(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 474832, 21588]*) (*NotebookOutlinePosition[ 475887, 21624]*) (* CellTagsIndexPosition[ 475843, 21620]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{Cell[TextData["Load the Package"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["<True], Cell[TextData["?RungeKutta"], "Input", AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["Check the RKConditions routine"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "First a 4-th order method with undefined Butcher array."], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Clear[a,b,c];\nRKConditions[4, c, a, b]"], "Input", AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["?cond"], "Input", AspectRatioFixed->True], Cell[TextData[ "Global`cond\n\nAttributes[cond] = {Listable}\n\ncond[1] = \ Unevaluated[Sum[b[[i]], {i, Length[b]}]] == 1\n\ncond[2] = b . c == 1/2\n\n\ cond[3] = b . (c^2) == 1/3\n\ncond[4] = b . a . c == 1/6\n\ncond[5] = b . \ (c^3) == 1/4\n\ncond[6] = b . (c*a . c) == 1/8\n\ncond[7] = b . a . (c^2) == \ 1/12\n\ncond[8] = b . a . a . c == 1/24"], "Info", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["cond[2]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ b . c == t^2/2\ \>", "\<\ 2 t b . c == -- 2\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["cond[{2,3,5}]"], "Input", AspectRatioFixed->True], Cell[OutputFormData[ "\<\ {b . c == t^2/2, b . (c^2) == t^3/3, b . (c^3) == t^4/4}\ \>", "\<\ 2 3 4 t 2 t 3 t {b . c == --, b . (c ) == --, b . (c ) == --} 2 3 4\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]]}, Open]], Cell[CellGroupData[{Cell[TextData[ "First a 3-rd order continuous method with 4-th order at the end points. \ Undefined Butcher array."], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Clear[a,b,c];\nRKConditions[3, c, a, b, Continuous->t]"], "Input", AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["?cond"], "Input", AspectRatioFixed->True], Cell[TextData[ "Global`cond\n\nAttributes[cond] = {Listable}\n\ncond[1] = \ Unevaluated[Sum[b[[i]], {i, Length[b]}]] == t\n\ncond[2] = b . c == t^2/2\n\n\ cond[3] = b . (c^2) == t^3/3\n\ncond[4] = b . a . c == t^3/6\n\ncond[5] = b . \ (c^3) == 1/4\n\ncond[6] = b . (c*a . c) == 1/8\n\ncond[7] = b . a . (c^2) == \ 1/12\n\ncond[8] = b . a . a . c == 1/24"], "Info", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["cond[2]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ b . c == t^2/2\ \>", "\<\ 2 t b . c == -- 2\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["cond[{2,3,5}]"], "Input", AspectRatioFixed->True], Cell[OutputFormData[ "\<\ {b . c == t^2/2, b . (c^2) == t^3/3, b . (c^3) == 1/4}\ \>", "\<\ 2 3 t 2 t 3 1 {b . c == --, b . (c ) == --, b . (c ) == -} 2 3 4\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]]}, Open]], Cell[CellGroupData[{Cell[TextData[ "Next a 6-th order ERK method (non-continuous/continuous)"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "a={a1,a2,a3,a4,a5,a6,a7};\nb={{0,0,0,0,0,0,0},\n {b21,0,0,0,0,0,0},\n \ {b31,b32,0,0,0,0,0},\n {b41,b42,b43,0,0,0,0},\n {b51,b52,b53,b54,0,0,0},\n\ {b61,b62,b63,b64,b65,0,0},\n {b71,b72,b73,b74,b75,b76,0}};\n\ c={c1,c2,c3,c4,c5,c6,c7};\nd={{0,1,d12,d13,d14,d15,d16},\n \ {0,0,d22,d23,d24,d25,d26},\n {0,0,d32,d33,d34,d35,d36},\n \ {0,0,d42,d43,d44,d45,d46},\n {0,0,d52,d53,d54,d55,d56},\n \ {0,0,d62,d63,d64,d65,d66},\n {0,0,d72,d73,d74,d75,d76}};\n\ w[i_,t_]:=d[[i,1]]+Sum[ d[[i,j]] t^(j-1),{j,2,7}];\n\ W[t_]:={w[1,t],w[2,t],w[3,t],w[4,t],w[5,t],w[6,t],w[7,t]};"], "Input", AspectRatioFixed->True], Cell[TextData["Here is the structure of W[t]"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["W[t]"], "Input", AspectRatioFixed->True], Cell[OutputFormData[ "\<\ {t + d12*t^2 + d13*t^3 + d14*t^4 + d15*t^5 + d16*t^6, d22*t^2 + d23*t^3 + d24*t^4 + d25*t^5 + d26*t^6, d32*t^2 + d33*t^3 + d34*t^4 + d35*t^5 + d36*t^6, d42*t^2 + d43*t^3 + d44*t^4 + d45*t^5 + d46*t^6, d52*t^2 + d53*t^3 + d54*t^4 + d55*t^5 + d56*t^6, d62*t^2 + d63*t^3 + d64*t^4 + d65*t^5 + d66*t^6, d72*t^2 + d73*t^3 + d74*t^4 + d75*t^5 + d76*t^6}\ \>", "\<\ 2 3 4 5 6 {t + d12 t + d13 t + d14 t + d15 t + d16 t , 2 3 4 5 6 d22 t + d23 t + d24 t + d25 t + d26 t , 2 3 4 5 6 d32 t + d33 t + d34 t + d35 t + d36 t , 2 3 4 5 6 d42 t + d43 t + d44 t + d45 t + d46 t , 2 3 4 5 6 d52 t + d53 t + d54 t + d55 t + d56 t , 2 3 4 5 6 d62 t + d63 t + d64 t + d65 t + d66 t , 2 3 4 5 6 d72 t + d73 t + d74 t + d75 t + d76 t }\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData["RKConditions[6, a, b, c]"], "Input", AspectRatioFixed->True], Cell[TextData["A look at the quadrature conditions"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["qcond[{1,2,3,4,5,6}]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {c1 + c2 + c3 + c4 + c5 + c6 + c7 == 1, a1*c1 + a2*c2 + a3*c3 + a4*c4 + a5*c5 + a6*c6 + a7*c7 == 1/2, a1^2*c1 + a2^2*c2 + a3^2*c3 + a4^2*c4 + a5^2*c5 + a6^2*c6 + a7^2*c7 == 1/3, a1^3*c1 + a2^3*c2 + a3^3*c3 + a4^3*c4 + a5^3*c5 + a6^3*c6 + a7^3*c7 == 1/4, a1^4*c1 + a2^4*c2 + a3^4*c3 + a4^4*c4 + a5^4*c5 + a6^4*c6 + a7^4*c7 == 1/5, a1^5*c1 + a2^5*c2 + a3^5*c3 + a4^5*c4 + a5^5*c5 + a6^5*c6 + a7^5*c7 == 1/6}\ \ \>", "\<\ {c1 + c2 + c3 + c4 + c5 + c6 + c7 == 1, 1 a1 c1 + a2 c2 + a3 c3 + a4 c4 + a5 c5 + a6 c6 + a7 c7 == -, 2 2 2 2 2 2 2 2 1 a1 c1 + a2 c2 + a3 c3 + a4 c4 + a5 c5 + a6 c6 + a7 c7 == -, 3 3 3 3 3 3 3 3 1 a1 c1 + a2 c2 + a3 c3 + a4 c4 + a5 c5 + a6 c6 + a7 c7 == -, 4 4 4 4 4 4 4 4 1 a1 c1 + a2 c2 + a3 c3 + a4 c4 + a5 c5 + a6 c6 + a7 c7 == -, 5 5 5 5 5 5 5 5 1 a1 c1 + a2 c2 + a3 c3 + a4 c4 + a5 c5 + a6 c6 + a7 c7 == -} 6\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData[ "A look at the first order row simplifying assumptions and then at the second \ of these."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["rcond[1]\nrcond[1,2]"], "Input", AspectRatioFixed->True], Cell[OutputFormData[ "\<\ {-a1 == 0, -a2 + b21 == 0, -a3 + b31 + b32 == 0, -a4 + b41 + b42 + b43 \ == 0, -a5 + b51 + b52 + b53 + b54 == 0, -a6 + b61 + b62 + b63 + b64 + b65 == 0, -a7 + b71 + b72 + b73 + b74 + b75 + b76 == 0}\ \>", "\<\ {-a1 == 0, -a2 + b21 == 0, -a3 + b31 + b32 == 0, -a4 + b41 + b42 + b43 \ == 0, -a5 + b51 + b52 + b53 + b54 == 0, -a6 + b61 + b62 + b63 + b64 + b65 == 0, -a7 + b71 + b72 + b73 + b74 + b75 + b76 == 0}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True], Cell[OutputFormData["\<\ -a2 + b21 == 0\ \>", "\<\ -a2 + b21 == 0\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData[ "A look at the squaring conditions and the squaring condition in the thrid \ column."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["ccond[2]\nccond[2,3]"], "Input", AspectRatioFixed->True], Cell[OutputFormData[ "\<\ {-((1 - a1^2)*c1)/2 + a2*b21*c2 + a3*b31*c3 + a4*b41*c4 + a5*b51*c5 + a6*b61*c6 + a7*b71*c7 == 0, -((1 - a2^2)*c2)/2 + a3*b32*c3 + a4*b42*c4 + a5*b52*c5 + a6*b62*c6 + a7*b72*c7 == 0, -((1 - a3^2)*c3)/2 + a4*b43*c4 + a5*b53*c5 + a6*b63*c6 + a7*b73*c7 == 0, -((1 - a4^2)*c4)/2 + a5*b54*c5 + a6*b64*c6 + a7*b74*c7 == 0, -((1 - a5^2)*c5)/2 + a6*b65*c6 + a7*b75*c7 == 0, -((1 - a6^2)*c6)/2 + a7*b76*c7 == 0, -((1 - a7^2)*c7)/2 == 0}\ \>", "\<\ 2 -((1 - a1 ) c1) {--------------- + a2 b21 c2 + a3 b31 c3 + a4 b41 c4 + a5 b51 c5 + a6 b61 c6 \ + 2 2 -((1 - a2 ) c2) a7 b71 c7 == 0, --------------- + a3 b32 c3 + a4 b42 c4 + a5 b52 c5 + 2 2 -((1 - a3 ) c3) a6 b62 c6 + a7 b72 c7 == 0, --------------- + a4 b43 c4 + a5 b53 c5 + 2 2 -((1 - a4 ) c4) a6 b63 c6 + a7 b73 c7 == 0, --------------- + a5 b54 c5 + a6 b64 c6 + 2 2 -((1 - a5 ) c5) a7 b74 c7 == 0, --------------- + a6 b65 c6 + a7 b75 c7 == 0, 2 2 2 -((1 - a6 ) c6) -((1 - a7 ) c7) --------------- + a7 b76 c7 == 0, --------------- == 0} 2 2\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True], Cell[OutputFormData[ "\<\ -((1 - a3^2)*c3)/2 + a4*b43*c4 + a5*b53*c5 + a6*b63*c6 + a7*b73*c7 == \ 0\ \>", "\<\ 2 -((1 - a3 ) c3) --------------- + a4 b43 c4 + a5 b53 c5 + a6 b63 c6 + a7 b73 c7 == 0 2\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData["Now the continuous version"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["RKConditions[5, a, b, W[t], Continuous->t]"], "Input", AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["qcond[{1,2,3,4,5,6}]"], "Input", AspectRatioFixed->True], Cell[OutputFormData[ "\<\ {t + d12*t^2 + d22*t^2 + d32*t^2 + d42*t^2 + d52*t^2 + d62*t^2 + d72*t^2 \ + d13*t^3 + d23*t^3 + d33*t^3 + d43*t^3 + d53*t^3 + d63*t^3 + d73*t^3 + d14*t^4 + d24*t^4 + d34*t^4 + d44*t^4 + d54*t^4 + d64*t^4 + d74*t^4 + d15*t^5 + d25*t^5 + d35*t^5 + d45*t^5 + d55*t^5 + d65*t^5 + d75*t^5 + d16*t^6 + d26*t^6 + d36*t^6 + d46*t^6 + d56*t^6 + d66*t^6 + d76*t^6 == t, a1*(t + d12*t^2 + d13*t^3 + d14*t^4 + d15*t^5 + d16*t^6) + a2*(d22*t^2 + d23*t^3 + d24*t^4 + d25*t^5 + d26*t^6) + a3*(d32*t^2 + d33*t^3 + d34*t^4 + d35*t^5 + d36*t^6) + a4*(d42*t^2 + d43*t^3 + d44*t^4 + d45*t^5 + d46*t^6) + a5*(d52*t^2 + d53*t^3 + d54*t^4 + d55*t^5 + d56*t^6) + a6*(d62*t^2 + d63*t^3 + d64*t^4 + d65*t^5 + d66*t^6) + a7*(d72*t^2 + d73*t^3 + d74*t^4 + d75*t^5 + d76*t^6) == t^2/2, a1^2*(t + d12*t^2 + d13*t^3 + d14*t^4 + d15*t^5 + d16*t^6) + a2^2*(d22*t^2 + d23*t^3 + d24*t^4 + d25*t^5 + d26*t^6) + a3^2*(d32*t^2 + d33*t^3 + d34*t^4 + d35*t^5 + d36*t^6) + a4^2*(d42*t^2 + d43*t^3 + d44*t^4 + d45*t^5 + d46*t^6) + a5^2*(d52*t^2 + d53*t^3 + d54*t^4 + d55*t^5 + d56*t^6) + a6^2*(d62*t^2 + d63*t^3 + d64*t^4 + d65*t^5 + d66*t^6) + a7^2*(d72*t^2 + d73*t^3 + d74*t^4 + d75*t^5 + d76*t^6) == t^3/3, a1^3*(t + d12*t^2 + d13*t^3 + d14*t^4 + d15*t^5 + d16*t^6) + a2^3*(d22*t^2 + d23*t^3 + d24*t^4 + d25*t^5 + d26*t^6) + a3^3*(d32*t^2 + d33*t^3 + d34*t^4 + d35*t^5 + d36*t^6) + a4^3*(d42*t^2 + d43*t^3 + d44*t^4 + d45*t^5 + d46*t^6) + a5^3*(d52*t^2 + d53*t^3 + d54*t^4 + d55*t^5 + d56*t^6) + a6^3*(d62*t^2 + d63*t^3 + d64*t^4 + d65*t^5 + d66*t^6) + a7^3*(d72*t^2 + d73*t^3 + d74*t^4 + d75*t^5 + d76*t^6) == t^4/4, a1^4*(t + d12*t^2 + d13*t^3 + d14*t^4 + d15*t^5 + d16*t^6) + a2^4*(d22*t^2 + d23*t^3 + d24*t^4 + d25*t^5 + d26*t^6) + a3^4*(d32*t^2 + d33*t^3 + d34*t^4 + d35*t^5 + d36*t^6) + a4^4*(d42*t^2 + d43*t^3 + d44*t^4 + d45*t^5 + d46*t^6) + a5^4*(d52*t^2 + d53*t^3 + d54*t^4 + d55*t^5 + d56*t^6) + a6^4*(d62*t^2 + d63*t^3 + d64*t^4 + d65*t^5 + d66*t^6) + a7^4*(d72*t^2 + d73*t^3 + d74*t^4 + d75*t^5 + d76*t^6) == t^5/5, a1^5*(1 + d12 + d13 + d14 + d15 + d16) + a2^5*(d22 + d23 + d24 + d25 + d26) \ + a3^5*(d32 + d33 + d34 + d35 + d36) + a4^5*(d42 + d43 + d44 + d45 + d46) + a5^5*(d52 + d53 + d54 + d55 + d56) + a6^5*(d62 + d63 + d64 + d65 + d66) + a7^5*(d72 + d73 + d74 + d75 + d76) == 1/6}\ \>", "\<\ 2 2 2 2 2 2 2 \ 3 {t + d12 t + d22 t + d32 t + d42 t + d52 t + d62 t + d72 t + d13 t + 3 3 3 3 3 3 4 4 d23 t + d33 t + d43 t + d53 t + d63 t + d73 t + d14 t + d24 t + 4 4 4 4 4 5 5 5 d34 t + d44 t + d54 t + d64 t + d74 t + d15 t + d25 t + d35 t + 5 5 5 5 6 6 6 6 d45 t + d55 t + d65 t + d75 t + d16 t + d26 t + d36 t + d46 t + 6 6 6 d56 t + d66 t + d76 t == t, 2 3 4 5 6 a1 (t + d12 t + d13 t + d14 t + d15 t + d16 t ) + 2 3 4 5 6 a2 (d22 t + d23 t + d24 t + d25 t + d26 t ) + 2 3 4 5 6 a3 (d32 t + d33 t + d34 t + d35 t + d36 t ) + 2 3 4 5 6 a4 (d42 t + d43 t + d44 t + d45 t + d46 t ) + 2 3 4 5 6 a5 (d52 t + d53 t + d54 t + d55 t + d56 t ) + 2 3 4 5 6 a6 (d62 t + d63 t + d64 t + d65 t + d66 t ) + 2 2 3 4 5 6 t a7 (d72 t + d73 t + d74 t + d75 t + d76 t ) == --, 2 2 2 3 4 5 6 a1 (t + d12 t + d13 t + d14 t + d15 t + d16 t ) + 2 2 3 4 5 6 a2 (d22 t + d23 t + d24 t + d25 t + d26 t ) + 2 2 3 4 5 6 a3 (d32 t + d33 t + d34 t + d35 t + d36 t ) + 2 2 3 4 5 6 a4 (d42 t + d43 t + d44 t + d45 t + d46 t ) + 2 2 3 4 5 6 a5 (d52 t + d53 t + d54 t + d55 t + d56 t ) + 2 2 3 4 5 6 a6 (d62 t + d63 t + d64 t + d65 t + d66 t ) + 3 2 2 3 4 5 6 t a7 (d72 t + d73 t + d74 t + d75 t + d76 t ) == --, 3 3 2 3 4 5 6 a1 (t + d12 t + d13 t + d14 t + d15 t + d16 t ) + 3 2 3 4 5 6 a2 (d22 t + d23 t + d24 t + d25 t + d26 t ) + 3 2 3 4 5 6 a3 (d32 t + d33 t + d34 t + d35 t + d36 t ) + 3 2 3 4 5 6 a4 (d42 t + d43 t + d44 t + d45 t + d46 t ) + 3 2 3 4 5 6 a5 (d52 t + d53 t + d54 t + d55 t + d56 t ) + 3 2 3 4 5 6 a6 (d62 t + d63 t + d64 t + d65 t + d66 t ) + 4 3 2 3 4 5 6 t a7 (d72 t + d73 t + d74 t + d75 t + d76 t ) == --, 4 4 2 3 4 5 6 a1 (t + d12 t + d13 t + d14 t + d15 t + d16 t ) + 4 2 3 4 5 6 a2 (d22 t + d23 t + d24 t + d25 t + d26 t ) + 4 2 3 4 5 6 a3 (d32 t + d33 t + d34 t + d35 t + d36 t ) + 4 2 3 4 5 6 a4 (d42 t + d43 t + d44 t + d45 t + d46 t ) + 4 2 3 4 5 6 a5 (d52 t + d53 t + d54 t + d55 t + d56 t ) + 4 2 3 4 5 6 a6 (d62 t + d63 t + d64 t + d65 t + d66 t ) + 5 4 2 3 4 5 6 t a7 (d72 t + d73 t + d74 t + d75 t + d76 t ) == --, 5 5 5 a1 (1 + d12 + d13 + d14 + d15 + d16) + a2 (d22 + d23 + d24 + d25 + d26) + 5 5 a3 (d32 + d33 + d34 + d35 + d36) + a4 (d42 + d43 + d44 + d45 + d46) + 5 5 a5 (d52 + d53 + d54 + d55 + d56) + a6 (d62 + d63 + d64 + d65 + d66) + 5 1 a7 (d72 + d73 + d74 + d75 + d76) == -} 6\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]]}, Open]], Cell[CellGroupData[{Cell[TextData[ "Finally a test of an actual method.\nThe method is the Dormand & Prince 5(4) \ method given in Hairer/Norsett/Wanner P.178/192"], "Subsection", CellMargins->{{Inherited, -57}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "c = {0,1/5,3/10,4/5,8/9,1,1};\na = {{0,0,0,0,0,0,0},\n\t \ {1/5,0,0,0,0,0,0},\n\t {3/40,9/40,0,0,0,0,0},\n\t \ 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