368, (228, 1) = 7.21690613401665, (228, 2) = 144.33967783665653, (229, 0) = 144.33967783665653, (229, 1) = .1257796598473918, (229, 2) = 7.361170935674048, (230, 0) = 7.361170935674048, (230, 1) = 148.75612357712635, (230, 2) = .1268172469555592, (231, 0) = .1268172469555592, (231, 1) = 7.51807450812241, (231, 2) = 153.7396889327021, (232, 0) = 153.7396889327021, (232, 1) = .12785483406372658, (232, 2) = 7.680325951679255, (233, 0) = 7.680325951679255, (233, 1) = 159.06511519388064, (233, 2) = .12889242117189398, (234, 0) = .12889242117189398, (234, 1) = 7.8482819474795305, (234, 2) = 164.7363690555636, (235, 0) = 164.7363690555636, (235, 1) = .12993000828006135, (235, 2) = 8.022303554742969, (236, 0) = 8.022303554742969, (236, 1) = 170.7575770697361, (236, 2) = .1310732720492926, (237, 0) = .1310732720492926, (237, 1) = 8.22151045371536, (237, 2) = 177.80245442810445, (238, 0) = 177.80245442810445, (238, 1) = .13221653581852383, (238, 2) = 8.42902017613363, (239, 0) = 8.42902017613363, (239, 1) = 185.28361500079794, (239, 2) = .13335979958775507, (240, 0) = .13335979958775507, (240, 1) = 8.645335283377188, (240, 2) = 193.20776934122952, (241, 0) = 193.20776934122952, (241, 1) = .1345030633569863, (241, 2) = 8.870966562920325, (242, 0) = 8.870966562920325, (242, 1) = 201.58225154204246, (242, 2) = .13583166632874244, (243, 0) = .13583166632874244, (243, 1) = 9.145568313730873, (243, 2) = 211.89070544352353, (244, 0) = 211.89070544352353, (244, 1) = .13716026930049857, (244, 2) = 9.434284292269332, (245, 0) = 9.434284292269332, (245, 1) = 222.83181478413974, (245, 2) = .1384888722722547, (246, 0) = .1384888722722547, (246, 1) = 9.73796490871873, (246, 2) = 234.4205989558001, (247, 0) = 234.4205989558001, (247, 1) = .13981747524401084, (247, 2) = 10.057481887484302, (248, 0) = 10.057481887484302, (248, 1) = 246.67343837697507, (248, 2) = .14126570192323573, (249, 0) = .14126570192323573, (249, 1) = 10.424856415235485, (249, 2) = 260.80671977602054, (250, 0) = 260.80671977602054, (250, 1) = .14271392860246063, (250, 2) = 10.813300084954669, (251, 0) = 10.813300084954669, (251, 1) = 275.7747872613617, (251, 2) = .14416215528168552, (252, 0) = .14416215528168552, (252, 1) = 11.224040500215104, (252, 2) = 291.60398100989613, (253, 0) = 291.60398100989613, (253, 1) = .14561038196091042, (253, 2) = 11.658345678368356, (254, 0) = 11.658345678368356, (254, 1) = 308.32213071351407, (254, 2) = .1470542655892233, (255, 0) = .1470542655892233, (255, 1) = 12.11610929862614, (255, 2) = 325.90408139498857, (256, 0) = 325.90408139498857, (256, 1) = .14849814921753618, (256, 2) = 12.59993579231736, (257, 0) = 12.59993579231736, (257, 1) = 344.42863639202915, (257, 2) = .14994203284584903, (258, 0) = .14994203284584903, (258, 1) = 13.111207115957406, (258, 2) = 363.9262453649068, (259, 0) = 363.9262453649068, (259, 1) = .1513859164741619, (259, 2) = 13.651351461777951, (260, 0) = 13.651351461777951, (260, 1) = 384.4274351705777, (260, 2) = .15273637101982201, (261, 0) = .15273637101982201, (261, 1) = 14.183979504179, (261, 2) = 404.53713366860484, (262, 0) = 404.53713366860484, (262, 1) = .1540868255654821, (262, 2) = 14.74438921728897, (263, 0) = 14.74438921728897, (263, 1) = 425.57539255977866, (263, 2) = .1554372801111422, (264, 0) = .1554372801111422, (264, 1) = 15.333848310421521, (264, 2) = 447.564832419329, (265, 0) = 447.564832419329, (265, 1) = .1567877346568023, (265, 2) = 15.953656490951275, (266, 0) = 15.953656490951275, (266, 1) = 470.52598814019404, (266, 2) = .15803238502745792, (267, 0) = .15803238502745792, (267, 1) = 16.552924714002508, (267, 2) = 492.5642742260262, (268, 0) = 492.5642742260262, (268, 1) = .15927703539811355, (268, 2) = 17.180152745720427, (269, 0) = 17.180152745720427, (269, 1) = 515.4549535242854, (269, 2) = .1605216857687692, (270, 0) = .1605216857687692, (270, 1) = 17.83640472392934, (270, 2) = 539.2061722828935, (271, 0) = 539.2061722828935, (271, 1) = .16176633613942484, (271, 2) = 18.522755529590363, (272, 0) = 18.522755529590363, (272, 1) = 563.8219498181656, (272, 2) = .1629444067877777, (273, 0) = .1629444067877777, (273, 1) = 19.201095343237206, (273, 2) = 587.9168832222862, (274, 0) = 587.9168832222862, (274, 1) = .16412247743613056, (274, 2) = 19.908275171843812, (275, 0) = 19.908275171843812, (275, 1) = 612.7808083809373, (275, 2) = .1653005480844834, (276, 0) = .1653005480844834, (276, 1) = 20.64519226178667, (276, 2) = 638.4029262950463, (277, 0) = 638.4029262950463, (277, 1) = .16647861873283626, (277, 2) = 21.412731052700696, (278, 0) = 21.412731052700696, (278, 1) = 664.7659493264136, (278, 2) = .167619413788986, (279, 0) = .167619413788986, (279, 1) = 22.185982309404032, (279, 2) = 690.9780140239461, (280, 0) = 690.9780140239461, (280, 1) = .16876020884513576, (280, 2) = 22.989507064119316, (281, 0) = 22.989507064119316, (281, 1) = 717.8327298815063, (281, 2) = .16990100390128554, (282, 0) = .16990100390128554, (282, 1) = 23.824013855367678, (282, 2) = 745.2921514856275, (283, 0) = 745.2921514856275, (283, 1) = .1710417989574353, (283, 2) = 24.690167136534775, (284, 0) = 24.690167136534775, (284, 1) = 773.3089169249303, (284, 2) = .17218559023634622, (285, 0) = .17218559023634622, (285, 1) = 25.5909791491693, (285, 2) = 801.9011608449952, (286, 0) = 801.9011608449952, (286, 1) = .17332938151525717, (286, 2) = 26.524751564280084, (287, 0) = 26.524751564280084, (287, 1) = 830.9269025227654, (287, 2) = .17447317279416813, (288, 0) = .17447317279416813, (288, 1) = 27.49192942258245, (288, 2) = 860.304273881365, (289, 0) = 860.304273881365, (289, 1) = .17561696407307906, (289, 2) = 28.492862417666576, (290, 0) = 28.492862417666576, (290, 1) = 889.9379720686193, (290, 2) = .1768775055273268, (291, 0) = .1768775055273268, (291, 1) = 29.63534563857709, (291, 2) = 922.7623378282941, (292, 0) = 922.7623378282941, (292, 1) = .17813804698157454, (292, 2) = 30.81923433844416, (293, 0) = 30.81923433844416, (293, 1) = 955.5977408132247, (293, 2) = .1793985884358223, (294, 0) = .1793985884358223, (294, 1) = 32.04441608995515, (294, 2) = 988.2544986125317, (295, 0) = 988.2544986125317, (295, 1) = .18065912989007005, (295, 2) = 33.310535666436095, (296, 0) = 33.310535666436095, (296, 1) = 1020.5212814694421, (296, 2) = .18159822222964242, (297, 0) = .18159822222964242, (297, 1) = 34.28003414533035, (297, 2) = 1044.1688120719527, (298, 0) = 1044.1688120719527, (298, 1) = .1825373145692148, (298, 2) = 35.27153795276432, (299, 0) = 35.27153795276432, (299, 1) = 1067.367699875199, (299, 2) = .1834764069087872, (300, 0) = .1834764069087872, (300, 1) = 36.28457413273854, (300, 2) = 1090.0097199000809, (301, 0) = 1090.0097199000809, (301, 1) = .18441549924835957, (301, 2) = 37.31856750855757, (302, 0) = 37.31856750855757, (302, 1) = 1111.9829744088815, (302, 2) = .18535459158793194, (303, 0) = .18535459158793194, (303, 1) = 38.37283744799259, (303, 2) = 1133.1727652582092, (304, 0) = 1133.1727652582092, (304, 1) = .18629368392750434, (304, 2) = 39.44659261929887, (305, 0) = 39.44659261929887, (305, 1) = 1153.4622562487607, (305, 2) = .1872327762670767, (306, 0) = .1872327762670767, (306, 1) = 40.53893434856023, (306, 2) = 1172.7363311853112, (307, 0) = 1172.7363311853112, (307, 1) = .18817186860664908, (307, 2) = 41.648856251960666, (308, 0) = 41.648856251960666, (308, 1) = 1190.883531466396, (308, 2) = .18903904916401412, (309, 0) = .18903904916401412, (309, 1) = 42.6884377178276, (309, 2) = 1206.548097329997, (310, 0) = 1206.548097329997, (310, 1) = .18990622972137916, (310, 2) = 43.74111753778738, (311, 0) = 43.74111753778738, (311, 1) = 1221.0825515938413, (311, 2) = .19077341027874423, (312, 0) = .19077341027874423, (312, 1) = 44.80588769991621, (312, 2) = 1234.417295466171, (313, 0) = 1234.417295466171, (313, 1) = .19164059083610926, (313, 2) = 45.88168115337819, (314, 0) = 45.88168115337819, (314, 1) = 1246.4931860015906, (314, 2) = .19248494983078182, (315, 0) = .19248494983078182, (315, 1) = 46.93869076177756, (315, 2) = 1256.9961872276042, (316, 0) = 1256.9961872276042, (316, 1) = .19332930882545438, (316, 2) = 48.004032490870486, (317, 0) = 48.004032490870486, (317, 1) = 1266.2287737370689, (317, 2) = .19417366782012693, (318, 0) = .19417366782012693, (318, 1) = 49.07662965795563, (318, 2) = 1274.1739317189606, (319, 0) = 1274.1739317189606, (319, 1) = .1950180268147995, (319, 2) = 50.15539326201277, (320, 0) = 50.15539326201277, (320, 1) = 1280.8304873246423, (320, 2) = .19593171007690702, (321, 0) = .19593171007690702, (321, 1) = 51.32841333799885, (321, 2) = 1286.5998165644587, (322, 0) = 1286.5998165644587, (322, 1) = .19684539333901457, (322, 2) = 52.50603620415373, (323, 0) = 52.50603620415373, (323, 1) = 1290.9209556664705, (323, 2) = .1977590766011221, (324, 0) = .1977590766011221, (324, 1) = 53.68697497233516, (324, 2) = 1293.861524247951, (325, 0) = 1293.861524247951, (325, 1) = .19867275986322963, (325, 2) = 54.870008115689004, (326, 0) = 54.870008115689004, (326, 1) = 1295.5140416096642, (326, 2) = .1995034946195085, (327, 0) = .1995034946195085, (327, 1) = 55.946500077992184, (327, 2) = 1295.9964795300593, (328, 0) = 1295.9964795300593, (328, 1) = .2003342293757874, (328, 2) = 57.02302513738758, (329, 0) = 57.02302513738758, (329, 1) = 1295.6157176008367, (329, 2) = .2011649641320663, (330, 0) = .2011649641320663, (330, 1) = 58.09891835026868, (330, 2) = 1294.4911717351215, (331, 0) = 1294.4911717351215, (331, 1) = .20199569888834518, (331, 2) = 59.173615670132186, (332, 0) = 59.173615670132186, (332, 1) = 1292.7550870025732, (332, 2) = .20282643364462405, (333, 0) = .20282643364462405, (333, 1) = 60.246663864443775, (333, 2) = 1290.5514499370029, (334, 0) = 1290.5514499370029, (334, 1) = .20365716840090295, (334, 2) = 61.317739859384154, (335, 0) = 61.317739859384154, (335, 1) = 1288.0345615349502, (335, 2) = .20448790315718185, (336, 0) = .20448790315718185, (336, 1) = 62.38665107837663, (336, 2) = 1285.3651107917551, (337, 0) = 1285.3651107917551, (337, 1) = .20531863791346072, (337, 2) = 63.45333902354682, (338, 0) = 63.45333902354682, (338, 1) = 1282.708253855004, (338, 2) = .20612405801648662, (339, 0) = .20612405801648662, (339, 1) = 64.48547318537929, (339, 2) = 1280.303582566655, (340, 0) = 1280.303582566655, (340, 1) = .2069294781195125, (340, 2) = 65.51579129069823, (341, 0) = 65.51579129069823, (341, 1) = 1278.2235912477088, (341, 2) = .2077348982225384, (342, 0) = .2077348982225384, (342, 1) = 66.54461563038882, (342, 2) = 1276.6207966271204, (343, 0) = 1276.6207966271204, (343, 1) = .2085403183255643, (343, 2) = 67.57239092098533, (344, 0) = 67.57239092098533, (344, 1) = 1275.6443593182291, (344, 2) = .20938722488283976, (345, 0) = .20938722488283976, (345, 1) = 68.65259563789651, (345, 2) = 1275.4523392920792, (346, 0) = 1275.4523392920792, (346, 1) = .2102341314401152, (346, 2) = 69.73305735698426, (347, 0) = 69.73305735698426, (347, 1) = 1276.2753522818593, (347, 2) = .21108103799739067, (348, 0) = .21108103799739067, (348, 1) = 70.81469871378525, (348, 2) = 1278.2668067981385, (349, 0) = 1278.2668067981385, (349, 1) = .21192794455466613, (349, 2) = 71.89857109721108, (350, 0) = 71.89857109721108, (350, 1) = 1281.5706038978306, (350, 2) = .21295764524180646, (351, 0) = .21295764524180646, (351, 1) = 73.22109116189871, (351, 2) = 1287.5482814321174, (352, 0) = 1287.5482814321174, (352, 1) = .2139873459289468, (352, 2) = 74.55096528026552, (353, 0) = 74.55096528026552, (353, 1) = 1295.8837462905256, (353, 2) = .21501704661608714, (354, 0) = .21501704661608714, (354, 1) = 75.89071685190187, (354, 2) = 1306.7741771082583, (355, 0) = 1306.7741771082583, (355, 1) = .21604674730322748, (355, 2) = 77.24306943476793, (356, 0) = 77.24306943476793, (356, 1) = 1320.3943609114253, (356, 2) = .21717402116645695, (357, 0) = .21717402116645695, (357, 1) = 78.74145507447734, (357, 2) = 1338.6160474891813, (358, 0) = 1338.6160474891813, (358, 1) = .2183012950296864, (358, 2) = 80.26241416307897, (359, 0) = 80.26241416307897, (359, 1) = 1360.4652038647664, (359, 2) = .21942856889291584, (360, 0) = .21942856889291584, (360, 1) = 81.81011217058614, (360, 2) = 1386.0898944837184, (361, 0) = 1386.0898944837184, (361, 1) = .22055584275614531, (361, 2) = 83.3888799792391, (362, 0) = 83.3888799792391, (362, 1) = 1415.616685305146, (362, 2) = .22152260329660595, (363, 0) = .22152260329660595, (363, 1) = 84.77098540231123, (363, 2) = 1444.1288298621394, (364, 0) = 1444.1288298621394, (364, 1) = .22248936383706658, (364, 2) = 86.18210478082088, (365, 0) = 86.18210478082088, (365, 1) = 1475.6477055972343, (365, 2) = .22345612437752718, (366, 0) = .22345612437752718, (366, 1) = 87.62516808175275, (366, 2) = 1510.2252424587884, (367, 0) = 1510.2252424587884, (367, 1) = .2244228849179878, (367, 2) = 89.10315626906832, (368, 0) = 89.10315626906832, (368, 1) = 1547.9086824256367, (368, 2) = .22540828260825657, (369, 0) = .22540828260825657, (369, 1) = 90.64871364512015, (369, 2) = 1589.5596995522403, (370, 0) = 1589.5596995522403, (370, 1) = .22639368029852536, (370, 2) = 92.23694344849108, (371, 0) = 92.23694344849108, (371, 1) = 1634.525517162914, (371, 2) = .22737907798879414, (372, 0) = .22737907798879414, (372, 1) = 93.87113198491402, (372, 2) = 1682.848538115276, (373, 0) = 1682.848538115276, (373, 1) = .2283644756790629, (373, 2) = 95.55460938903121, (374, 0) = 95.55460938903121, (374, 1) = 1734.5715917055431, (374, 2) = .22942284966539345, (375, 0) = .22942284966539345, (375, 1) = 97.42151172941401, (375, 2) = 1793.9622111708384, (376, 0) = 1793.9622111708384, (376, 1) = .23048122365172402, (376, 2) = 99.353397921403, (377, 0) = 99.353397921403, (377, 1) = 1857.3823437737597, (377, 2) = .23153959763805457, (378, 0) = .23153959763805457, (378, 1) = 101.35456469712467, (378, 2) = 1924.8931514955382, (379, 0) = 1924.8931514955382, (379, 1) = .2325979716243851, (379, 2) = 103.42937762283626, (380, 0) = 103.42937762283626, (380, 1) = 1996.5611639019062, (380, 2) = .23377535328326376, (381, 0) = .23377535328326376, (381, 1) = 105.82943696151227, (381, 2) = 2081.260815683415, (382, 0) = 2081.260815683415, (382, 1) = .23495273494214242, (382, 2) = 108.33235228573695, (383, 0) = 108.33235228573695, (383, 1) = 2171.303691744349, (383, 2) = .2361301166010211, (384, 0) = .2361301166010211, (384, 1) = 110.9444857299478, (384, 2) = 2266.8102500301125, (385, 0) = 2266.8102500301125, (385, 1) = .23730749825989975, (385, 2) = 113.67234967727556, (386, 0) = 113.67234967727556, (386, 1) = 2367.9131863593725, (386, 2) = .2386833748678849, (387, 0) = .2386833748678849, (387, 1) = 117.01568788717357, (387, 2) = 2493.3471570575184, (388, 0) = 2493.3471570575184, (388, 1) = .2400592514758701, (388, 2) = 120.53714085395411, (389, 0) = 120.53714085395411, (389, 1) = 2626.8762520165606, (389, 2) = .24143512808385525, (390, 0) = .24143512808385525, (390, 1) = 124.24803847929122, (390, 2) = 2768.779201376044, (391, 0) = 2768.779201376044, (391, 1) = .24281100469184042, (391, 2) = 128.1601150748452, (392, 0) = 128.1601150748452, (392, 1) = 2919.359626486487, (392, 2) = .24426739284110574, (393, 0) = .24426739284110574, (393, 1) = 132.5338117425088, (393, 2) = 3088.5701746547243, (394, 0) = 3088.5701746547243, (394, 1) = .24572378099037107, (394, 2) = 137.16153809285197, (395, 0) = 137.16153809285197, (395, 1) = 3268.290219497763, (395, 2) = .2471801691396364, (396, 0) = .2471801691396364, (396, 1) = 142.0589217966556, (396, 2) = 3458.966242643104, (397, 0) = 3458.966242643104, (397, 1) = .24863655728890172, (397, 2) = 147.24227167053613, (398, 0) = 147.24227167053613, (398, 1) = 3661.070381733814, (398, 2) = .2500765585801312, (399, 0) = .2500765585801312, (399, 1) = 152.6651164790699, (399, 2) = 3872.621727579742, (400, 0) = 3872.621727579742, (400, 1) = .2515165598713608, (400, 2) = 158.40128305061194, (401, 0) = 158.40128305061194, (401, 1) = 4096.329743530422, (401, 2) = .2529565611625903, (402, 0) = .2529565611625903, (402, 1) = 164.46863296259593, (402, 2) = 4332.70543426198, (403, 0) = 4332.70543426198, (403, 1) = .2543965624538198, (403, 2) = 170.88579462923676, (404, 0) = 170.88579462923676, (404, 1) = 4582.265838758704, (404, 2) = .2557033760070104, (405, 0) = .2557033760070104, (405, 1) = 177.02844437227205, (405, 2) = 4820.590809225206, (406, 0) = 4820.590809225206, (406, 1) = .25701018956020094, (406, 2) = 183.4901249912879, (407, 0) = 183.4901249912879, (407, 1) = 5070.585063707809, (407, 2) = .25831700311339156, (408, 0) = .25831700311339156, (408, 1) = 190.28630805111408, (408, 2) = 5332.615709721448, (409, 0) = 5332.615709721448, (409, 1) = .25962381666658213, (409, 2) = 197.4329626802184, (410, 0) = 197.4329626802184, (410, 1) = 5607.026212412612, (410, 2) = .26081352425295345, (411, 0) = .26081352425295345, (411, 1) = 204.25780742886354, (411, 2) = 5867.877140140835, (412, 0) = 5867.877140140835, (412, 1) = .2620032318393247, (412, 2) = 211.3993572807448, (413, 0) = 211.3993572807448, (413, 1) = 6139.457733737788, (413, 2) = .26319293942569605, (414, 0) = .26319293942569605, (414, 1) = 218.8704544688837, (414, 2) = 6421.934409407223, (415, 0) = 6421.934409407223, (415, 1) = .2643826470120674, (415, 2) = 226.68414650421124, (416, 0) = 226.68414650421124, (416, 1) = 6715.423218471578, (416, 2) = .26549843909379744, (417, 0) = .26549843909379744, (417, 1) = 234.33544169599776, (417, 2) = 7000.738159594487, (418, 0) = 7000.738159594487, (418, 1) = .26661423117552746, (418, 2) = 242.31053140030951, (419, 0) = 242.31053140030951, (419, 1) = 7295.7857570945425, (419, 2) = .2677300232572575, (420, 0) = .2677300232572575, (420, 1) = 250.62020513575055, (420, 2) = 7600.489262405348, (421, 0) = 7600.489262405348, (421, 1) = .26884581533898755, (421, 2) = 259.2751659772788, (422, 0) = 259.2751659772788, (422, 1) = 7914.691831851301, (422, 2) = .2699201675756128, (423, 0) = .2699201675756128, (423, 1) = 267.94480062751086, (423, 2) = 8225.974105956382, (424, 0) = 8225.974105956382, (424, 1) = .2709945198122381, (424, 2) = 276.9533521032163, (425, 0) = 276.9533521032163, (425, 1) = 8545.530141033778, (425, 2) = .27206887204886343, (426, 0) = .27206887204886343, (426, 1) = 286.3094526901696, (426, 2) = 8872.946721857088, (427, 0) = 8872.946721857088, (427, 1) = .2731432242854887, (427, 2) = 296.02128185149564, (428, 0) = 296.02128185149564, (428, 1) = 9207.694081441576, (428, 2) = .27421750880923573, (429, 0) = .27421750880923573, (429, 1) = 306.09579494125023, (429, 2) = 9549.096517875558, (430, 0) = 9549.096517875558, (430, 1) = .27529179333298276, (430, 2) = 316.5403293808994, (431, 0) = 316.5403293808994, (431, 1) = 9896.38125232285, (431, 2) = .2763660778567298, (432, 0) = .2763660778567298, (432, 1) = 327.360646261209, (432, 2) = 10248.596282604158, (433, 0) = 10248.596282604158, (433, 1) = .2774403623804768, (433, 2) = 338.56146478216914, (434, 0) = 338.56146478216914, (434, 1) = 10604.622596180883, (434, 2) = .278623524144049, (435, 0) = .278623524144049, (435, 1) = 351.3419181080146, (435, 2) = 10999.59288444602, (436, 0) = 10999.59288444602, (436, 1) = .27980668590762114, (436, 2) = 364.59056175607543, (437, 0) = 364.59056175607543, (437, 1) = 11395.624073131614, (437, 2) = .2809898476711933, (438, 0) = .2809898476711933, (438, 1) = 378.30722618467775, (438, 2) = 11790.445045360359, (439, 0) = 11790.445045360359, (439, 1) = .2821730094347655, (439, 2) = 392.48901256387603, (440, 0) = 392.48901256387603, (440, 1) = 12181.511606887298, (440, 2) = .28306300632280784, (441, 0) = .28306300632280784, (441, 1) = 403.4598958049087, (441, 2) = 12471.515073341732, (442, 0) = 12471.515073341732, (442, 1) = .28395300321085015, (442, 2) = 414.6867583578623, (443, 0) = 414.6867583578623, (443, 1) = 12756.51254334674, (443, 2) = .2848430000988925, (444, 0) = .2848430000988925, (444, 1) = 426.16452881540476, (444, 2) = 13035.14473323342, (445, 0) = 13035.14473323342, (445, 1) = .2857329969869349, (445, 2) = 437.88691854692394, (446, 0) = 437.88691854692394, (446, 1) = 13306.000292351384, (446, 2) = .28662299387497725, (447, 0) = .28662299387497725, (447, 1) = 449.8463779594025, (447, 2) = 13567.626235895515, (448, 0) = 13567.626235895515, (448, 1) = .2875129907630196, (448, 2) = 462.0340224226368, (449, 0) = 462.0340224226368, (449, 1) = 13818.536152804525, (449, 2) = .2884029876510619, (450, 0) = .2884029876510619, (450, 1) = 474.43966678509946, (450, 2) = 14057.256391893045, (451, 0) = 14057.256391893045, (451, 1) = .2892929845391043, (451, 2) = 487.05181783169627, (452, 0) = 487.05181783169627, (452, 1) = 14282.3491540896, (452, 2) = .2901174155375901, (453, 0) = .2901174155375901, (453, 1) = 498.9079869518289, (453, 2) = 14477.493311942455, (454, 0) = 14477.493311942455, (454, 1) = .2909418465360758, (454, 2) = 510.91935620029915, (455, 0) = 510.91935620029915, (455, 1) = 14658.712486080716, (455, 2) = .2917662775345616, (456, 0) = .2917662775345616, (456, 1) = 523.0740809508618, (456, 2) = 14825.064977964885, (457, 0) = 14825.064977964885, (457, 1) = .29259070853304736, (457, 2) = 535.3595544708189, (458, 0) = 535.3595544708189, (458, 1) = 14975.72890452385, (458, 2) = .29339687483362936, (459, 0) = .29339687483362936, (459, 1) = 547.486547316057, (459, 2) = 15107.220766809107, (460, 0) = 15107.220766809107, (460, 1) = .29420304113421136, (460, 2) = 559.7130401765992, (461, 0) = 559.7130401765992, (461, 1) = 15222.550670138033, (461, 2) = .2950092074347934, (462, 0) = .2950092074347934, (462, 1) = 572.0259077337587, (462, 2) = 15321.391153589044, (463, 0) = 15321.391153589044, (463, 1) = .2958153737353754, (463, 2) = 584.4117837163598, (464, 0) = 584.4117837163598, (464, 1) = 15403.601619801591, (464, 2) = .2966899563165042, (465, 0) = .2966899563165042, (465, 1) = 597.91570989303, (465, 2) = 15474.047093638455, (466, 0) = 15474.047093638455, (466, 1) = .2975645388976329, (466, 2) = 611.4727997297264, (467, 0) = 611.4727997297264, (467, 1) = 15525.331075179789, (467, 2) = .29843912147876167, (468, 0) = .29843912147876167, (468, 1) = 625.066638226198, (468, 2) = 15558.09742708208, (469, 0) = 15558.09742708208, (469, 1) = .29931370405989044, (469, 2) = 638.6814144485186, (470, 0) = 638.6814144485186, (470, 1) = 15573.28728516297, (470, 2) = .3001170867491531, (471, 0) = .3001170867491531, (471, 1) = 651.1934092666994, (471, 2) = 15572.803120325827, (472, 0) = 15572.803120325827, (472, 1) = .30092046943841577, (472, 2) = 663.6998433334561, (473, 0) = 663.6998433334561, (473, 1) = 15559.694701119168, (473, 2) = .30172385212767844, (474, 0) = .30172385212767844, (474, 1) = 676.1911465983748, (474, 2) = 15535.30941709627, (475, 0) = 15535.30941709627, (475, 1) = .3025272348169411, (475, 2) = 688.6588520401392, (476, 0) = 688.6588520401392, (476, 1) = 15501.155443680387, (476, 2) = .30333061750620377, (477, 0) = .30333061750620377, (477, 1) = 701.095715768795, (477, 2) = 15458.888436167957, (478, 0) = 15458.888436167957, (478, 1) = .30413400019546644, (478, 2) = 713.4959520571451, (479, 0) = 713.4959520571451, (479, 1) = 15410.29398557977, (479, 2) = .3049373828847291, (480, 0) = .3049373828847291, (480, 1) = 725.8552395837582, (480, 2) = 15357.240163698494, (481, 0) = 15357.240163698494, (481, 1) = .30574076557399177, (481, 2) = 738.1707652724408, (482, 0) = 738.1707652724408, (482, 1) = 15301.654246800575, (482, 2) = .3065175361266368, (483, 0) = .3065175361266368, (483, 1) = 750.0355283522397, (483, 2) = 15247.356586202899, (484, 0) = 15247.356586202899, (484, 1) = .30729430667928187, (484, 2) = 761.858496849936, (485, 0) = 761.858496849936, (485, 1) = 15194.334578144251, (485, 2) = .30807107723192695, (486, 0) = .30807107723192695, (486, 1) = 773.6413465226273, (486, 2) = 15144.368039545147, (487, 0) = 15144.368039545147, (487, 1) = .30884784778457197, (487, 2) = 785.3871312509232, (488, 0) = 785.3871312509232, (488, 1) = 15099.201053955634, (488, 2) = .30965936965210406, (489, 0) = .30965936965210406, (489, 1) = 797.6236048421247, (489, 2) = 15058.982024424002, (490, 0) = 15058.982024424002, (490, 1) = .3104708915196362, (490, 2) = 809.8309625979832, (491, 0) = 809.8309625979832, (491, 1) = 15027.723349842658, (491, 2) = .31128241338716833, (492, 0) = .31128241338716833, (492, 1) = 822.0171734578327, (492, 2) = 15007.195353822883, (493, 0) = 15007.195353822883, (493, 1) = .3120939352547004, (493, 2) = 834.1916302987474, (494, 0) = 834.1916302987474, (494, 1) = 14999.061962340247, (494, 2) = .3130541054927739, (495, 0) = .3130541054927739, (495, 1) = 848.5957228722291, (495, 2) = 15007.57066818521, (496, 0) = 15007.57066818521, (496, 1) = .3140142757308474, (496, 2) = 863.0183382740088, (497, 0) = 863.0183382740088, (497, 1) = 15037.961925615751, (497, 2) = .3149744459689209, (498, 0) = .3149744459689209, (498, 1) = 877.4814515260163, (498, 2) = 15092.361885833592, (499, 0) = 15092.361885833592, (499, 1) = .31593461620699437, (499, 2) = 892.0090507357161, (500, 0) = 892.0090507357161, (500, 1) = 15172.665135904816, (500, 2) = .3171181184244708, (501, 0) = .3171181184244708, (501, 1) = 910.0428235324766, (501, 2) = 15309.751983198272, (502, 0) = 15309.751983198272, (502, 1) = .3183016206419472, (502, 2) = 928.2650182847754, (503, 0) = 928.2650182847754, (503, 1) = 15491.397764545662, (503, 2) = .3194851228594236, (504, 0) = .3194851228594236, (504, 1) = 946.7295840473157, (504, 2) = 15719.864919018473, (505, 0) = 15719.864919018473, (505, 1) = .3206686250769, (505, 2) = 965.4931178902214, (506, 0) = 965.4931178902214, (506, 1) = 15997.051207352286, (506, 2) = .32161592485079343, (507, 0) = .32161592485079343, (507, 1) = 980.7667759168829, (507, 2) = 16255.082091512126, (508, 0) = 16255.082091512126, (508, 1) = .32256322462468684, (508, 2) = 996.3004062556182, (509, 0) = 996.3004062556182, (509, 1) = 16546.01323610053, (509, 2) = .32351052439858025, (510, 0) = .32351052439858025, (510, 1) = 1012.1254375612506, (510, 2) = 16870.448014180103, (511, 0) = 16870.448014180103, (511, 1) = .32445782417247365, (511, 2) = 1028.2738788774566, (512, 0) = 1028.2738788774566, (512, 1) = 17228.929584615216, (512, 2) = .325412472953582, (513, 0) = .325412472953582, (513, 1) = 1044.9077733568358, (513, 2) = 17625.13560807134, (514, 0) = 17625.13560807134, (514, 1) = .3263671217346904, (514, 2) = 1061.9368351474347, (515, 0) = 1061.9368351474347, (515, 1) = 18056.888431676427, (515, 2) = .3273217705157988, (516, 0) = .3273217705157988, (516, 1) = 1079.395207922994, (516, 2) = 18524.64878919646, (517, 0) = 18524.64878919646, (517, 1) = .32827641929690715, (517, 2) = 1097.3174970396299, (518, 0) = 1097.3174970396299, (518, 1) = 19028.880763230063, (518, 2) = .3292940936370929, (519, 0) = .3292940936370929, (519, 1) = 1116.9733437102818, (519, 2) = 19607.098853079293, (520, 0) = 19607.098853079293, (520, 1) = .3303117679772786, (520, 2) = 1137.2392363009244, (521, 0) = 1137.2392363009244, (521, 1) = 20227.90404567929, (521, 2) = .33132944231746436, (522, 0) = .33132944231746436, (522, 1) = 1158.1588441139947, (522, 2) = 20891.948795879827, (523, 0) = 20891.948795879827, (523, 1) = .3323471166576501, (523, 2) = 1179.7765423303706, (524, 0) = 1179.7765423303706, (524, 1) = 21599.946806741205, (524, 2) = .33346471160992264, (525, 0) = .33346471160992264, (525, 1) = 1204.3747611205104, (525, 2) = 22429.02884153289, (526, 0) = 22429.02884153289, (526, 1) = .3345823065621951, (526, 2) = 1229.9302080477764, (527, 0) = 1229.9302080477764, (527, 1) = 23313.200285838815, (527, 2) = .3356999015144676, (528, 0) = .3356999015144676, (528, 1) = 1256.5051674076922, (528, 2) = 24253.730147645714, (529, 0) = 24253.730147645714, (529, 1) = .33681749646674014, (529, 2) = 1284.1634220271753, (530, 0) = 1284.1634220271753, (530, 1) = 25252.02692356115, (530, 2) = .3380808996164253, (531, 0) = .3380808996164253, (531, 1) = 1316.816912433403, (531, 2) = 26452.087601389172, (532, 0) = 26452.087601389172, (532, 1) = .3393443027661105, (532, 2) = 1351.0356937723843, (533, 0) = 1351.0356937723843, (533, 1) = 27730.434898521577, (533, 2) = .3406077059157957, (534, 0) = .3406077059157957, (534, 1) = 1386.9204251038007, (534, 2) = 29089.807711098583, (535, 0) = 29089.807711098583, (535, 1) = .34187110906548085, (535, 2) = 1424.5753959216659, (536, 0) = 1424.5753959216659, (536, 1) = 30533.21605534698, (536, 2) = .3433164580940943, (537, 0) = .3433164580940943, (537, 1) = 1469.9634118381111, (537, 2) = 32291.785431042274, (538, 0) = 32291.785431042274, (538, 1) = .34476180712270765, (538, 2) = 1517.9788646680142, (539, 0) = 1517.9788646680142, (539, 1) = 34170.00977464647, (539, 2) = .3462071561513211, (540, 0) = .3462071561513211, (540, 1) = 1568.7989470495604, (540, 2) = 36173.73374188717, (541, 0) = 36173.73374188717, (541, 1) = .3476525051799345, (541, 2) = 1622.6096816465574, (542, 0) = 1622.6096816465574, (542, 1) = 38309.27663116045, (542, 2) = .34907803842842466, (543, 0) = .34907803842842466, (543, 1) = 1678.8025237639504, (543, 2) = 40551.27455105201, (544, 0) = 40551.27455105201, (544, 1) = .3505035716769148, (544, 2) = 1738.2915223308364, (545, 0) = 1738.2915223308364, (545, 1) = 42935.0552429717, (545, 2) = .351929104925405, (546, 0) = .351929104925405, (546, 1) = 1801.283979758192, (546, 2) = 45467.95500763046, (547, 0) = 45467.95500763046, (547, 1) = .3533546381738952, (547, 2) = 1867.998066988907, (548, 0) = 1867.998066988907, (548, 1) = 48157.65818560454, (548, 2) = .35473854794379023, (549, 0) = .35473854794379023, (549, 1) = 1936.5415537589574, (549, 2) = 50926.39905454716, (550, 0) = 50926.39905454716, (550, 1) = .3561224577136853, (550, 2) = 2009.0285096606297, (551, 0) = 2009.0285096606297, (551, 1) = 53858.023276338005, (551, 2) = .35750636748358033, (552, 0) = .35750636748358033, (552, 1) = 2085.6894723307983, (552, 2) = 56960.18911693143, (553, 0) = 56960.18911693143, (553, 1) = .3588902772534754, (553, 2) = 2166.7659571058302, (554, 0) = 2166.7659571058302, (554, 1) = 60240.55699150072, (554, 2) = .3600933027376366, (555, 0) = .3600933027376366, (555, 1) = 2241.028586649002, (555, 2) = 63242.84857739971, (556, 0) = 63242.84857739971, (556, 1) = .3612963282217979, (556, 2) = 2318.989976628418, (557, 0) = 2318.989976628418, (557, 1) = 66390.4022363483, (557, 2) = .36249935370595915, (558, 0) = .36249935370595915, (558, 1) = 2400.8274063114004, (558, 2) = 69687.78929362164, (559, 0) = 69687.78929362164, (559, 1) = .3637023791901204, (559, 2) = 2486.723823658532, (560, 0) = 2486.723823658532, (560, 1) = 73139.19128437438, (560, 2) = .3648003966532263, (561, 0) = .3648003966532263, (561, 1) = 2568.8248926430197, (561, 2) = 76426.9207324087, (562, 0) = 76426.9207324087, (562, 1) = .36589841411633217, (562, 2) = 2654.609272153401, (563, 0) = 2654.609272153401, (563, 1) = 79848.4243681124, (563, 2) = .366996431579438, (564, 0) = .366996431579438, (564, 1) = 2744.224526482528, (564, 2) = 83405.44589057959, (565, 0) = 83405.44589057959, (565, 1) = .3680944490425439, (565, 2) = 2837.820190129137, (566, 0) = 2837.820190129137, (566, 1) = 87098.98421762414, (566, 2) = .3691284129633801, (567, 0) = .3691284129633801, (567, 1) = 2929.7297871147584, (567, 2) = 90702.01622474691, (568, 0) = 90702.01622474691, (568, 1) = .37016237688421627, (568, 2) = 3025.4273229553983, (569, 0) = 3025.4273229553983, (569, 1) = 94425.57764170738, (569, 2) = .3711963408050525, (570, 0) = .3711963408050525, (570, 1) = 3125.036191492129, (570, 2) = 98267.969772945, (571, 0) = 98267.969772945, (571, 1) = .3722303047258887, (571, 2) = 3228.677993029887, (572, 0) = 3228.677993029887, (572, 1) = 102226.34558054504, (572, 2) = .37323468226852086, (573, 0) = .37323468226852086, (573, 1) = 3333.328118268732, (573, 2) = 106178.63081522759, (574, 0) = 106178.63081522759, (574, 1) = .37423905981115296, (574, 2) = 3441.9989827065892, (575, 0) = 3441.9989827065892, (575, 1) = 110231.44500229633, (575, 2) = .37524343735378507, (576, 0) = .37524343735378507, (576, 1) = 3554.787794153935, (576, 2) = 114378.20297006432, (577, 0) = 114378.20297006432, (577, 1) = .37624781489641723, (577, 2) = 3671.7850103267597, (578, 0) = 3671.7850103267597, (578, 1) = 118610.66036158372, (578, 2) = .37727362393560887, (579, 0) = .37727362393560887, (579, 1) = 3795.7080365397373, (579, 2) = 123011.48743036397, (580, 0) = 123011.48743036397, (580, 1) = .3782994329748005, (580, 2) = 3924.180895342468, (581, 0) = 3924.180895342468, (581, 1) = 127478.49628678852, (581, 2) = .37932524201399215, (582, 0) = .37932524201399215, (582, 1) = 4057.26307063306, (582, 2) = 131996.5602971787, (583, 0) = 131996.5602971787, (583, 1) = .3803510510531838, (583, 2) = 4194.998247672558, (584, 0) = 4194.998247672558, (584, 1) = 136548.11437297158, (584, 2) = .381330263426524, (585, 0) = .381330263426524, (585, 1) = 4330.841281073362, (585, 2) = 140905.7729020593, (586, 0) = 140905.7729020593, (586, 1) = .38230947579986413, (586, 2) = 4470.949666693183, (587, 0) = 4470.949666693183, (587, 1) = 145255.95668233032, (587, 2) = .3832886881732043, (588, 0) = .3832886881732043, (588, 1) = 4615.304894581068, (588, 2) = 149576.81589856252, (589, 0) = 149576.81589856252, (589, 1) = .38426790054654447, (589, 2) = 4763.866791415513, (590, 0) = 4763.866791415513, (590, 1) = 153844.48609861216, (590, 2) = .38524711291988467, (591, 0) = .38524711291988467, (591, 1) = 4916.57163014111, (591, 2) = 158033.21151349985, (592, 0) = 158033.21151349985, (592, 1) = .3862263252932248, (592, 2) = 5073.328336485354, (593, 0) = 5073.328336485354, (593, 1) = 162115.29865896175, (593, 2) = .38720553766656496, (594, 0) = .38720553766656496, (594, 1) = 5234.018171480541, (594, 2) = 166061.88242961364, (595, 0) = 166061.88242961364, (595, 1) = .38818475003990516, (595, 2) = 5398.493999768764, (596, 0) = 5398.493999768764, (596, 1) = 169843.3202499081, (596, 2) = .38901444570731164, (597, 0) = .38901444570731164, (597, 1) = 5540.68883964467, (597, 2) = 172895.7336606606, (598, 0) = 172895.7336606606, (598, 1) = .3898441413747181, (598, 2) = 5685.3516583251085, (599, 0) = 5685.3516583251085, (599, 1) = 175789.60841241817, (599, 2) = .39067383704212455, (600, 0) = .39067383704212455, (600, 1) = 5832.343743309921, (600, 2) = 178507.2112040326, (601, 0) = 178507.2112040326, (601, 1) = .39150353270953103, (601, 2) = 5981.51176619617, (602, 0) = 5981.51176619617, (602, 1) = 181031.7135476214, (602, 2) = .39227383053365095, (603, 0) = .39227383053365095, (603, 1) = 6121.802786593649, (603, 2) = 183188.8601064289, (604, 0) = 183188.8601064289, (604, 1) = .3930441283577708, (604, 2) = 6263.682240733006, (605, 0) = 6263.682240733006, (605, 1) = 185154.53666966816, (605, 2) = .3938144261818907, (606, 0) = .3938144261818907, (606, 1) = 6406.999148839278, (606, 2) = 186918.8337183744, (607, 0) = 186918.8337183744, (607, 1) = .39458472400601063, (607, 2) = 6551.595098707301, (608, 0) = 6551.595098707301, (608, 1) = 188473.60595733408, (608, 2) = .395367151649, (609, 0) = .395367151649, (609, 1) = 6699.608022810343, (609, 2) = 189831.9254757487, (610, 0) = 189831.9254757487, (610, 1) = .39614957929198935, (610, 2) = 6848.594905665068, (611, 0) = 6848.594905665068, (611, 1) = 190963.32593814595, (611, 2) = .39693200693497865, (612, 0) = .39693200693497865, (612, 1) = 6998.378033942487, (612, 2) = 191866.0979862694, (613, 0) = 191866.0979862694, (613, 1) = .397714434577968, (613, 2) = 7148.7786850174825, (614, 0) = 7148.7786850174825, (614, 1) = 192541.2366130295, (614, 2) = .39863461057546223, (615, 0) = .39863461057546223, (615, 1) = 7326.2073364539265, (615, 2) = 193049.1494859998, (616, 0) = 193049.1494859998, (616, 1) = .3995547865729565, (616, 2) = 7503.964440477295, (617, 0) = 7503.964440477295, (617, 1) = 193258.6815640299, (617, 2) = .4004749625704508, (618, 0) = .4004749625704508, (618, 1) = 7681.784111963161, (618, 2) = 193186.29695455692, (619, 0) = 193186.29695455692, (619, 1) = .401395138567945, (619, 2) = 7859.4163550230705, (620, 0) = 7859.4163550230705, (620, 1) = 192853.50810753563, (620, 2) = .40230131027445476, (621, 0) = .40230131027445476, (621, 1) = 8033.938489358477, (621, 2) = 192296.8284892497, (622, 0) = 192296.8284892497, (622, 1) = .40320748198096446, (622, 2) = 8207.863966494268, (623, 0) = 8207.863966494268, (623, 1) = 191542.46882079964, (623, 2) = .4041136536874742, (624, 0) = .4041136536874742, (624, 1) = 8381.028921904035, (624, 2) = 190622.9121209392, (625, 0) = 190622.9121209392, (625, 1) = .40501982539398396, (625, 2) = 8553.299286574245, (626, 0) = 8553.299286574245, (626, 1) = 189573.15404288913, (626, 2) = .4057760884030213, (627, 0) = .4057760884030213, (627, 1) = 8696.31101575892, (627, 2) = 188624.21782385476, (628, 0) = 188624.21782385476, (628, 1) = .4065323514120587, (628, 2) = 8838.58761217999, (629, 0) = 8838.58761217999, (629, 1) = 187632.78121750007, (629, 2) = .4072886144210961, (630, 0) = .4072886144210961, (630, 1) = 8980.105434338377, (630, 2) = 186621.345257642, (631, 0) = 186621.345257642, (631, 1) = .40804487743013346, (631, 2) = 9120.8578453654, (632, 0) = 9120.8578453654, (632, 1) = 185612.33678222485, (632, 2) = .4088037043169399, (633, 0) = .4088037043169399, (633, 1) = 9261.32852520306, (633, 2) = 184624.70852176158, (634, 0) = 184624.70852176158, (634, 1) = .4095625312037463, (634, 2) = 9401.066237971821, (635, 0) = 9401.066237971821, (635, 1) = 183683.8749277787, (635, 2) = .41032135809055276, (636, 0) = .41032135809055276, (636, 1) = 9540.114352595157, (636, 2) = 182811.0061950597, (637, 0) = 182811.0061950597, (637, 1) = .4110801849773592, (637, 2) = 9678.532190022992, (638, 0) = 9678.532190022992, (638, 1) = 182026.3131001884, (638, 2) = .4119208777415202, (639, 0) = .4119208777415202, (639, 1) = 9831.237979248861, (639, 2) = 181283.10079772305, (640, 0) = 181283.10079772305, (640, 1) = .41276157050568113, (640, 2) = 9983.383159750172, (641, 0) = 9983.383159750172, (641, 1) = 180696.0721675369, (641, 2) = .41360226326984206, (642, 0) = .41360226326984206, (642, 1) = 10135.108085273418, (642, 2) = 180287.73815282772, (643, 0) = 180287.73815282772, (643, 1) = .41444295603400305, (643, 2) = 10286.571805005451, (644, 0) = 10286.571805005451, (644, 1) = 180078.63698366084, (644, 2) = .41561541600071666, (645, 0) = .41561541600071666, (645, 1) = 10497.708203368571, (645, 2) = 180154.5300606661, (646, 0) = 180154.5300606661, (646, 1) = .4167878759674303, (646, 2) = 10709.204531591728, (647, 0) = 10709.204531591728, (647, 1) = 180697.8566161173, (647, 2) = .41796033593414395, (648, 0) = .41796033593414395, (648, 1) = 10921.628783075435, (648, 2) = 181745.9809416163, (649, 0) = 181745.9809416163, (649, 1) = .41913279590085756, (649, 2) = 11135.59203859185, (650, 0) = 11135.59203859185, (650, 1) = 183330.11707759026, (650, 2) = .4200916433418562, (651, 0) = .4200916433418562, (651, 1) = 11312.168046306244, (651, 2) = 185042.01411778506, (652, 0) = 185042.01411778506, (652, 1) = .4210504907828549, (652, 2) = 11490.570441377818, (653, 0) = 11490.570441377818, (653, 1) = 187141.09811548103, (653, 2) = .42200933822385356, (654, 0) = .42200933822385356, (654, 1) = 11671.174755698541, (654, 2) = 189637.25156728327, (655, 0) = 189637.25156728327, (655, 1) = .4229681856648522, (655, 2) = 11854.36603202153, (656, 0) = 11854.36603202153, (656, 1) = 192538.95325778413, (656, 2) = .4238800546175746, (657, 0) = .4238800546175746, (657, 1) = 12031.34056555869, (657, 2) = 195681.33208446053, (658, 0) = 195681.33208446053, (658, 1) = .42479192357029705, (658, 2) = 12211.352868942044, (659, 0) = 12211.352868942044, (659, 1) = 199202.4943979387, (659, 2) = .4257037925230195, (660, 0) = .4257037925230195, (660, 1) = 12394.750445463551, (660, 2) = 203107.4538790396, (661, 0) = 203107.4538790396, (661, 1) = .42661566147574187, (661, 2) = 12581.885555932258, (662, 0) = 12581.885555932258, (662, 1) = 207400.95002316975, (662, 2) = .4275676783187452, (663, 0) = .4275676783187452, (663, 1) = 12781.634070839402, (663, 2) = 212302.9913972964, (664, 0) = 212302.9913972964, (664, 1) = .42851969516174854, (664, 2) = 12986.255360434041, (665, 0) = 12986.255360434041, (665, 1) = 217638.6864817161, (665, 2) = .42947171200475187, (666, 0) = .42947171200475187, (666, 1) = 13196.164791914423, (666, 2) = 223413.48209564734, (667, 0) = 223413.48209564734, (667, 1) = .4304237288477552, (667, 2) = 13411.783274955606, (668, 0) = 13411.783274955606, (668, 1) = 229633.22474015655, (668, 2) = .4314515743466773, (669, 0) = .4314515743466773, (669, 1) = 13651.476984978788, (669, 2) = 236855.19068897408, (670, 0) = 236855.19068897408, (670, 1) = .43247941984559946, (670, 2) = 13898.867633549973, (671, 0) = 13898.867633549973, (671, 1) = 244611.93039674312, (671, 2) = .43350726534452155, (672, 0) = .43350726534452155, (672, 1) = 14154.509978579445, (672, 2) = 252913.26075607553, (673, 0) = 252913.26075607553, (673, 1) = .43453511084344365, (673, 2) = 14418.969512379377, (674, 0) = 14418.969512379377, (674, 1) = 261770.1877424041, (674, 2) = .4356691201419871, (675, 0) = .4356691201419871, (675, 1) = 14721.667853463778, (675, 2) = 272201.32273425965, (676, 0) = 272201.32273425965, (676, 1) = .43680312944053057, (676, 2) = 15036.595357485892, (677, 0) = 15036.595357485892, (677, 1) = 283342.0934942626, (677, 2) = .43793713873907403, (678, 0) = .43793713873907403, (678, 1) = 15364.568688831576, (678, 2) = 295213.1438524385, (679, 0) = 295213.1438524385, (679, 1) = .4390711480376175, (679, 2) = 15706.429128271218, (680, 0) = 15706.429128271218, (680, 1) = 307837.4885943271, (680, 2) = .4403473719088535, (681, 0) = .4403473719088535, (681, 1) = 16108.853234188395, (681, 2) = 322977.7671754702, (682, 0) = 322977.7671754702, (682, 1) = .4416235957800895, (682, 2) = 16531.24946587439, (683, 0) = 16531.24946587439, (683, 1) = 339144.5539821098, (683, 2) = .4428998196513255, (684, 0) = .4428998196513255, (684, 1) = 16974.956765061128, (684, 2) = 356382.29932073434, (685, 0) = 356382.29932073434, (685, 1) = .4441760435225615, (685, 2) = 17441.373295632584, (686, 0) = 17441.373295632584, (686, 1) = 374739.9004271949, (686, 2) = .4455969367367128, (687, 0) = .4455969367367128, (687, 1) = 17989.16551162714, (687, 2) = 396561.29976193444, (688, 0) = 396561.29976193444, (688, 1) = .44701782995086403, (688, 2) = 18569.04260931924, (689, 0) = 18569.04260931924, (689, 1) = 419917.66210097715, (689, 2) = .4484387231650153, (690, 0) = .4484387231650153, (690, 1) = 19183.248743728465, (690, 2) = 444896.7775019398, (691, 0) = 444896.7775019398, (691, 1) = .4498596163791666, (691, 2) = 19834.15785079536, (692, 0) = 19834.15785079536, (692, 1) = 471593.7439439525, (692, 2) = .45125251282242607, (693, 0) = .45125251282242607, (693, 1) = 20510.28797297615, (693, 2) = 499530.75580993295, (694, 0) = 499530.75580993295, (694, 1) = .4526454092656855, (694, 2) = 21226.607868765277, (695, 0) = 21226.607868765277, (695, 1) = 529320.3141688858, (695, 2) = .45403830570894504, (696, 0) = .45403830570894504, (696, 1) = 21985.774196707043, (696, 2) = 561072.0274526618, (697, 0) = 561072.0274526618, (697, 1) = .4554312021522045, (697, 2) = 22790.601773832852, (698, 0) = 22790.601773832852, (698, 1) = 594901.3781271209, (698, 2) = .4567782777812973, (699, 0) = .4567782777812973, (699, 1) = 23615.186876052514, (699, 2) = 629707.8153118583, (700, 0) = 629707.8153118583, (700, 1) = .4581253534103902, (700, 2) = 24488.10686977899, (701, 0) = 24488.10686977899, (701, 1) = 666684.5235425337, (701, 2) = .459472429039483, (702, 0) = .459472429039483, (702, 1) = 25412.361802979594, (702, 2) = 705947.9568045158, (703, 0) = 705947.9568045158, (703, 1) = .4608195046685758, (703, 2) = 26391.113820688017, (704, 0) = 26391.113820688017, (704, 1) = 747615.8805216381, (704, 2) = .4619717361737435, (705, 0) = .4619717361737435, (705, 1) = 27274.044897660162, (705, 2) = 785253.9951342004, (706, 0) = 785253.9951342004, (706, 1) = .4631239676789112, (706, 2) = 28201.442511616624, (707, 0) = 28201.442511616624, (707, 1) = 824810.820501115, (707, 2) = .46427619918407886, (708, 0) = .46427619918407886, (708, 1) = 29175.55517018994, (708, 2) = 866356.3707222887, (709, 0) = 866356.3707222887, (709, 1) = .46542843068924655, (709, 2) = 30198.714339906397, (710, 0) = 30198.714339906397, (710, 1) = 909956.3028938789, (710, 2) = .4664711763537427, (711, 0) = .4664711763537427, (711, 1) = 31168.93769716251, (711, 2) = 951234.541486685, (712, 0) = 951234.541486685, (712, 1) = .4675139220182388, (712, 2) = 32183.124997614716, (713, 0) = 32183.124997614716, (713, 1) = 994284.1249598789, (713, 2) = .46855666768273496, (714, 0) = .46855666768273496, (714, 1) = 33243.13657373136, (714, 2) = 1039136.8836424553, (715, 0) = 1039136.8836424553, (715, 1) = .4695994133472311, (715, 2) = 34350.866785010156, (716, 0) = 34350.866785010156, (716, 1) = 1085815.5802836039, (716, 2) = .4705768653175893, (717, 0) = .4705768653175893, (717, 1) = 35434.26941785946, (717, 2) = 1131240.6349408436, (718, 0) = 1131240.6349408436, (718, 1) = .4715543172879475, (718, 2) = 36562.86443732506, (719, 0) = 36562.86443732506, (719, 1) = 1178281.5088189342, (719, 2) = .47253176925830576, (720, 0) = .47253176925830576, (720, 1) = 37738.220996416574, (720, 2) = 1226925.860575558, (721, 0) = 1226925.860575558, (721, 1) = .47350922122866396, (721, 2) = 38961.89589674573, (722, 0) = 38961.89589674573, (722, 1) = 1277147.0668902271, (722, 2) = .4744560340732418, (723, 0) = .4744560340732418, (723, 1) = 40194.727798964785, (723, 2) = 1327257.7930490347, (724, 0) = 1327257.7930490347, (724, 1) = .47540284691781953, (724, 2) = 41475.66823881862, (725, 0) = 41475.66823881862, (725, 1) = 1378753.9324396157, (725, 2) = .47634965976239735, (726, 0) = .47634965976239735, (726, 1) = 42805.988561419486, (726, 2) = 1431561.6988717201, (727, 0) = 1431561.6988717201, (727, 1) = .47729647260697516, (727, 2) = 44186.888782916576, (728, 0) = 44186.888782916576, (728, 1) = 1485586.4621425974, (728, 2) = .47825971940891704, (729, 0) = .47825971940891704, (729, 1) = 45644.806094191226, (729, 2) = 1541677.186802131, (730, 0) = 1541677.186802131, (730, 1) = .4792229662108589, (730, 2) = 47157.24559539322, (731, 0) = 47157.24559539322, (731, 1) = 1598757.0928827166, (731, 2) = .48018621301280073, (732, 0) = .48018621301280073, (732, 1) = 48725.066126749414, (732, 2) = 1656647.1306359037, (733, 0) = 1656647.1306359037, (733, 1) = .4811494598147426, (733, 2) = 50348.95087936517, (734, 0) = 50348.95087936517, (734, 1) = 1715137.4684068859, (734, 2) = .4822922672114153, (735, 0) = .4822922672114153, (735, 1) = 52348.90594157632, (735, 2) = 1784975.6632171874, (736, 0) = 1784975.6632171874, (736, 1) = .483435074608088, (736, 2) = 54428.75010190249, (737, 0) = 54428.75010190249, (737, 1) = 1854851.264579692, (737, 2) = .4845778820047607, (738, 0) = .4845778820047607, (738, 1) = 56588.20469968647, (738, 2) = 1924232.696236686, (739, 0) = 1924232.696236686, (739, 1) = .4857206894014334, (739, 2) = 58826.3744385191, (740, 0) = 58826.3744385191, (740, 1) = 1992527.1090407248, (740, 2) = .4866823893522817, (741, 0) = .4866823893522817, (741, 1) = 60769.70429748154, (741, 2) = 2048680.2744089572, (742, 0) = 2048680.2744089572, (742, 1) = .48764408930312997, (742, 2) = 62766.27972231602, (743, 0) = 62766.27972231602, (743, 1) = 2103185.325716239, (743, 2) = .4886057892539782, (744, 0) = .4886057892539782, (744, 1) = 64814.305330263516, (744, 2) = 2155608.7709725127, (745, 0) = 2155608.7709725127, (745, 1) = .48956748920482646, (745, 2) = 66911.5673076788, (746, 0) = 66911.5673076788, (746, 1) = 2205508.7084854813, (746, 2) = .4903427105022228, (747, 0) = .4903427105022228, (747, 1) = 68636.21771829756, (747, 2) = 2243592.0632263464, (748, 0) = 2243592.0632263464, (748, 1) = .49111793179961916, (748, 2) = 70389.56723064935, (749, 0) = 70389.56723064935, (749, 1) = 2279514.6768680294, (749, 2) = .49189315309701553, (750, 0) = .49189315309701553, (750, 1) = 72169.85735053786, (750, 2) = 2313054.8726457693, (751, 0) = 2313054.8726457693, (751, 1) = .49266837439441186, (751, 2) = 73975.15907420326, (752, 0) = 73975.15907420326, (752, 1) = 2344004.096652902, (752, 2) = .49340063272668777, (753, 0) = .49340063272668777, (753, 1) = 75701.49926366823, (753, 2) = 2370684.196119692, (754, 0) = 2370684.196119692, (754, 1) = .49413289105896374, (754, 2) = 77446.41929254166, (755, 0) = 77446.41929254166, (755, 1) = 2394733.5523626907, (755, 2) = .49486514939123966, (756, 0) = .49486514939123966, (756, 1) = 79207.95112309034, (756, 2) = 2416026.942414744, (757, 0) = 2416026.942414744, (757, 1) = .49559740772351557, (757, 2) = 80984.03762366119, (758, 0) = 80984.03762366119, (758, 1) = 2434463.048047884, (758, 2) = .49635413645459847, (759, 0) = .49635413645459847, (759, 1) = 82832.50613911272, (759, 2) = 2450432.585254802, (760, 0) = 2450432.585254802, (760, 1) = .4971108651856813, (760, 2) = 84691.85320636658, (761, 0) = 84691.85320636658, (761, 1) = 2463217.6598100783, (761, 2) = .49786759391676416, (762, 0) = .49786759391676416, (762, 1) = 86559.66953122207, (762, 2) = 2472801.9733032454, (763, 0) = 2472801.9733032454, (763, 1) = .49862432264784706, (763, 2) = 88433.53785254294, (764, 0) = 88433.53785254294, (764, 1) = 2479206.2162393183, (764, 2) = .49955548249885123, (765, 0) = .49955548249885123, (765, 1) = 90744.11010231251, (765, 2) = 2482810.5534116165, (766, 0) = 2482810.5534116165, (766, 1) = .5004866423498554, (766, 2) = 93055.89940831932, (767, 0) = 93055.89940831932, (767, 1) = 2481877.8273741947, (767, 2) = .5014178022008595, (768, 0) = .5014178022008595, (768, 1) = 95364.82409794425, (768, 2) = 2476678.6688390323, (769, 0) = 2476678.6688390323, (769, 1) = .5023489620518637, (769, 2) = 97667.06599132507, (770, 0) = 97667.06599132507, (770, 1) = 2467560.9019755763, (770, 2) = .5031815175180465, (771, 0) = .5031815175180465, (771, 1) = 99716.99627264048, (771, 2) = 2456432.3449352058, (772, 0) = 2456432.3449352058, (772, 1) = .5040140729842293, (772, 2) = 101756.61266607692, (773, 0) = 101756.61266607692, (773, 1) = 2442850.6696412507, (773, 2) = .5048466284504121, (774, 0) = .5048466284504121, (774, 1) = 103784.03315632256, (774, 2) = 2427189.766294855, (775, 0) = 2427189.766294855, (775, 1) = .5056791839165948, (775, 2) = 105797.690007021, (776, 0) = 105797.690007021, (776, 1) = 2409846.6749335346, (776, 2) = .5064020435736993, (777, 0) = .5064020435736993, (777, 1) = 107533.90120100992, (777, 2) = 2393746.998957796, (778, 0) = 2393746.998957796, (778, 1) = .5071249032308038, (778, 2) = 109258.21395214874, (779, 0) = 109258.21395214874, (779, 1) = 2376972.615348099, (779, 2) = .5078477628879083, (780, 0) = .5078477628879083, (780, 1) = 110970.24147898442, (780, 2) = 2359803.7022914793, (781, 0) = 2359803.7022914793, (781, 1) = .5085706225450128, (781, 2) = 112669.79918864815, (782, 0) = 112669.79918864815, (782, 1) = 2342518.08165685, (782, 2) = .509303228745844, (783, 0) = .509303228745844, (783, 1) = 114379.56659500377, (783, 2) = 2325161.0637650983, (784, 0) = 2325161.0637650983, (784, 1) = .5100358349466751, (784, 2) = 116076.76356673951, (785, 0) = 116076.76356673951, (785, 1) = 2308243.4499668465, (785, 2) = .5107684411475062, (786, 0) = .5107684411475062, (786, 1) = 117761.80738405435, (786, 2) = 2292031.8095945218, (787, 0) = 2292031.8095945218, (787, 1) = .5115010473483373, (787, 2) = 119435.30910668496, (788, 0) = 119435.30910668496, (788, 1) = 2276779.5370274214, (788, 2) = .5123152306537342, (789, 0) = .5123152306537342, (789, 1) = 121282.59134982148, (789, 2) = 2261245.4451005063, (790, 0) = 2261245.4451005063, (790, 1) = .5131294139591311, (790, 2) = 123117.93598410983, (791, 0) = 123117.93598410983, (791, 1) = 2247498.160966972, (791, 2) = .513943597264528, (792, 0) = .513943597264528, (792, 1) = 124942.90692294508, (792, 2) = 2235818.655022103, (793, 0) = 2235818.655022103, (793, 1) = .5147577805699248, (793, 2) = 126759.29393135739, (794, 0) = 126759.29393135739, (794, 1) = 2226461.882032, (794, 2) = .5156852989370676, (795, 0) = .5156852989370676, (795, 1) = 128820.61721089299, (795, 2) = 2218924.394517269, (796, 0) = 2218924.394517269, (796, 1) = .5166128173042104, (796, 2) = 130876.606219624, (797, 0) = 130876.606219624, (797, 1) = 2214997.08204408, (797, 2) = .5175403356713533, (798, 0) = .5175403356713533, (798, 1) = 132930.72053106458, (798, 2) = 2214939.456412788, (799, 0) = 2214939.456412788, (799, 1) = .518467854038496, (799, 2) = 134986.6569858542, (800, 0) = 134986.6569858542, (800, 1) = 2218973.980858906, (800, 2) = .5193953724056388, (801, 0) = .5193953724056388, (801, 1) = 137048.3170801267, (801, 2) = 2227288.4279843275, (802, 0) = 2227288.4279843275, (802, 1) = .5203228907727817, (802, 2) = 139119.73782627395, (803, 0) = 139119.73782627395, (803, 1) = 2240039.4726961623, (803, 2) = .5212504091399245, (804, 0) = .5212504091399245, (804, 1) = 141205.08890251035, (804, 2) = 2257358.9267314617, (805, 0) = 2257358.9267314617, (805, 1) = .5221779275070673, (805, 2) = 143308.66213750857, (806, 0) = 143308.66213750857, (806, 1) = 2279356.9771170667, (806, 2) = .523057778769193, (807, 0) = .523057778769193, (807, 1) = 145324.9602616193, (807, 2) = 2304631.259281105, (808, 0) = 2304631.259281105, (808, 1) = .5239376300313187, (808, 2) = 147365.40883212382, (809, 0) = 147365.40883212382, (809, 1) = 2334262.3608617564, (809, 2) = .5248174812934443, (810, 0) = .5248174812934443, (810, 1) = 149433.86343190988, (810, 2) = 2368307.110829698, (811, 0) = 2368307.110829698, (811, 1) = .52569733255557, (811, 2) = 151534.23157209848, (812, 0) = 151534.23157209848, (812, 1) = 2406816.3419337682, (812, 2) = .5266033139204701, (813, 0) = .5266033139204701, (813, 1) = 153734.49846087283, (813, 2) = 2451183.2597282063, (814, 0) = 2451183.2597282063, (814, 1) = .5275092952853702, (814, 2) = 155977.1450867287, (815, 0) = 155977.1450867287, (815, 1) = 2500380.5939776404, (815, 2) = .5284152766502701, (816, 0) = .5284152766502701, (816, 1) = 158266.568556324, (816, 2) = 2554457.1187881995, (817, 0) = 2554457.1187881995, (817, 1) = .5293212580151702, (817, 2) = 160607.21384801329, (818, 0) = 160607.21384801329, (818, 1) = 2613464.1636620755, (818, 2) = .5302908621158346, (819, 0) = .5302908621158346, (819, 1) = 163174.07032014275, (819, 2) = 2682139.377357365, (820, 0) = 2682139.377357365, (820, 1) = .5312604662164992, (820, 2) = 165810.31092560725, (821, 0) = 165810.31092560725, (821, 1) = 2756598.9194873516, (821, 2) = .5322300703171639, (822, 0) = .5322300703171639, (822, 1) = 168521.58510912926, (822, 2) = 2836926.896521667, (823, 0) = 2836926.896521667, (823, 1) = .5331996744178283, (823, 2) = 171313.63025013503, (824, 0) = 171313.63025013503, (824, 1) = 2923218.931613156, (824, 2) = .5342599349107775, (825, 0) = .5342599349107775, (825, 1) = 174466.0670099217, (825, 2) = 3024533.5690231323, (826, 0) = 3024533.5690231323, (826, 1) = .5353201954037268, (826, 2) = 177729.83922581174, (827, 0) = 177729.83922581174, (827, 1) = 3133269.239295125, (827, 2) = .5363804558966759, (828, 0) = .5363804558966759, (828, 1) = 181112.9143363488, (828, 2) = 3249608.8533992744, (829, 0) = 3249608.8533992744, (829, 1) = .5374407163896251, (829, 2) = 184623.46497668576, (830, 0) = 184623.46497668576, (830, 1) = 3373759.505229698, (830, 2) = .5386135537258843, (831, 0) = .5386135537258843, (831, 1) = 188665.39496550645, (831, 2) = 3520472.300292947, (832, 0) = 3520472.300292947, (832, 1) = .5397863910621433, (832, 2) = 192885.32991194693, (833, 0) = 192885.32991194693, (833, 1) = 3677379.6050648983, (833, 2) = .5409592283984024, (834, 0) = .5409592283984024, (834, 1) = 197295.46381657728, (834, 2) = 3844876.46239003, (835, 0) = 3844876.46239003, (835, 1) = .5421320657346616, (835, 2) = 201908.47494226779, (836, 0) = 201908.47494226779, (836, 1) = 4023402.686403325, (836, 2) = .5434419944330378, (837, 0) = .5434419944330378, (837, 1) = 207316.7699960603, (837, 2) = 4236433.5400127, (838, 0) = 4236433.5400127, (838, 1) = .5447519231314139, (838, 2) = 213013.91822502235, (839, 0) = 213013.91822502235, (839, 1) = 4464577.47697332, (839, 2) = .54606185182979, (840, 0) = .54606185182979, (840, 1) = 219020.27020882504, (840, 2) = 4708661.039454781, (841, 0) = 4708661.039454781, (841, 1) = .5473717805281662, (841, 2) = 225357.30192202446, (842, 0) = 225357.30192202446, (842, 1) = 4969592.309012137, (842, 2) = .548760188092277, (843, 0) = .548760188092277, (843, 1) = 232460.2658363775, (843, 2) = 5265653.275573642, (844, 0) = 5265653.275573642, (844, 1) = .5501485956563877, (844, 2) = 239988.91077437944, (845, 0) = 239988.91077437944, (845, 1) = 5583042.529162049, (845, 2) = .5515370032204985, (846, 0) = .5515370032204985, (846, 1) = 247973.8349462834, (846, 2) = 5923159.054895745, (847, 0) = 5923159.054895745, (847, 1) = .5529254107846093, (847, 2) = 256447.64957661123, (848, 0) = 256447.64957661123, (848, 1) = 6287515.097824822, (848, 2) = .5542713696116297, (849, 0) = .5542713696116297, (849, 1) = 265161.8992740871, (849, 2) = 6665405.404698245, (850, 0) = 6665405.404698245, (850, 1) = .5556173284386501, (850, 2) = 274401.99531968776, (851, 0) = 274401.99531968776, (851, 1) = 7069188.538214139, (851, 2) = .5569632872656706, (852, 0) = .5569632872656706, (852, 1) = 284203.9199077336, (852, 2) = 7500544.277317812, (853, 0) = 7500544.277317812, (853, 1) = .558309246092691, (853, 2) = 294605.99180152785, (854, 0) = 294605.99180152785, (854, 1) = 7961241.008936638, (854, 2) = .5595643130745056, (855, 0) = .5595643130745056, (855, 1) = 304882.17587205674, (855, 2) = 8418891.357378418, (856, 0) = 8418891.357378418, (856, 1) = .5608193800563204, (856, 2) = 315750.56584833364, (857, 0) = 315750.56584833364, (857, 1) = 8905223.496618355, (857, 2) = .562074447038135, (858, 0) = .562074447038135, (858, 1) = 327248.1354089397, (858, 2) = 9421829.445939634, (859, 0) = 9421829.445939634, (859, 1) = .5633295140199497, (859, 2) = 339413.91804284294, (860, 0) = 339413.91804284294, (860, 1) = 9970316.698193232, (860, 2) = .5644060113400358, (861, 0) = .5644060113400358, (861, 1) = 350412.2458842593, (861, 2) = 10467387.096050642, (862, 0) = 10467387.096050642, (862, 1) = .5654825086601218, (862, 2) = 361959.38492086413, (863, 0) = 361959.38492086413, (863, 1) = 10990099.042246759, (863, 2) = .5665590059802078, (864, 0) = .5665590059802078, (864, 1) = 374083.42130715796, (864, 2) = 11539411.853922885, (865, 0) = 11539411.853922885, (865, 1) = .5676355033002939, (865, 2) = 386813.501554873, (866, 0) = 386813.501554873, (866, 1) = 12116224.098543864, (866, 2) = .5686113882970171, (867, 0) = .5686113882970171, (867, 1) = 398902.7220091268, (867, 2) = 12663580.546635803, (868, 0) = 12663580.546635803, (868, 1) = .5695872732937403, (868, 2) = 411537.69151517417, (869, 0) = 411537.69151517417, (869, 1) = 13234753.877589338, (869, 2) = .5705631582904634, (870, 0) = .5705631582904634, (870, 1) = 424741.822255563, (870, 2) = 13830179.094357671, (871, 0) = 13830179.094357671, (871, 1) = .5715390432871866, (871, 2) = 438538.9611108468, (872, 0) = 438538.9611108468, (872, 1) = 14450166.356453564, (872, 2) = .5724566164259799, (873, 0) = .5724566164259799, (873, 1) = 452074.1843127429, (873, 2) = 15055664.622474603, (874, 0) = 15055664.622474603, (874, 1) = .5733741895647733, (874, 2) = 466175.05313442595, (875, 0) = 466175.05313442595, (875, 1) = 15683029.30924399, (875, 2) = .5742917627035665, (876, 0) = .5742917627035665, (876, 1) = 480861.4977879311, (876, 2) = 16332087.0213859, (877, 0) = 16332087.0213859, (877, 1) = .5752093358423598, (877, 2) = 496153.28567024565, (878, 0) = 496153.28567024565, (878, 1) = 17002467.99487122, (878, 2) = .5761028077363959, (879, 0) = .5761028077363959, (879, 1) = 511643.6182055378, (879, 2) = 17675175.89701508, (880, 0) = 17675175.89701508, (880, 1) = .576996279630432, (880, 2) = 527743.5339106533, (881, 0) = 527743.5339106533, (881, 1) = 18366782.49346146, (881, 2) = .5778897515244681, (882, 0) = .5778897515244681, (882, 1) = 544469.3855057943, (882, 2) = 19076254.915227078, (883, 0) = 19076254.915227078, (883, 1) = .5787832234185042, (883, 2) = 561836.5840089574, (884, 0) = 561836.5840089574, (884, 1) = 19802272.075619206, (884, 2) = .5797012995453732, (885, 0) = .5797012995453732, (885, 1) = 580365.0495749065, (885, 2) = 20563804.374263, (886, 0) = 20563804.374263, (886, 1) = .5806193756722422, (886, 2) = 599599.130769846, (887, 0) = 599599.130769846, (887, 1) = 21338942.023530975, (887, 2) = .5815374517991111, (888, 0) = .5815374517991111, (888, 1) = 619550.0372367561, (888, 2) = 22125121.645739153, (889, 0) = 22125121.645739153, (889, 1) = .5824555279259801, (889, 2) = 640226.5828135118, (890, 0) = 640226.5828135118, (890, 1) = 22919345.695353188, (890, 2) = .5833353567630933, (891, 0) = .5833353567630933, (891, 1) = 660728.2726912699, (891, 2) = 23684852.24736674, (892, 0) = 23684852.24736674, (892, 1) = .5842151856002067, (892, 2) = 681904.1288654808, (893, 0) = 681904.1288654808, (893, 1) = 24451143.448663123, (893, 2) = .5850950144373199, (894, 0) = .5850950144373199, (894, 1) = 703753.0748713388, (894, 2) = 25214388.488195904, (895, 0) = 25214388.488195904, (895, 1) = .5859748432744332, (895, 2) = 726270.620793598, (896, 0) = 726270.620793598, (896, 1) = 25970387.55220225, (896, 2) = .5868546721115464, (897, 0) = .5868546721115464, (897, 1) = 749448.5510912753, (897, 2) = 26714590.66829398, (898, 0) = 26714590.66829398, (898, 1) = .5877345009486596, (898, 2) = 773274.2947274544, (899, 0) = 773274.2947274544, (899, 1) = 27442089.016618382, (899, 2) = .588614329785773, (900, 0) = .588614329785773, (900, 1) = 797730.8554812501, (900, 2) = 28147739.402497742, (901, 0) = 28147739.402497742, (901, 1) = .5894941586228862, (901, 2) = 822796.6864998975, (902, 0) = 822796.6864998975, (902, 1) = 28826227.487679962, (902, 2) = .590247309111109, (903, 0) = .590247309111109, (903, 1) = 844717.8153048842, (903, 2) = 29381335.288557332, (904, 0) = 29381335.288557332, (904, 1) = .5910004595993318, (904, 2) = 867046.9365361712, (905, 0) = 867046.9365361712, (905, 1) = 29909111.282221675, (905, 2) = .5917536100875546, (906, 0) = .5917536100875546, (906, 1) = 889762.220142879, (906, 2) = 30406187.67815824, (907, 0) = 30406187.67815824, (907, 1) = .5925067605757773, (907, 2) = 912839.3104830986, (908, 0) = 912839.3104830986, (908, 1) = 30869320.441891216, (908, 2) = .5932095200295708, (909, 0) = .5932095200295708, (909, 1) = 934675.0977728957, (909, 2) = 31268127.22683726, (910, 0) = 31268127.22683726, (910, 1) = .5939122794833642, (910, 2) = 956779.1065854105, (911, 0) = 956779.1065854105, (911, 1) = 31632350.839763343, (911, 2) = .5946150389371577, (912, 0) = .5946150389371577, (912, 1) = 979126.3433053305, (912, 2) = 31959893.61013816, (913, 0) = 31959893.61013816, (913, 1) = .5953177983909512, (913, 2) = 1001690.3685198026, (914, 0) = 1001690.3685198026, (914, 1) = 32248936.321538616, (914, 2) = .5960301708683019, (915, 0) = .5960301708683019, (915, 1) = 1024755.8989038268, (915, 2) = 32501082.027372226, (916, 0) = 32501082.027372226, (916, 1) = .5967425433456526, (916, 2) = 1047985.992214978, (917, 0) = 1047985.992214978, (917, 1) = 32710884.010969214, (917, 2) = .5974549158230033, (918, 0) = .5974549158230033, (918, 1) = 1071350.2710074517, (918, 2) = 32877535.76644551, (919, 0) = 32877535.76644551, (919, 1) = .5981672883003539, (919, 2) = 1094817.831619019, (920, 0) = 1094817.831619019, (920, 1) = 33000669.58467279, (920, 2) = .5989802994744349, (921, 0) = .5989802994744349, (921, 1) = 1121687.083353007, (921, 2) = 33088142.69303308, (922, 0) = 33088142.69303308, (922, 1) = .5997933106485158, (922, 2) = 1148604.6492325969, (923, 0) = 1148604.6492325969, (923, 1) = 33119920.741777036, (923, 2) = .6006063218225968, (924, 0) = .6006063218225968, (924, 1) = 1175526.0703071225, (924, 2) = 33097661.960229594, (925, 0) = 33097661.960229594, (925, 1) = .6014193329966777, (925, 2) = 1202408.3351718138, (926, 0) = 1202408.3351718138, (926, 1) = 33023791.513904985, (926, 2) = .6021289933312894, (927, 0) = .6021289933312894, (927, 1) = 1225809.1351349268, (927, 2) = 32919583.77204987, (928, 0) = 32919583.77204987, (928, 1) = .6028386536659011, (928, 2) = 1249123.5965927865, (929, 0) = 1249123.5965927865, (929, 1) = 32780996.881875437, (929, 2) = .6035483140005127, (930, 0) = .6035483140005127, (930, 1) = 1272328.3947542657, (930, 2) = 32610911.307000533, (931, 0) = 32610911.307000533, (931, 1) = .6042579743351244, (931, 2) = 1295402.2822260754, (932, 0) = 1295402.2822260754, (932, 1) = 32412504.81413907, (932, 2) = .6049676346697361, (933, 0) = .6049676346697361, (933, 1) = 1318326.2847813761, (933, 2) = 32189222.94927621, (934, 0) = 32189222.94927621, (934, 1) = .6056772950043477, (934, 2) = 1341084.0787530504, (935, 0) = 1341084.0787530504, (935, 1) = 31944735.297750823, (935, 2) = .6063869553389594, (936, 0) = .6063869553389594, (936, 1) = 1363661.9782368145, (936, 2) = 31682840.293128483, (937, 0) = 31682840.293128483, (937, 1) = .6070966156735711, (937, 2) = 1386049.0007964408, (938, 0) = 1386049.0007964408, (938, 1) = 31407418.542974517, (938, 2) = .6077875371304292, (939, 0) = .6077875371304292, (939, 1) = 1407653.6518001645, (939, 2) = 31130021.58985029, (940, 0) = 31130021.58985029, (940, 1) = .6084784585872873, (940, 2) = 1429064.5575309035, (941, 0) = 1429064.5575309035, (941, 1) = 30847167.977542955, (941, 2) = .6091693800441454, (942, 0) = .6091693800441454, (942, 1) = 1450279.1667823375, (942, 2) = 30562428.375809748, (943, 0) = 30562428.375809748, (943, 1) = .6098603015010036, (943, 2) = 1471297.3837971212, (944, 0) = 1471297.3837971212, (944, 1) = 30279277.958339807, (944, 2) = .6105846400362738, (945, 0) = .6105846400362738, (945, 1) = 1493123.836826349, (945, 2) = 29987808.40560109, (946, 0) = 29987808.40560109, (946, 1) = .611308978571544, (946, 2) = 1514742.2844668033, (947, 0) = 1514742.2844668033, (947, 1) = 29705492.927968826, (947, 2) = .6120333171068142, (948, 0) = .6120333171068142, (948, 1) = 1536160.5839480364, (948, 2) = 29435828.9871662, (949, 0) = 29435828.9871662, (949, 1) = .6127576556420844, (949, 2) = 1557389.1003185986, (950, 0) = 1557389.1003185986, (950, 1) = 29182082.350863084, (950, 2) = .613597707323476, (951, 0) = .613597707323476, (951, 1) = 1581788.0830446754, (951, 2) = 28911719.40315091, (952, 0) = 28911719.40315091, (952, 1) = .6144377590048675, (952, 2) = 1605972.2164699656, (953, 0) = 1605972.2164699656, (953, 1) = 28671140.02925078, (953, 2) = .6152778106862592, (954, 0) = .6152778106862592, (954, 1) = 1629968.0433079954, (954, 2) = 28464199.01034319, (955, 0) = 28464199.01034319, (955, 1) = .6161178623676508, (955, 2) = 1653805.296681478, (956, 0) = 1653805.296681478, (956, 1) = 28294301.1851779, (956, 2) = .6169830648536256, (957, 0) = .6169830648536256, (957, 1) = 1678224.8679791281, (957, 2) = 28161174.932313368, (958, 0) = 28161174.932313368, (958, 1) = .6178482673396006, (958, 2) = 1702548.6797149887, (959, 0) = 1702548.6797149887, (959, 1) = 28073281.373162888, (959, 2) = .6187134698255755, (960, 0) = .6187134698255755, (960, 1) = 1726816.8026484002, (960, 2) = 28032998.16395125, (961, 0) = 28032998.16395125, (961, 1) = .6195786723115504, (961, 2) = 1751071.34063465, (962, 0) = 1751071.34063465, (962, 1) = 28042323.96274925, (962, 2) = .6204438747975253, (963, 0) = .6204438747975253, (963, 1) = 1775356.1140148335, (963, 2) = 28102906.518885482, (964, 0) = 28102906.518885482, (964, 1) = .6213090772835002, (964, 2) = 1799715.991757361, (965, 0) = 1799715.991757361, (965, 1) = 28216089.827601116, (965, 2) = .6221742797694751, (966, 0) = .6221742797694751, (966, 1) = 1824196.9005557466, (966, 2) = 28382980.33942945, (967, 0) = 28382980.33942945, (967, 1) = .62303948225545, (967, 2) = 1848845.7338705657, (968, 0) = 1848845.7338705657, (968, 1) = 28604481.401182715, (968, 2) = .6238897528686292, (969, 0) = .6238897528686292, (969, 1) = 1873278.9539422377, (969, 2) = 28876066.78764884, (970, 0) = 28876066.78764884, (970, 1) = .6247400234818086, (970, 2) = 1897966.021485055, (971, 0) = 1897966.021485055, (971, 1) = 29201655.44238633, (971, 2) = .6255902940949878, (972, 0) = .6255902940949878, (972, 1) = 1922953.0294000506, (972, 2) = 29581732.501414653, (973, 0) = 29581732.501414653, (973, 1) = .626440564708167, (973, 2) = 1948286.5133714026, (974, 0) = 1948286.5133714026, (974, 1) = 30016731.3728635, (974, 2) = .627326269018945, (975, 0) = .627326269018945, (975, 1) = 1975094.7526411116, (975, 2) = 30528689.431895357, (976, 0) = 30528689.431895357, (976, 1) = .6282119733297229, (976, 2) = 2002383.173061685, (977, 0) = 2002383.173061685, (977, 1) = 31101115.39945869, (977, 2) = .6290976776405008, (978, 0) = .6290976776405008, (978, 1) = 2030205.5281563932, (978, 2) = 31734483.73365151, (979, 0) = 31734483.73365151, (979, 1) = .6299833819512788, (979, 2) = 2058616.0441003412, (980, 0) = 2058616.0441003412, (980, 1) = 32429321.584762845, (980, 2) = .6309365960594568, (981, 0) = .6309365960594568, (981, 1) = 2089911.8674892907, (981, 2) = 33246477.660809543, (982, 0) = 33246477.660809543, (982, 1) = .6318898101676349, (982, 2) = 2122021.186370865, (983, 0) = 2122021.186370865, (983, 1) = 34136369.749654695, (983, 2) = .6328430242758128, (984, 0) = .6328430242758128, (984, 1) = 2155013.8184670024, (984, 2) = 35100002.87323617, (985, 0) = 35100002.87323617, (985, 1) = .6337962383839909, (985, 2) = 2188960.6269531627, (986, 0) = 2188960.6269531627, (986, 1) = 36138553.56052189, (986, 2) = .6348406231966771, (987, 0) = .6348406231966771, (987, 1) = 2227335.0980526856, (987, 2) = 37364061.72124663, (988, 0) = 37364061.72124663, (988, 1) = .6358850080093633, (988, 2) = 2267038.1250409554, (989, 0) = 2267038.1250409554, (989, 1) = 38683205.11295852, (989, 2) = .6369293928220495, (990, 0) = .6369293928220495, (990, 1) = 2308168.7851424264, (990, 2) = 40098376.878768764, (991, 0) = 40098376.878768764, (991, 1) = .6379737776347357, (991, 2) = 2350828.8052038597, (992, 0) = 2350828.8052038597, (992, 1) = 41612316.521659456, (992, 2) = .639123098120588, (993, 0) = .639123098120588, (993, 1) = 2399667.8337340765, (993, 2) = 43396236.29411974, (994, 0) = 43396236.29411974, (994, 1) = .6402724186064404, (994, 2) = 2450630.1626505423, (995, 0) = 2450630.1626505423, (995, 1) = 45308189.194686495, (995, 2) = .6414217390922927, (996, 0) = .6414217390922927, (996, 1) = 2503866.0587969813, (996, 2) = 47353438.82219228, (997, 0) = 47353438.82219228, (997, 1) = .642571059578145, (997, 2) = 2559532.106788657, (998, 0) = 2559532.106788657, (998, 1) = 49537878.18957457, (998, 2) = .643829074349073, (999, 0) = .643829074349073, (999, 1) = 2623441.9548407616, (999, 2) = 52096219.3651618, (1000, 0) = 52096219.3651618, (1000, 1) = .6450870891200011, (1000, 2) = 2690684.81349888, (1001, 0) = 2690684.81349888, (1001, 1) = 54838684.984271534, (1001, 2) = .6463451038909291, (1002, 0) = .6463451038909291, (1002, 1) = 2761499.139304321, (1002, 2) = 57775836.308509015, (1003, 0) = 57775836.308509015, (1003, 1) = .6476031186618572, (1003, 2) = 2836137.1719181924, (1004, 0) = 2836137.1719181924, (1004, 1) = 60919325.79498723, (1004, 2) = .6489722834394382, (1005, 0) = .6489722834394382, (1005, 1) = 2922028.131637721, (1005, 2) = 64590021.002394065, (1006, 0) = 64590021.002394065, (1006, 1) = .6503414482170193, (1006, 2) = 3013132.4283512607, (1007, 0) = 3013132.4283512607, (1007, 1) = 68538341.41484345, (1007, 2) = .6517106129946005, (1008, 0) = .6517106129946005, (1008, 1) = 3109844.045479892, (1008, 2) = 72784103.77576719, (1009, 0) = 72784103.77576719, (1009, 1) = .6530797777721816, (1009, 2) = 3212585.061952949, (1010, 0) = 3212585.061952949, (1010, 1) = 77348931.7961268, (1010, 2) = .6544024911453019, (1011, 0) = .6544024911453019, (1011, 1) = 3317990.725449382, (1011, 2) = 82083913.80500948, (1012, 0) = 82083913.80500948, (1012, 1) = .6557252045184223, (1012, 2) = 3429883.0067536877, (1013, 0) = 3429883.0067536877, (1013, 1) = 87161437.74513753, (1013, 2) = .6570479178915426, (1014, 0) = .6570479178915426, (1014, 1) = 3548731.3735707747, (1014, 2) = 92606029.91648005, (1015, 0) = 92606029.91648005, (1015, 1) = .6583706312646629, (1015, 2) = 3675038.8029821897, (1016, 0) = 3675038.8029821897, (1016, 1) = 98443850.13870624, (1016, 2) = .6596274674222308, (1017, 0) = .6596274674222308, (1017, 1) = 3802456.453844456, (1017, 2) = 104380518.64089516, (1018, 0) = 104380518.64089516, (1018, 1) = .6608843035797987, (1018, 2) = 3937587.255961265, (1019, 0) = 3937587.255961265, (1019, 1) = 110722409.88274199, (1019, 2) = .6621411397373667, (1020, 0) = .6621411397373667, (1020, 1) = 4080956.771071618, (1020, 2) = 117495665.22209361, (1021, 0) = 117495665.22209361, (1021, 1) = .6633979758949345, (1021, 2) = 4233124.449461746, (1022, 0) = 4233124.449461746, (1022, 1) = 124727248.37811223, (1022, 2) = .6644668807137524, (1023, 0) = .6644668807137524, (1023, 1) = 4369904.962413958, (1023, 2) = 131258799.7388615, (1024, 0) = 131258799.7388615, (1024, 1) = .6655357855325702, (1024, 2) = 4513862.58750542, (1025, 0) = 4513862.58750542, (1025, 1) = 138159045.9490277, (1025, 2) = .666604690351388, (1026, 0) = .666604690351388, (1026, 1) = 4665400.134116391, (1026, 2) = 145444992.28698072, (1027, 0) = 145444992.28698072, (1027, 1) = .6676735951702059, (1027, 2) = 4824939.091014138, (1028, 0) = 4824939.091014138, (1028, 1) = 153133211.92122102, (1028, 2) = .6686250297159377, (1029, 0) = .6686250297159377, (1029, 1) = 4974031.441406206, (1029, 2) = 160327834.6440145, (1030, 0) = 160327834.6440145, (1030, 1) = .6695764642616695, (1030, 2) = 5130130.967951499, (1031, 0) = 5130130.967951499, (1031, 1) = 167864392.98095107, (1031, 2) = .6705278988074014, (1032, 0) = .6705278988074014, (1032, 1) = 5293567.011307463, (1032, 2) = 175752378.3918431, (1033, 0) = 175752378.3918431, (1033, 1) = .6714793333531333, (1033, 2) = 5464678.170587061, (1034, 0) = 5464678.170587061, (1034, 1) = 183999938.84142354, (1034, 2) = .6723632182075568, (1035, 0) = .6723632182075568, (1035, 1) = 5630820.832039482, (1035, 2) = 191989910.78768885, (1036, 0) = 191989910.78768885, (1036, 1) = .6732471030619804, (1036, 2) = 5804167.022217236, (1037, 0) = 5804167.022217236, (1037, 1) = 200299663.84245694, (1037, 2) = .674130987916404, (1038, 0) = .674130987916404, (1038, 1) = 5984999.521903008, (1038, 2) = 208930954.40219155, (1039, 0) = 208930954.40219155, (1039, 1) = .6750148727708276, (1039, 2) = 6173602.704654874, (1040, 0) = 6173602.704654874, (1040, 1) = 217883260.13390142, (1040, 2) = .6758651288344302, (1041, 0) = .6758651288344302, (1041, 1) = 6362627.498651925, (1041, 2) = 226795073.0405768, (1042, 0) = 226795073.0405768, (1042, 1) = .6767153848980327, (1042, 2) = 6559353.243233879, (1043, 0) = 6559353.243233879, (1043, 1) = 235995820.84757015, (1043, 2) = .6775656409616352, (1044, 0) = .6775656409616352, (1044, 1) = 6764021.177117888, (1044, 2) = 245477056.95237303, (1045, 0) = 245477056.95237303, (1045, 1) = .6784158970252377, (1045, 2) = 6976865.213117, (1046, 0) = 6976865.213117, (1046, 1) = 255226908.46668133, (1046, 2) = .6792702935346726, (1047, 0) = .6792702935346726, (1047, 1) = 7199207.278361248, (1047, 2) = 265279124.7038306, (1048, 0) = 265279124.7038306, (1048, 1) = .6801246900441076, (1048, 2) = 7430240.604065027, (1049, 0) = 7430240.604065027, (1049, 1) = 275566822.329839, (1049, 2) = .6809790865535424, (1050, 0) = .6809790865535424, (1050, 1) = 7670154.584449086, (1050, 2) = 286065119.09425527, (1051, 0) = 286065119.09425527, (1051, 1) = .6818334830629773, (1051, 2) = 7919116.960588606, (1052, 0) = 7919116.960588606, (1052, 1) = 296744042.8686687, (1052, 2) = .6827814777026986, (1053, 0) = .6827814777026986, (1053, 1) = 8206113.411022565, (1053, 2) = 308761292.9846274, (1054, 0) = 308761292.9846274, (1054, 1) = .6837294723424198, (1054, 2) = 8504565.896482315, (1055, 0) = 8504565.896482315, (1055, 1) = 320900736.96524835, (1055, 2) = .6846774669821409, (1056, 0) = .6846774669821409, (1056, 1) = 8814556.989623716, (1056, 2) = 333096748.46220505, (1057, 0) = 333096748.46220505, (1057, 1) = .6856254616218622, (1057, 2) = 9136106.043626556, (1058, 0) = 9136106.043626556, (1058, 1) = 345274822.1453267, (1058, 2) = .6863655855700008, (1059, 0) = .6863655855700008, (1059, 1) = 9395151.496801715, (1059, 2) = 354716716.14980835, (1060, 0) = 354716716.14980835, (1060, 1) = .6871057095181393, (1060, 2) = 9661149.24879759, (1061, 0) = 9661149.24879759, (1061, 1) = 364053581.2876393, (1061, 2) = .6878458334662778, (1062, 0) = .6878458334662778, (1062, 1) = 9934004.049161136, (1062, 2) = 373239244.6552961, (1063, 0) = 373239244.6552961, (1063, 1) = .6885859574144164, (1063, 2) = 10213586.203658294, (1064, 0) = 10213586.203658294, (1064, 1) = 382225152.85291684, (1064, 2) = .689326081362555, (1065, 0) = .689326081362555, (1065, 1) = 10499729.903703546, (1065, 2) = 390960658.18897945, (1066, 0) = 390960658.18897945, (1066, 1) = .6900662053106935, (1066, 2) = 10792230.141745001, (1067, 0) = 10792230.141745001, (1067, 1) = 399393236.671029, (1067, 2) = .690806329258832, (1068, 0) = .690806329258832, (1068, 1) = 11090843.108131163, (1068, 2) = 407469820.8510915, (1069, 0) = 407469820.8510915, (1069, 1) = .6915464532069706, (1069, 2) = 11395285.681034338, (1070, 0) = 11395285.681034338, (1070, 1) = 415137457.64857996, (1070, 2) = .6922329169348227, (1071, 0) = .6922329169348227, (1071, 1) = 11682586.070238896, (1071, 2) = 421837912.5095289, (1072, 0) = 421837912.5095289, (1072, 1) = .6929193806626748, (1072, 2) = 11974337.933538118, (1073, 0) = 11974337.933538118, (1073, 1) = 428100714.8082822, (1073, 2) = .693605844390527}, datatype = float[8], order = C_order)), ( 15 ) = ("rkf45"), ( 14 ) = ([0, 0]), ( 19 ) = (0), ( 16 ) = ([0, 0, 0, []]), ( 17 ) = ([proc (N, X, Y, YP) option `[Y[1] = x(t), Y[2] = diff(x(t),t)]`; YP[2] := (9.86960440108934*(1.57079632679490+200*sin(31.4159265358979*X)-19.73920881*cos(31.4159265358979*X))*sin(31.4159265358979*X)-125.663706143592*X*sin(31.4159265358979*X)*cos(31.4159265358979*X)*Y[2]-(9.86960440108934-1973.92088021787*sin(31.4159265358979*X)^2+62.01255338*sin(31.4159265358979*X)*cos(31.4159265358979*X))*Y[1])/(1+sin(31.4159265358979*X)^2); YP[1] := Y[2]; 0 end proc, -1, 0, 0, 0, 0, 0, 0]), ( 22 ) = (0), ( 23 ) = (0), ( 20 ) = ([]), ( 21 ) = (0), ( 26 ) = (Array(1..0, {})), ( 25 ) = (Array(1..0, {})), ( 24 ) = (0) ] )), ( 4 ) = (3) ] ); _y0 := Array(0..2, {(1) = 0., (2) = 0.}); _vmap := array( 1 .. 2, [( 1 ) = (1), ( 2 ) = (2) ] ); _x0 := _dtbl[1][5][5]; _n := _dtbl[1][4][1]; _ne := _dtbl[1][4][3]; _nd := _dtbl[1][4][4]; _nv := _dtbl[1][4][16]; if not type(_xout, 'numeric') then if member(_xout, ["start", "left", "right"]) then if _Env_smart_dsolve_numeric = true or _dtbl[1][4][10] = 1 then if _xout = "left" then if type(_dtbl[2], 'table') then return _dtbl[2][5][1] end if elif _xout = "right" then if type(_dtbl[3], 'table') then return _dtbl[3][5][1] end if end if end if; return _dtbl[1][5][5] elif _xout = "method" then return _dtbl[1][15] elif _xout = "storage" then return evalb(_dtbl[1][4][10] = 1) elif _xout = "leftdata" then if not type(_dtbl[2], 'array') then return NULL else return eval(_dtbl[2]) end if elif _xout = "rightdata" then if not type(_dtbl[3], 'array') then return NULL else return eval(_dtbl[3]) end if elif _xout = "enginedata" then return eval(_dtbl[1]) elif _xout = "enginereset" then _dtbl[2] := evaln(_dtbl[2]); _dtbl[3] := evaln(_dtbl[3]); return NULL elif _xout = "initial" then return procname(_y0[0]) elif _xout = "laxtol" then return _dtbl[`if`(member(_dtbl[4], {2, 3}), _dtbl[4], 1)][5][18] elif _xout = "numfun" then return `if`(member(_dtbl[4], {2, 3}), _dtbl[_dtbl[4]][4][18], 0) elif _xout = "parameters" then return [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] elif _xout = "initial_and_parameters" then return procname(_y0[0]), [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] elif _xout = "last" then if _dtbl[4] <> 2 and _dtbl[4] <> 3 or _x0-_dtbl[_dtbl[4]][5][1] = 0. then error "no information is available on last computed point" else _xout := _dtbl[_dtbl[4]][5][1] end if elif _xout = "function" then if _dtbl[1][4][33]-2. = 0 then return eval(_dtbl[1][10], 1) else return eval(_dtbl[1][10][1], 1) end if elif _xout = "map" then return copy(_vmap) elif type(_xin, `=`) and type(rhs(_xin), 'list') and member(lhs(_xin), {"initial", "parameters", "initial_and_parameters"}) then _ini, _par := [], []; if lhs(_xin) = "initial" then _ini := rhs(_xin) elif lhs(_xin) = "parameters" then _par := rhs(_xin) elif select(type, rhs(_xin), `=`) <> [] then _par, _ini := selectremove(type, rhs(_xin), `=`) elif nops(rhs(_xin)) < nops(_pars)+1 then error "insufficient data for specification of initial and parameters" else _par := rhs(_xin)[-nops(_pars) .. -1]; _ini := rhs(_xin)[1 .. -nops(_pars)-1] end if; _xout := lhs(_xout); if _par <> [] then `dsolve/numeric/process_parameters`(_n, _pars, _par, _y0) end if; if _ini <> [] then `dsolve/numeric/process_initial`(_n-_ne, _ini, _y0, _pars, _vmap) end if; `dsolve/numeric/SC/reinitialize`(_dtbl, _y0, _n, procname, _pars); if _Env_smart_dsolve_numeric = true and type(_y0[0], 'numeric') and _dtbl[1][4][10] <> 1 then procname("right") := _y0[0]; procname("left") := _y0[0] end if; if _xout = "initial" then return [_y0[0], seq(_y0[_vmap[_i]], _i = 1 .. _n-_ne)] elif _xout = "parameters" then return [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] else return [_y0[0], seq(_y0[_vmap[_i]], _i = 1 .. _n-_ne)], [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] end if elif _xin = "eventstop" then if _nv = 0 then error "this solution has no events" end if; _i := _dtbl[4]; if _i <> 2 and _i <> 3 then return 0 end if; if _dtbl[_i][4][10] = 1 and assigned(_dtbl[5-_i]) and _dtbl[_i][4][9] < 100 and 100 <= _dtbl[5-_i][4][9] then _i := 5-_i; _dtbl[4] := _i; _j := round(_dtbl[_i][4][17]); return round(_dtbl[_i][3][1][_j, 1]) elif 100 <= _dtbl[_i][4][9] then _j := round(_dtbl[_i][4][17]); return round(_dtbl[_i][3][1][_j, 1]) else return 0 end if elif _xin = "eventstatus" then if _nv = 0 then error "this solution has no events" end if; _i := [selectremove(proc (a) options operator, arrow; _dtbl[1][3][1][a, 7] = 1 end proc, {seq(_j, _j = 1 .. round(_dtbl[1][3][1][_nv+1, 1]))})]; return ':-enabled' = _i[1], ':-disabled' = _i[2] elif _xin = "eventclear" then if _nv = 0 then error "this solution has no events" end if; _i := _dtbl[4]; if _i <> 2 and _i <> 3 then error "no events to clear" end if; if _dtbl[_i][4][10] = 1 and assigned(_dtbl[5-_i]) and _dtbl[_i][4][9] < 100 and 100 < _dtbl[5-_i][4][9] then _dtbl[4] := 5-_i; _i := 5-_i end if; if _dtbl[_i][4][9] < 100 then error "no events to clear" elif _nv < _dtbl[_i][4][9]-100 then error "event error condition cannot be cleared" else _j := _dtbl[_i][4][9]-100; if irem(round(_dtbl[_i][3][1][_j, 4]), 2) = 1 then error "retriggerable events cannot be cleared" end if; _j := round(_dtbl[_i][3][1][_j, 1]); for _k to _nv do if _dtbl[_i][3][1][_k, 1] = _j then if _dtbl[_i][3][1][_k, 2] = 3 then error "range events cannot be cleared" end if; _dtbl[_i][3][1][_k, 8] := _dtbl[_i][3][1][_nv+1, 8] end if end do; _dtbl[_i][4][17] := 0; _dtbl[_i][4][9] := 0; if _dtbl[1][4][10] = 1 then if _i = 2 then try procname(procname("left")) catch: end try else try procname(procname("right")) catch: end try end if end if end if; return elif type(_xin, `=`) and member(lhs(_xin), {"eventdisable", "eventenable"}) then if _nv = 0 then error "this solution has no events" end if; if type(rhs(_xin), {('list')('posint'), ('set')('posint')}) then _i := {op(rhs(_xin))} elif type(rhs(_xin), 'posint') then _i := {rhs(_xin)} else error "event identifiers must be integers in the range 1..%1", round(_dtbl[1][3][1][_nv+1, 1]) end if; if select(proc (a) options operator, arrow; _nv < a end proc, _i) <> {} then error "event identifiers must be integers in the range 1..%1", round(_dtbl[1][3][1][_nv+1, 1]) end if; _k := {}; for _j to _nv do if member(round(_dtbl[1][3][1][_j, 1]), _i) then _k := `union`(_k, {_j}) end if end do; _i := _k; if lhs(_xin) = "eventdisable" then _dtbl[4] := 0; _j := [evalb(assigned(_dtbl[2]) and member(_dtbl[2][4][17], _i)), evalb(assigned(_dtbl[3]) and member(_dtbl[3][4][17], _i))]; for _k in _i do _dtbl[1][3][1][_k, 7] := 0; if assigned(_dtbl[2]) then _dtbl[2][3][1][_k, 7] := 0 end if; if assigned(_dtbl[3]) then _dtbl[3][3][1][_k, 7] := 0 end if end do; if _j[1] then for _k to _nv+1 do if _k <= _nv and not type(_dtbl[2][3][4][_k, 1], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to defined init `, _dtbl[2][3][4][_k, 1]); _dtbl[2][3][1][_k, 8] := _dtbl[2][3][4][_k, 1] elif _dtbl[2][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[2][3][1][_k, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to rate hysteresis init `, _dtbl[2][5][24]); _dtbl[2][3][1][_k, 8] := _dtbl[2][5][24] elif _dtbl[2][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[2][3][1][_k, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to initial init `, _x0); _dtbl[2][3][1][_k, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to fireinitial init `, _x0-1); _dtbl[2][3][1][_k, 8] := _x0-1 end if end do; _dtbl[2][4][17] := 0; _dtbl[2][4][9] := 0; if _dtbl[1][4][10] = 1 then procname(procname("left")) end if end if; if _j[2] then for _k to _nv+1 do if _k <= _nv and not type(_dtbl[3][3][4][_k, 2], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to defined init `, _dtbl[3][3][4][_k, 2]); _dtbl[3][3][1][_k, 8] := _dtbl[3][3][4][_k, 2] elif _dtbl[3][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[3][3][1][_k, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to rate hysteresis init `, _dtbl[3][5][24]); _dtbl[3][3][1][_k, 8] := _dtbl[3][5][24] elif _dtbl[3][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[3][3][1][_k, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to initial init `, _x0); _dtbl[3][3][1][_k, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to fireinitial init `, _x0+1); _dtbl[3][3][1][_k, 8] := _x0+1 end if end do; _dtbl[3][4][17] := 0; _dtbl[3][4][9] := 0; if _dtbl[1][4][10] = 1 then procname(procname("right")) end if end if else for _k in _i do _dtbl[1][3][1][_k, 7] := 1 end do; _dtbl[2] := evaln(_dtbl[2]); _dtbl[3] := evaln(_dtbl[3]); _dtbl[4] := 0; if _dtbl[1][4][10] = 1 then if _x0 <= procname("right") then try procname(procname("right")) catch: end try end if; if procname("left") <= _x0 then try procname(procname("left")) catch: end try end if end if end if; return elif type(_xin, `=`) and lhs(_xin) = "eventfired" then if not type(rhs(_xin), 'list') then error "'eventfired' must be specified as a list" end if; if _nv = 0 then error "this solution has no events" end if; if _dtbl[4] <> 2 and _dtbl[4] <> 3 then error "'direction' must be set prior to calling/setting 'eventfired'" end if; _i := _dtbl[4]; _val := NULL; if not assigned(_EnvEventRetriggerWarned) then _EnvEventRetriggerWarned := false end if; for _k in rhs(_xin) do if type(_k, 'integer') then _src := _k elif type(_k, 'integer' = 'anything') and type(evalf(rhs(_k)), 'numeric') then _k := lhs(_k) = evalf[max(Digits, 18)](rhs(_k)); _src := lhs(_k) else error "'eventfired' entry is not valid: %1", _k end if; if _src < 1 or round(_dtbl[1][3][1][_nv+1, 1]) < _src then error "event identifiers must be integers in the range 1..%1", round(_dtbl[1][3][1][_nv+1, 1]) end if; _src := {seq(`if`(_dtbl[1][3][1][_j, 1]-_src = 0., _j, NULL), _j = 1 .. _nv)}; if nops(_src) <> 1 then error "'eventfired' can only be set/queried for root-finding events and time/interval events" end if; _src := _src[1]; if _dtbl[1][3][1][_src, 2] <> 0. and _dtbl[1][3][1][_src, 2]-2. <> 0. then error "'eventfired' can only be set/queried for root-finding events and time/interval events" elif irem(round(_dtbl[1][3][1][_src, 4]), 2) = 1 then if _EnvEventRetriggerWarned = false then WARNING(`'eventfired' has no effect on events that retrigger`) end if; _EnvEventRetriggerWarned := true end if; if _dtbl[_i][3][1][_src, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_src, 4]), 32), 2) = 1 then _val := _val, undefined elif type(_dtbl[_i][3][4][_src, _i-1], 'undefined') or _i = 2 and _dtbl[2][3][1][_src, 8] < _dtbl[2][3][4][_src, 1] or _i = 3 and _dtbl[3][3][4][_src, 2] < _dtbl[3][3][1][_src, 8] then _val := _val, _dtbl[_i][3][1][_src, 8] else _val := _val, _dtbl[_i][3][4][_src, _i-1] end if; if type(_k, `=`) then if _dtbl[_i][3][1][_src, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_src, 4]), 32), 2) = 1 then error "cannot set event code for a rate hysteresis event" end if; userinfo(3, {'events', 'eventreset'}, `manual set event code `, _src, ` to value `, rhs(_k)); _dtbl[_i][3][1][_src, 8] := rhs(_k); _dtbl[_i][3][4][_src, _i-1] := rhs(_k) end if end do; return [_val] elif type(_xin, `=`) and lhs(_xin) = "direction" then if not member(rhs(_xin), {-1, 1, ':-left', ':-right'}) then error "'direction' must be specified as either '1' or 'right' (positive) or '-1' or 'left' (negative)" end if; _src := `if`(_dtbl[4] = 2, -1, `if`(_dtbl[4] = 3, 1, undefined)); _i := `if`(member(rhs(_xin), {1, ':-right'}), 3, 2); _dtbl[4] := _i; _dtbl[_i] := `dsolve/numeric/SC/IVPdcopy`(_dtbl[1], `if`(assigned(_dtbl[_i]), _dtbl[_i], NULL)); if 0 < _nv then for _j to _nv+1 do if _j <= _nv and not type(_dtbl[_i][3][4][_j, _i-1], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to defined init `, _dtbl[_i][3][4][_j, _i-1]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][3][4][_j, _i-1] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to rate hysteresis init `, _dtbl[_i][5][24]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][5][24] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to initial init `, _x0); _dtbl[_i][3][1][_j, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to fireinitial init `, _x0-2*_i+5.0); _dtbl[_i][3][1][_j, 8] := _x0-2*_i+5.0 end if end do end if; return _src elif _xin = "eventcount" then if _dtbl[1][3][1] = 0 or _dtbl[4] <> 2 and _dtbl[4] <> 3 then return 0 else return round(_dtbl[_dtbl[4]][3][1][_nv+1, 12]) end if else return "procname" end if end if; if _xout = _x0 then return [_x0, seq(evalf(_dtbl[1][6][_vmap[_i]]), _i = 1 .. _n-_ne)] end if; _i := `if`(_x0 <= _xout, 3, 2); if _xin = "last" and 0 < _dtbl[_i][4][9] and _dtbl[_i][4][9] < 100 then _dat := eval(_dtbl[_i], 2); _j := _dat[4][20]; return [_dat[11][_j, 0], seq(_dat[11][_j, _vmap[_i]], _i = 1 .. _n-_ne-_nd), seq(_dat[8][1][_vmap[_i]], _i = _n-_ne-_nd+1 .. _n-_ne)] end if; if not type(_dtbl[_i], 'array') then _dtbl[_i] := `dsolve/numeric/SC/IVPdcopy`(_dtbl[1], `if`(assigned(_dtbl[_i]), _dtbl[_i], NULL)); if 0 < _nv then for _j to _nv+1 do if _j <= _nv and not type(_dtbl[_i][3][4][_j, _i-1], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to defined init `, _dtbl[_i][3][4][_j, _i-1]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][3][4][_j, _i-1] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to rate hysteresis init `, _dtbl[_i][5][24]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][5][24] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to initial init `, _x0); _dtbl[_i][3][1][_j, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to fireinitial init `, _x0-2*_i+5.0); _dtbl[_i][3][1][_j, 8] := _x0-2*_i+5.0 end if end do end if end if; if _xin <> "last" then if 0 < 0 then if `dsolve/numeric/checkglobals`(op(_dtbl[1][14]), _pars, _n, _y0) then `dsolve/numeric/SC/reinitialize`(_dtbl, _y0, _n, procname, _pars, _i) end if end if; if _dtbl[1][4][7] = 0 then error "parameters must be initialized before solution can be computed" end if end if; _dat := eval(_dtbl[_i], 2); _dtbl[4] := _i; try _src := `dsolve/numeric/SC/IVPrun`(_dat, _xout) catch: userinfo(2, `dsolve/debug`, print(`Exception in solnproc:`, [lastexception][2 .. -1])); error end try; if _dat[17] <> _dtbl[1][17] then _dtbl[1][17] := _dat[17]; _dtbl[1][10] := _dat[10] end if; if _src = 0 and 100 < _dat[4][9] then _val := _dat[3][1][_nv+1, 8] else _val := _dat[11][_dat[4][20], 0] end if; if _src <> 0 or _dat[4][9] <= 0 then _dtbl[1][5][1] := _xout else _dtbl[1][5][1] := _val end if; if _i = 3 and _val < _xout then Rounding := -infinity; if _dat[4][9] = 1 then error "cannot evaluate the solution further right of %1, probably a singularity", evalf[8](_val) elif _dat[4][9] = 2 then error "cannot evaluate the solution further right of %1, maxfun limit exceeded (see ?dsolve,maxfun for details)", evalf[8](_val) elif _dat[4][9] = 3 then if _dat[4][25] = 3 then error "cannot evaluate the solution past the initial point, problem may be initially singular or improperly set up" else error "cannot evaluate the solution past the initial point, problem may be complex, initially singular or improperly set up" end if elif _dat[4][9] = 4 then error "cannot evaluate the solution further right of %1, accuracy goal cannot be achieved with specified 'minstep'", evalf[8](_val) elif _dat[4][9] = 5 then error "cannot evaluate the solution further right of %1, too many step failures, tolerances may be too loose for problem", evalf[8](_val) elif _dat[4][9] = 6 then error "cannot evaluate the solution further right of %1, cannot downgrade delay storage for problems with delay derivative order > 1, try increasing delaypts", evalf[8](_val) elif _dat[4][9] = 10 then error "cannot evaluate the solution further right of %1, interrupt requested", evalf[8](_val) elif 100 < _dat[4][9] then if _dat[4][9]-100 = _nv+1 then error "constraint projection failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-100 = _nv+2 then error "index-1 and derivative evaluation failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-100 = _nv+3 then error "maximum number of event iterations reached (%1) at t=%2", round(_dat[3][1][_nv+1, 3]), evalf[8](_val) else if _Env_dsolve_nowarnstop <> true then `dsolve/numeric/warning`(StringTools:-FormatMessage("cannot evaluate the solution further right of %1, event #%2 triggered a halt", evalf[8](_val), round(_dat[3][1][_dat[4][9]-100, 1]))) end if; Rounding := 'nearest'; _xout := _val end if else error "cannot evaluate the solution further right of %1", evalf[8](_val) end if elif _i = 2 and _xout < _val then Rounding := infinity; if _dat[4][9] = 1 then error "cannot evaluate the solution further left of %1, probably a singularity", evalf[8](_val) elif _dat[4][9] = 2 then error "cannot evaluate the solution further left of %1, maxfun limit exceeded (see ?dsolve,maxfun for details)", evalf[8](_val) elif _dat[4][9] = 3 then if _dat[4][25] = 3 then error "cannot evaluate the solution past the initial point, problem may be initially singular or improperly set up" else error "cannot evaluate the solution past the initial point, problem may be complex, initially singular or improperly set up" end if elif _dat[4][9] = 4 then error "cannot evaluate the solution further left of %1, accuracy goal cannot be achieved with specified 'minstep'", evalf[8](_val) elif _dat[4][9] = 5 then error "cannot evaluate the solution further left of %1, too many step failures, tolerances may be too loose for problem", evalf[8](_val) elif _dat[4][9] = 6 then error "cannot evaluate the solution further left of %1, cannot downgrade delay storage for problems with delay derivative order > 1, try increasing delaypts", evalf[8](_val) elif _dat[4][9] = 10 then error "cannot evaluate the solution further right of %1, interrupt requested", evalf[8](_val) elif 100 < _dat[4][9] then if _dat[4][9]-100 = _nv+1 then error "constraint projection failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-100 = _nv+2 then error "index-1 and derivative evaluation failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-100 = _nv+3 then error "maximum number of event iterations reached (%1) at t=%2", round(_dat[3][1][_nv+1, 3]), evalf[8](_val) else if _Env_dsolve_nowarnstop <> true then `dsolve/numeric/warning`(StringTools:-FormatMessage("cannot evaluate the solution further left of %1, event #%2 triggered a halt", evalf[8](_val), round(_dat[3][1][_dat[4][9]-100, 1]))) end if; Rounding := 'nearest'; _xout := _val end if else error "cannot evaluate the solution further left of %1", evalf[8](_val) end if end if; if _EnvInFsolve = true then _dig := _dat[4][26]; if type(_EnvDSNumericSaveDigits, 'posint') then _dat[4][26] := _EnvDSNumericSaveDigits else _dat[4][26] := Digits end if; _Env_dsolve_SC_native := true; if _dat[4][25] = 1 then _i := 1; _dat[4][25] := 2 else _i := _dat[4][25] end if; _val := `dsolve/numeric/SC/IVPval`(_dat, _xout, _src); _dat[4][25] := _i; _dat[4][26] := _dig; [_xout, seq(_val[_vmap[_i]], _i = 1 .. _n-_ne)] else Digits := _dat[4][26]; _val := `dsolve/numeric/SC/IVPval`(eval(_dat, 2), _xout, _src); [_xout, seq(_val[_vmap[_i]], _i = 1 .. _n-_ne)] end if end proc, (2) = Array(0..0, {}), (3) = [t, x(t), diff(x(t), t)], (4) = []}); _vars := _dat[3]; _pars := map(rhs, _dat[4]); _n := nops(_vars)-1; _solnproc := _dat[1]; if not type(_xout, 'numeric') then if member(x_rkf45, ["start", 'start', "method", 'method', "left", 'left', "right", 'right', "leftdata", "rightdata", "enginedata", "eventstop", 'eventstop', "eventclear", 'eventclear', "eventstatus", 'eventstatus', "eventcount", 'eventcount', "laxtol", 'laxtol', "numfun", 'numfun', NULL]) then _res := _solnproc(convert(x_rkf45, 'string')); if 1 < nops([_res]) then return _res elif type(_res, 'array') then return eval(_res, 1) elif _res <> "procname" then return _res end if elif member(x_rkf45, ["last", 'last', "initial", 'initial', "parameters", 'parameters', "initial_and_parameters", 'initial_and_parameters', NULL]) then _xout := convert(x_rkf45, 'string'); _res := _solnproc(_xout); if _xout = "parameters" then return [seq(_pars[_i] = _res[_i], _i = 1 .. nops(_pars))] elif _xout = "initial_and_parameters" then return [seq(_vars[_i+1] = [_res][1][_i+1], _i = 0 .. _n), seq(_pars[_i] = [_res][2][_i], _i = 1 .. nops(_pars))] else return [seq(_vars[_i+1] = _res[_i+1], _i = 0 .. _n)] end if elif type(_xout, `=`) and member(lhs(_xout), ["initial", 'initial', "parameters", 'parameters', "initial_and_parameters", 'initial_and_parameters', NULL]) then _xout := convert(lhs(x_rkf45), 'string') = rhs(x_rkf45); if type(rhs(_xout), 'list') then _res := _solnproc(_xout) else error "initial and/or parameter values must be specified in a list" end if; if lhs(_xout) = "initial" then return [seq(_vars[_i+1] = _res[_i+1], _i = 0 .. _n)] elif lhs(_xout) = "parameters" then return [seq(_pars[_i] = _res[_i], _i = 1 .. nops(_pars))] else return [seq(_vars[_i+1] = [_res][1][_i+1], _i = 0 .. _n), seq(_pars[_i] = [_res][2][_i], _i = 1 .. nops(_pars))] end if elif type(_xout, `=`) and member(lhs(_xout), ["eventdisable", 'eventdisable', "eventenable", 'eventenable', "eventfired", 'eventfired', "direction", 'direction', NULL]) then return _solnproc(convert(lhs(x_rkf45), 'string') = rhs(x_rkf45)) elif _xout = "solnprocedure" then return eval(_solnproc) elif _xout = "sysvars" then return _vars end if; if procname <> unknown then return ('procname')(x_rkf45) else _ndsol := 1; _ndsol := _ndsol; _ndsol := pointto(_dat[2][0]); return ('_ndsol')(x_rkf45) end if end if; try _res := _solnproc(_xout); [seq(_vars[_i+1] = _res[_i+1], _i = 0 .. _n)] catch: error end try end proc

  

 

 
> 

#analytical solution
dsolve(sys):
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Download odeProb.mw

I can get close with...

tomleslie 13876
November 11 2018
1 0

the following - which does integration by parts twice with the meanings of u and v' swapped round. Since these should give the same answer, subtract them and set the answer to zero

  restart;
  with(IntegrationTools):
  expr:=int(GAMMA(a,s)*exp(-b*s), s= 0..infinity);
  simplify
  ( `assuming`
     ( Parts(expr, GAMMA(a,s), exp(-b*s))-Parts(expr, exp(-b*s), GAMMA(a,s)),
       b > 0,
       a > 0
     )=0,
     power,
     symbolic
  );

 

Integral in the final expression is just (-b) times the original, so further manipulation *ought* to be simple.

What you have...

tomleslie 13876
November 10 2018
1 3

is a simple substitutional code which converts a supplied integer to a text message. Pretty sure this has nothing to do with any RSA cryptosystem.

See the examples in the attached.

The algorithm you are using will fail whenever the required message starts with the character "A", becuase this is represented as 00 - and the two leading zeros will be 'stripped' from the passed integer. This is illustrated in the attached when the returned message should be "APROBLEM", not "PROBLEM"

I have included a more efficient coder+decoder, using commands from Maple's StringTools() package, which does not suffer from this issue. For convenience these code "A".."Z" as integers in the range 65-90 (their ASCII positions) rather than 0..25. I was too lazy to add//subtract 65 everywhere ;-)

> 

toletter := proc (num)
                 local cs, cn, sl, ltable, ans, ii;
                 ltable := table([0 = "A", 1 = "B", 2 = "C", 3 = "D", 4 = "E", 5 = "F", 6 = "G", 7 = "H", 8 = "I", 9 = "J", 10 = "K", 11 = "L", 12 = "M", 13 = "N", 14 = "O", 15 = "P", 16 = "Q", 17 = "R", 18 = "S", 19 = "T", 20 = "U", 21 = "V", 22 = "W", 23 = "X", 24 = "Y", 25 = "Z"]);
                 cn := num;
                 sl := floor((1/2)*trunc(evalf(log10(cn))))+1;
                 ans := " ";
                 for ii to sl do
                     cn := (1/100)*cn;
                     cs := ltable[100*frac(cn)];
                     ans := cat(cs, ans);
                     cn := trunc(cn)
                 end do;
                 return ans;
            end proc:
toletter(151100081319042319); # PLAINTEXT
toletter(19070818081819070412041818000604); # THISISTHEMESSAGE
toletter(0015171401110412);# APROBLEM - misses first character

"PLAINTEXT "

  

"THISISTHEMESSAGE "

  

"PROBLEM "

 (1)
> 

  restart;
  decode:= proc(x::integer)
                uses StringTools:
                return Implode
                       ( Char~
                         ( parse~
                           ( [ LengthSplit(convert(x,string),2) ])
                         )
                       )
           end proc:
  coder:= proc( x::string)
                uses StringTools:
                return parse
                       ( convert
                         ( cat
                           ( Ord~
                             ( Explode(x))[]
                           ),
                           string
                         )
                       ):
          end proc:
#
# Code a message
#
  myCode:=coder("THIS IS A MESSAGE");
#
# Decode the message
#
  decode(myCode);

8472738332738332653277698383657169

  

"THIS IS A MESSAGE"

 (2)
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Download decoder.mw

 

For both examples cited...

tomleslie 13876
November 08 2018
2 0

you can, if you want (and I wouldn't generally recommend it!) just use simplify with the 'symbolic' option as in the attached

> 

restart;
((1-t)^2)^q;
simplify(%, power, symbolic);
exp(I*t)^q;
simplify(%, power, symbolic);

((1-t)^2)^q

  

(-1+t)^(2*q)

  

(exp(I*t))^q

  

exp(I*q*t)

 (1)
> 

 

Download simpPow.mw

As has been already observed, such 'simplification' is not generally true. As the associated help page states

The symbolic option can be used if it does not matter whether the transformations are correct over the whole complex plane.

so use with care!

One of the options...

tomleslie 13876
November 07 2018
0 1

in the attached perhaps?

> 

  restart;
#
# Define a random matrix with positve entries
# for test purposes.
#
# Since OP is importing data from Excel, this
# data will probably(?)) be a matrix
#
  M:= LinearAlgebra:-RandomMatrix(8,8)+~100:
#
# Define a procedure
#
  F1:= proc(lambda, alpha:=0.7)
           uses Statistics;
           return Quantile
                  ( RandomVariable
                    ( Poisson(lambda)
                    ),
                    alpha
                  );
       end proc:
#
# Procedure value for a single value with
# default value of 'lambda'
#
  F1(5);
#
# Procedure value for a single value with
# supplied value of 'lambda'
#
  F1(5, 0.1);
#
# Procedure applied to matrix with
# default value of 'lambda'
#
  F1~(M);
#
# Procedure applied to matrix with
# supplied value of 'lambda'
#
  F1~(M, 0.1);

6.

  

2.

  

Matrix(8, 8, {(1, 1) = 41., (1, 2) = 206., (1, 3) = 130., (1, 4) = 137., (1, 5) = 156., (1, 6) = 66., (1, 7) = 8., (1, 8) = 113., (2, 1) = 71., (2, 2) = 167., (2, 3) = 172., (2, 4) = 54., (2, 5) = 115., (2, 6) = 87., (2, 7) = 26., (2, 8) = 176., (3, 1) = 35., (3, 2) = 6., (3, 3) = 193., (3, 4) = 22., (3, 5) = 89., (3, 6) = 194., (3, 7) = 31., (3, 8) = 206., (4, 1) = 36., (4, 2) = 85., (4, 3) = 126., (4, 4) = 149., (4, 5) = 96., (4, 6) = 139., (4, 7) = 103., (4, 8) = 135., (5, 1) = 128., (5, 2) = 79., (5, 3) = 42., (5, 4) = 131., (5, 5) = 54., (5, 6) = 3., (5, 7) = 72., (5, 8) = 150., (6, 1) = 119., (6, 2) = 157., (6, 3) = 56., (6, 4) = 201., (6, 5) = 83., (6, 6) = 25., (6, 7) = 29., (6, 8) = 199., (7, 1) = 122., (7, 2) = 183., (7, 3) = 184., (7, 4) = 117., (7, 5) = 151., (7, 6) = 163., (7, 7) = 101., (7, 8) = 73., (8, 1) = 114., (8, 2) = 60., (8, 3) = 114., (8, 4) = 103., (8, 5) = 21., (8, 6) = 133., (8, 7) = 133., (8, 8) = 174.})

  

Matrix(%id = 18446744074335351142)

 (1)
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  restart;
#
# As above, but with a simple 'function'
#
  with(Statistics):
  M:= LinearAlgebra:-RandomMatrix(8,8)+~100:
  alpha:= 0.7:
  F2:= lambda-> Quantile
               ( RandomVariable
                 ( Poisson(lambda)
                 ),
                 alpha
               ):
  F2(5);
  F2~(M)

6.

  

Matrix(%id = 18446744074391591022)

 (2)
> 

 

Download quan.mw

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