diff --git a/CMMExamples.nb b/CMMExamples.nb index 89d15ec..6b3fee5 100644 --- a/CMMExamples.nb +++ b/CMMExamples.nb @@ -10,10 +10,10 @@ NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] -NotebookDataLength[ 423905, 8567] -NotebookOptionsPosition[ 409266, 8329] -NotebookOutlinePosition[ 409638, 8345] -CellTagsIndexPosition[ 409595, 8342] +NotebookDataLength[ 570715, 12283] +NotebookOptionsPosition[ 548984, 11942] +NotebookOutlinePosition[ 549330, 11957] +CellTagsIndexPosition[ 549287, 11954] WindowFrame->Normal*) (* Beginning of Notebook Content *) @@ -25,12 +25,35 @@ Cell["Compound Matrix Method", "Title", 3.714379176498895*^9}},ExpressionUUID->"f088044e-1e1d-4a4a-9586-\ 300fbeac4765"], +Cell[CellGroupData[{ + +Cell["Installation", "Chapter", + CellChangeTimes->{{3.73131153331354*^9, + 3.731311536142145*^9}},ExpressionUUID->"79c0595c-62c3-499b-9713-\ +e05df243f56c"], + +Cell["\<\ +The following command will install the version 0.5 from the website. Check \ +the github page for more recent versions.\ +\>", "Text", + CellChangeTimes->{{3.7276876542665205`*^9, 3.7276876568016253`*^9}, { + 3.731736383675373*^9, + 3.731736409434847*^9}},ExpressionUUID->"6250ab7b-a45e-47fe-9be2-\ +2fd6c776c7df"], + +Cell[BoxData[{ + RowBox[{"Needs", "[", "\"\\"", "]"}], "\n", + RowBox[{"PacletInstall", "[", + "\"\\"", "]"}]}], "Input", + CellChangeTimes->{{3.731128189868894*^9, 3.731128190041904*^9}, { + 3.7313027297251663`*^9, + 3.73130273203397*^9}},ExpressionUUID->"45366525-f94a-4847-845c-\ +e1e0ed862433"], + Cell["\<\ -Implementation of the Compound Matrix Method for solving boundary value \ -eigenvalue problems, as well as a function to generate the linear matrix form \ -from a set of equations. -The first use for a particular dimension stores the coefficients of the \ -matrix in the ODE of the minors, so will take a bit longer.\ +Implementation of the Compound Matrix Method to calculate the Evans function, \ +in order to solve boundary value eigenvalue problems.\ \>", "Text", CellChangeTimes->{{3.7143790902875557`*^9, 3.7143791234270654`*^9}, { 3.7143791889180803`*^9, 3.7143791890340824`*^9}, {3.714379417593276*^9, @@ -38,12 +61,14 @@ matrix in the ODE of the minors, so will take a bit longer.\ 3.7148834350923166`*^9, 3.714883435529117*^9}, {3.716781344008561*^9, 3.716781376658428*^9}, {3.717128824152254*^9, 3.7171288286575117`*^9}, { 3.722071573414387*^9, 3.722071574462447*^9}, {3.722084405807358*^9, - 3.7220844135698023`*^9}},ExpressionUUID->"6c4922d8-9fe7-4914-91d0-\ + 3.7220844135698023`*^9}, {3.7317364200984564`*^9, + 3.731736439283554*^9}},ExpressionUUID->"6c4922d8-9fe7-4914-91d0-\ 6136b008c9be"], Cell[BoxData[ RowBox[{"Needs", "[", "\"\\"", "]"}]], "Input",Expres\ -sionUUID->"1ef829d4-5ce2-4ebc-add5-eace84d521ff"], +sionUUID->"1ef829d4-5ce2-4ebc-add5-eace84d521ff"] +}, Open ]], Cell[CellGroupData[{ @@ -54,10 +79,12 @@ Cell["Examples", "Chapter", Cell[CellGroupData[{ -Cell["Example 1", "Subchapter", +Cell["Example 1: Second order equation with constant coefficients ", \ +"Subchapter", CellChangeTimes->{{3.7143818912972*^9, 3.7143818999653144`*^9}, { - 3.7171308795933895`*^9, 3.7171308952752867`*^9}, - 3.722084486294962*^9},ExpressionUUID->"82a46fee-76db-4504-80b2-\ + 3.7171308795933895`*^9, 3.7171308952752867`*^9}, 3.722084486294962*^9, { + 3.7313110919405375`*^9, + 3.7313111315698085`*^9}},ExpressionUUID->"82a46fee-76db-4504-80b2-\ 43b15ea879db"], Cell[TextData[{ @@ -73,7 +100,7 @@ Cell[TextData[{ RowBox[{ SuperscriptBox["\[Lambda]", "2"], " ", RowBox[{"y", "(", "x", ")"}]}]}], " ", "=", "0"}]}], TraditionalForm]], - ExpressionUUID->"36bed721-d354-4fc8-b142-35046c82e948"], + ExpressionUUID->"9fa6b3be-4575-4b42-9e2b-117ec2baf0b6"], ", subject to ", Cell[BoxData[ FormBox[ @@ -83,7 +110,7 @@ Cell[TextData[{ RowBox[{ RowBox[{"y", RowBox[{"(", "L", ")"}]}], "=", "0"}]}], TraditionalForm]], - ExpressionUUID->"0b7485cd-fa61-485a-8680-a5a2d61b8956"], + ExpressionUUID->"c765dcd2-0fb7-4f98-9241-e9dbe9b015c7"], ". The roots to this can be found analytically to be", Cell[BoxData[ FormBox[ @@ -91,8 +118,8 @@ Cell[TextData[{ RowBox[{ FractionBox["n\[Pi]", "L"], ",", " ", RowBox[{"n", " ", "\[Element]", " ", "\[DoubleStruckCapitalZ]"}]}]}], - TraditionalForm]],ExpressionUUID->"0fa74df2-6cb5-4840-a83a-90df5eaafc6d"], - ". \nConverting to the matrix equations, we let ", + TraditionalForm]],ExpressionUUID->"3cfc6735-2386-4597-a825-3a093de79788"], + ". \nConverting to a matrix equation, we let ", Cell[BoxData[ FormBox[ RowBox[{ @@ -162,7 +189,7 @@ Cell[TextData[{ {"0", "0"} }], "\[NoBreak]", ")"}]}]}], TraditionalForm]],ExpressionUUID-> "cb1a390d-2943-46c2-b36c-ff62347c400d"], - ". \nWe can then evaluate the CMM function at values of ", + ". \nWe can then evaluate the Evans function at a given value of ", Cell[BoxData[ FormBox[ RowBox[{"\[Lambda]", "=", @@ -177,9 +204,9 @@ Cell[TextData[{ " (with ", Cell[BoxData[ FormBox[ - RowBox[{"L", "=", "3"}], TraditionalForm]],ExpressionUUID-> + RowBox[{"L", "=", "2"}], TraditionalForm]],ExpressionUUID-> "5547a692-5234-4269-ace4-038a7cdca5d9"], - "). 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Roots of this function \ -correspond to eigenvalues of the original system\ +This function is analytic, and its zeroes correspond to eigenvalues of the \ +original system:\ \>", "Text", CellChangeTimes->{{3.7220731683286114`*^9, 3.722073202599571*^9}, { - 3.7220734325517235`*^9, - 3.7220734681667604`*^9}},ExpressionUUID->"6211ca2d-79e9-4588-82b5-\ + 3.7220734325517235`*^9, 3.7220734681667604`*^9}, {3.731732124931304*^9, + 3.731732140890217*^9}},ExpressionUUID->"6211ca2d-79e9-4588-82b5-\ 04abb72ef9a9"], Cell[CellGroupData[{ @@ -247,7 +282,7 @@ Cell[BoxData[ RowBox[{"With", "[", RowBox[{ RowBox[{"{", - RowBox[{"L", "=", "2"}], "}"}], ",", + RowBox[{"L", "=", "2"}], "}"}], ",", "\[IndentingNewLine]", RowBox[{"Plot", "[", RowBox[{ RowBox[{"CompoundMatrixMethod", "[", @@ -275,10 +310,11 @@ Cell[BoxData[ RowBox[{"\[Lambda]0", ",", RowBox[{"-", "12"}], ",", "12"}], "}"}]}], "]"}]}], "]"}]], "Input", CellChangeTimes->{{3.714785551046855*^9, 3.7147856233009872`*^9}, { - 3.714786054824669*^9, 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6, 32, "Output",ExpressionUUID->"abafbfdb-f34c-4085-a6bd-cd3e18200a5d"] }, Open ]] +}, Closed]] }, Open ]] }, Open ]] } diff --git a/CompoundMatrixMethod-0.5.paclet b/CompoundMatrixMethod-0.5.paclet new file mode 100644 index 0000000..d44fa55 Binary files /dev/null and b/CompoundMatrixMethod-0.5.paclet differ diff --git a/CompoundMatrixMethod/CompoundMatrixMethod.m b/CompoundMatrixMethod/CompoundMatrixMethod.m index 9d31374..94ce524 100644 --- a/CompoundMatrixMethod/CompoundMatrixMethod.m +++ b/CompoundMatrixMethod/CompoundMatrixMethod.m @@ -5,8 +5,8 @@ (* :Title: CompoundMatrixMethod *) (* :Author: Simon Pearce *) (* :Context: CompoundMatrixMethod` *) -(* :Version: 0.4 *) -(* :Date: 2018-03-09 *) +(* :Version: 0.5 *) +(* :Date: 2018-03-29 *) (* :Mathematica Version: 9+ *) (* :Copyright: (c) 2017-18 Simon Pearce *) @@ -18,11 +18,15 @@ CompoundMatrixMethod::usage = "\ CompoundMatrixMethod[{k,k0},sys] evaluates the Evans function from the Compound Matrix Method with potential eigenvalue k=k0, for the system defined from ToLinearMatrixForm. CompoundMatrixMethod[{k,k0},A,B,C,{x,x0,x1}] evaluates the Evans function from the Compound Matrix Method with k=k0. Here the linear matrix ODE is given by dy/dx=A.y, with B.y=0 at x=x0 and C.y=0 at x=x1. -For either case a complex number is returned, zeroes of this function correspond to zeroes of the original eigenvalue equation."; +For either case a complex number is returned, zeroes of this function correspond to zeroes of the original eigenvalue equation. +CompoundMatrixMethodInterface[{k,k0},{A1,A2},B,C,{F,G},{x,x0,xmatch,x1}] evaluates the Evans function for potential eigenvalue k=k0 for an interface problem. +Here there are two linear matrix ODE is given on the left by dy/dx=A1.y, B.y=0 at x=x0, dz/dx=A2.z, and C.z=0 at x=x1, and the interface conditions are given by +F.y + G.z = 0 at the interface given by x=xmatch."; ToLinearMatrixForm::usage = "\ ToLinearMatrixForm[eqn,{},depvars,x] takes a list of differential equations in the dependent variables depvars and independent variable x and puts the equations into linear matrix form dy/dx = A.y -ToLinearMatrixForm[eqn,bcs,depvars,{x,x0,x1}] also includes the boundary conditions evaluated at x=x0 and x=x1."; +ToLinearMatrixForm[eqn,bcs,depvars,{x,x0,x1}] also includes the boundary conditions evaluated at x=x0 and x=x1. +ToLinearMatrixForm[{eqns1,eqns2},bcs,depvars,{x,x0,xmatch,x1}] sets up the function with an interface at x=xmatch, with eqns1 in x0 Signature[a] \[FormalPhi][Sort[a]]; +reprules = \[FormalPhi][a_?ListQ] :> Signature[a] \[FormalPhi][Sort[a]]; +reprules2= {\[FormalPhi]L[a_?ListQ][q_] :> Signature[a] \[FormalPhi]L[Sort[a]][q], \[FormalPhi]R[a_?ListQ][q_] :> Signature[a] \[FormalPhi]R[Sort[a]][q]}; (* Generation of the derivatives of the matrix minors, looping over rule to sort the lists of indices *) minorsDerivs[list_?VectorQ, len_?NumericQ] := minorsDerivs[list, len] = - Sum[ - Sum[\[FormalCapitalA][y, z] \[FormalPhi][list /. y -> z], {z, Union[Complement[Range[len], list], {y}]}], + Sum[ Sum[\[FormalCapitalA][y, z] \[FormalPhi][list /. y -> z], {z, Union[Complement[Range[len], list], {y}]}], {y, list}] /. reprules qMatrix[len_?NumericQ, len2_?NumericQ] := qMatrix[len, len2] = @@ -73,15 +82,48 @@ Transpose[Table[Coefficient[minorsTab, \[FormalPhi][i]], {i, 1, Binomial[len, len2]}]] ] +rulesFG[len_?NumericQ] := rulesFG[len] = {\[FormalPhi]\[FormalPhi]L[{l_}][x_] :> Sum[F[l, a] \[FormalPhi]L[a][x], {a, 1, len}], + \[FormalPhi]\[FormalPhi]L[{l_, m_}][x_] :> Sum[F[l, a] F[m, b] \[FormalPhi]L[{a, b}][x], {a, 1, len}, {b, 1, len}], + \[FormalPhi]\[FormalPhi]L[{l_, m_, n_}][x_] :> Sum[F[l, a] F[m, b] F[n, c] \[FormalPhi]L[{a, b, c}][x], {a, 1, len}, {b, 1, len}, {c, 1, len}], + \[FormalPhi]\[FormalPhi]L[{l_, m_, n_, o_}][x_] :> Sum[F[l, a] F[m, b] F[n, c] F[o, d] + \[FormalPhi]L[{a, b, c, d}][x], {a, 1, len}, {b, 1, len}, {c, 1, len}, {d, 1, len}], + \[FormalPhi]\[FormalPhi]L[{l_, m_, n_, o_, p_}][x_] :> Sum[F[l, a] F[m, b] F[n, c] F[o, d] F[p,e] + \[FormalPhi]L[{a, b, c, d, e}][x], {a, 1, len}, {b, 1, len}, {c, 1, len}, {d, 1, len}, {e, 1, len}], + \[FormalPhi]\[FormalPhi]R[{l_}][x_] :> Sum[G[l, a] \[FormalPhi]R[a][x], {a, 1, len}], + \[FormalPhi]\[FormalPhi]R[{l_, m_}][x_] :> Sum[G[l, a] G[m, b] \[FormalPhi]R[{a, b}][x], {a, 1, len}, {b, 1, len}], + \[FormalPhi]\[FormalPhi]R[{l_, m_, n_}][x_] :> Sum[G[l, a] G[m, b] G[n, c] \[FormalPhi]R[{a, b, c}][x], {a, 1, len}, {b, 1, len}, {c, 1, len}], + \[FormalPhi]\[FormalPhi]R[{l_, m_, n_, o_}][x_] :> Sum[G[l, a] G[m, b] G[n, c] G[o, d] \[FormalPhi]R[{a, b, c, d}][ + x], {a, 1, len}, {b, 1, len}, {c, 1, len}, {d, 1, len}], + \[FormalPhi]\[FormalPhi]R[{l_, m_, n_, o_, p_}][x_] :> Sum[G[l, a] G[m, b] G[n, c] G[o, d] G[p, e] + \[FormalPhi]R[{a, b, c, d, e}][x], {a, 1, len}, {b, 1, len}, {c, 1, len}, {d, 1, len}, {e, 1, len}]}; + +CompoundMatrixMethod[{\[FormalLambda]_ /; !NumericQ[\[FormalLambda]], \[FormalLambda]0_?NumericQ}, {AMatrix_?MatrixQ, {}, rightBCMatrix_?MatrixQ, {x_ /; ! NumericQ[x], xa_, xb_, xm_ : False}}] := + CompoundMatrixMethod[{\[FormalLambda], \[FormalLambda]0}, AMatrix, N@SelectNegativeEigenvectors[AMatrix /. \[FormalLambda] -> \[FormalLambda]0, x], rightBCMatrix, {x, xa, xb, xm}] + +CompoundMatrixMethod[{\[FormalLambda]_ /; !NumericQ[\[FormalLambda]], \[FormalLambda]0_?NumericQ}, {AMatrix_?MatrixQ, {}, {}, {x_ /; ! NumericQ[x], xa_, xb_, xm_ : False}}] := + CompoundMatrixMethod[{\[FormalLambda], \[FormalLambda]0}, AMatrix, N@SelectNegativeEigenvectors[AMatrix /. \[FormalLambda] -> \[FormalLambda]0, x], + N@SelectPositiveEigenvectors[AMatrix /. \[FormalLambda] -> \[FormalLambda]0, x], {x, xa, xb, xm}] + +CompoundMatrixMethod[{\[FormalLambda]_ /; !NumericQ[\[FormalLambda]], \[FormalLambda]0_?NumericQ}, {AMatrix_?MatrixQ, leftBCMatrix_?MatrixQ, {}, {x_ /; ! NumericQ[x], xa_, xb_, xm_ : False}}] := + CompoundMatrixMethod[{\[FormalLambda], \[FormalLambda]0}, AMatrix, leftBCMatrix, N@SelectPositiveEigenvectors[AMatrix /. \[FormalLambda] -> \[FormalLambda]0, x], {x, xa, xb, xm}] + +CompoundMatrixMethod[{\[FormalLambda]_ /; !NumericQ[\[FormalLambda]], \[FormalLambda]0_?NumericQ}, AMatrix_?MatrixQ, {}, rightBCMatrix_?MatrixQ, {x_ /; ! NumericQ[x], xa_, xb_, xm_ : False}] := + CompoundMatrixMethod[{\[FormalLambda], \[FormalLambda]0}, AMatrix, N@SelectNegativeEigenvectors[AMatrix /. \[FormalLambda] -> \[FormalLambda]0, x], rightBCMatrix, {x, xa, xb, xm}] + +CompoundMatrixMethod[{\[FormalLambda]_ /; !NumericQ[\[FormalLambda]], \[FormalLambda]0_?NumericQ}, AMatrix_?MatrixQ, {}, {}, {x_ /; ! NumericQ[x], xa_, xb_, xm_ : False}] := + CompoundMatrixMethod[{\[FormalLambda], \[FormalLambda]0}, + AMatrix, N@SelectNegativeEigenvectors[AMatrix /. \[FormalLambda] -> \[FormalLambda]0, x], N@SelectPositiveEigenvectors[AMatrix /. \[FormalLambda] -> \[FormalLambda]0, x], {x, xa,xb, xm}] + +CompoundMatrixMethod[{\[FormalLambda]_ /; !NumericQ[\[FormalLambda]], \[FormalLambda]0_?NumericQ}, AMatrix_?MatrixQ, leftBCMatrix_?MatrixQ, {{}, {}}, {x_ /; ! NumericQ[x], xa_, xb_, xm_ : False}] := + CompoundMatrixMethod[{\[FormalLambda], \[FormalLambda]0}, AMatrix, leftBCMatrix, N@SelectPositiveEigenvectors[AMatrix /. \[FormalLambda] -> \[FormalLambda]0, x], {x, xa, xb, xm}] + + + CompoundMatrixMethod[{\[FormalLambda]_ /; !NumericQ[\[FormalLambda]], \[FormalLambda]0_?NumericQ}, {AMatrix_?MatrixQ, leftBCMatrix_?MatrixQ, rightBCMatrix_?MatrixQ, {x_ /; !NumericQ[x], xa_, xb_, xm_ : False}}] := - CompoundMatrixMethod[{\[FormalLambda], \[FormalLambda]0}, AMatrix, {leftBCMatrix, 0}, {rightBCMatrix, 0}, {x, xa, xb, xm}] -CompoundMatrixMethod[{\[FormalLambda]_ /; !NumericQ[\[FormalLambda]], \[FormalLambda]0_?NumericQ}, {AMatrix_?MatrixQ, {leftBCMatrix_?MatrixQ, leftBCVector_}, {rightBCMatrix_?MatrixQ, rightBCVector_}, {x_ /; !NumericQ[x], xa_, xb_, xm_ : False}}] := - CompoundMatrixMethod[{\[FormalLambda], \[FormalLambda]0}, AMatrix, {leftBCMatrix, leftBCVector}, {rightBCMatrix, rightBCVector}, {x, xa, xb, xm}] -CompoundMatrixMethod[{\[FormalLambda]_ /; !NumericQ[\[FormalLambda]], \[FormalLambda]0_?NumericQ}, AMatrix_?MatrixQ, leftBCMatrix_?MatrixQ, rightBCMatrix_?MatrixQ, {x_ /; !NumericQ[x], xa_, xb_, xm_ : False}] := - CompoundMatrixMethod[{\[FormalLambda], \[FormalLambda]0}, AMatrix, {leftBCMatrix, 0}, {rightBCMatrix, 0}, {x, xa, xb, xm}] - -CompoundMatrixMethod[{\[FormalLambda]_ /; !NumericQ[\[FormalLambda]], \[FormalLambda]0_?NumericQ}, AMatrix_?MatrixQ, {leftBCMatrix_?MatrixQ, leftBCVector_}, {rightBCMatrix_?MatrixQ, rightBCVector_}, {x_ /; !NumericQ[x], xaa_, xbb_, xm_ : False}] := Module[ - {len, subsets, newYs, leftYICs, rightYICs, phiLeftVector, phiRightVector, LeftBCSolution, RightBCSolution, yLeft, yRight, + CompoundMatrixMethod[{\[FormalLambda], \[FormalLambda]0}, AMatrix, leftBCMatrix, rightBCMatrix, {x, xa, xb, xm}] + +CompoundMatrixMethod[{\[FormalLambda]_ /; !NumericQ[\[FormalLambda]], \[FormalLambda]0_?NumericQ}, AMatrix_?MatrixQ, leftBCMatrix_?MatrixQ, rightBCMatrix_?MatrixQ, {x_ /; !NumericQ[x], xaa_, xbb_, xm_ : False}] := + Module[{len, subsets, newYs, leftYICs, rightYICs, phiLeftVector, phiRightVector, LeftBCSolution, RightBCSolution, yLeft, yRight, phiLeft, phiRight, LeftPositiveEigenvalues, RightNegativeEigenvalues, phiLeftICs, phiRightICs, QQ, solutionFromRight, solutionFromLeft, det, matchPoint,lenLeft,lenRight,subsetsLeft,subsetsRight,QLeft,QRight,xa,xb}, @@ -100,9 +142,6 @@ (* Actually, don't necessarily need full rank, try without. If[Length[AMatrix] != MatrixRank[AMatrix], Message[CompoundMatrixMethod::matrixRank];Return[$Failed]]; *) - (* Check the equations are inhomogeneous *) - If[Max@Abs@leftBCVector != 0 || Max@Abs@rightBCVector != 0, Message[CompoundMatrixMethod::nonZeroBoundaryConditions];Return[$Failed]]; - (* Check that the eigenvalue appears in the matrix A, give warning otherwise*) If[MatrixQ[AMatrix, FreeQ[#, \[FormalLambda]] &],Message[CompoundMatrixMethod::noEigenvalue,AMatrix,\[FormalLambda]]]; @@ -114,14 +153,13 @@ newYs = Through[Array[\[FormalY], {len}][x]]; (* Initial conditions for shooting from the LHS *) - LeftBCSolution = Quiet@Solve[leftBCMatrix.newYs == leftBCVector, newYs]; + LeftBCSolution = Quiet@Solve[leftBCMatrix.newYs == 0, newYs]; leftYICs = NullSpace[leftBCMatrix /. x -> xa, Method -> "DivisionFreeRowReduction"]; lenLeft = Length[leftYICs]; subsetsLeft = Subsets[Range[len], {lenLeft}]; - (* Initial conditions for shooting from the RHS *) - RightBCSolution = Quiet@Solve[rightBCMatrix.newYs == rightBCVector, newYs]; + RightBCSolution = Quiet@Solve[rightBCMatrix.newYs == 0, newYs]; rightYICs = NullSpace[rightBCMatrix /. x -> xb, Method -> "DivisionFreeRowReduction"]; lenRight = Length[rightYICs]; subsetsRight = Subsets[Range[len], {lenRight}]; @@ -173,12 +211,12 @@ Exp[-Integrate[Tr[AMatrix /. \[FormalLambda] -> \[FormalLambda]0], {x, xa, matchPoint}]] det /. x -> matchPoint /. solutionFromRight /. solutionFromLeft] -ToLinearMatrixForm[eqn_, bcs_?ListQ, depvars_, x_] := +ToLinearMatrixForm[eqn_, bcs_?ListQ, depvars_, x_ /; (!NumericQ[x] && !ListQ[x])] := If[bcs == {}, ToLinearMatrixForm[eqn, bcs, depvars, {x, 0, 0}], Message[ToLinearMatrixForm::noLimits];Return[$Failed]] -ToLinearMatrixForm[eqn_, BCs_?ListQ, depvars_, {x_, xa_, xb_}] := +ToLinearMatrixForm[eqn_, BCs_?ListQ, depvars_, {x_ /; !NumericQ[x], xa_, xb_}] := Module[{allVariables, nDepVars, allVariablesInd, originalYVariablesInd, nODEi, eqns, newYs, linearisedEqn, YDerivativeVector, FVector, originalYVariables, depVariables, epsilonEqn, newYSubs, - newYDerivSubs, AMatrix, leftBCs, rightBCs, leftBCMatrix, rightBCMatrix, leftBCVector, rightBCVector, undifferentiatedVariables = {}, sol, undifsol}, + newYDerivSubs, AMatrix, leftBCs, rightBCs, leftBCMatrix, rightBCMatrix, leftBCVector={}, rightBCVector={}, undifferentiatedVariables = {}, sol, undifsol}, (* Cover for the case of a single variable or multiple: *) depVariables = Flatten[{depvars}]; @@ -209,14 +247,14 @@ epsilonEqn = eqns /. Thread[allVariables -> (allVariables \[FormalE])]; - + (* Linearise the equation in case it isn't already *) linearisedEqn = Normal@Series[epsilonEqn, {\[FormalE], 0, 1}]; - + (* Check if the linearised equation is the same as the original, if not throw a warning to make it explicit *) If[! (Thread[(((linearisedEqn /. Equal -> Subtract) - (epsilonEqn /.Equal -> Subtract) // Simplify) == Table[0, nDepVars])] === True), Message[ToLinearMatrixForm::linearised, linearisedEqn /. \[FormalE] -> 1]]; - + (* Replace all the original variables with a set indexed by Y *) newYs = Through[Array[\[FormalY], {Length[originalYVariables]}][x]]; newYSubs = Thread[originalYVariables -> newYs]; @@ -240,9 +278,6 @@ (* If the equation is inhomogeneous, print message to say the CMM doesn't work *) If[AnyTrue[FVector, (!PossibleZeroQ[#])&], Message[ToLinearMatrixForm::inhomogeneous]]; - (* If no BCs were given, just return the Matrix and give a warning *) - If[Length[BCs] != Length[AMatrix], Message[ToLinearMatrixForm::matrixOnly,Length[BCs],Length[AMatrix]];Return[AMatrix]]; - (* Don't think this is doing anything... If[Length[undifferentiatedVariables] > 0, undifsol = (DSolve[Take[sol[[1]], -Length[undifferentiatedVariables]] /. Rule -> Equal, depvars, x] // Quiet), undifsol = {}]; @@ -251,15 +286,71 @@ (*Generate the boundary condition matrices and vectors *) leftBCs = Select[BCs /. (Thread[originalYVariables -> newYs] /. x -> xa) /. (sol[[1]] /. x -> xa), !FreeQ[#, \[FormalY]]&]; rightBCs = Select[BCs /. (Thread[originalYVariables -> newYs] /. x -> xb) /. (sol[[1]] /. x -> xb), !FreeQ[#, \[FormalY]]&]; - leftBCMatrix = Coefficient[#, newYs /. x -> xa]& /@ (leftBCs /. Equal -> Subtract); - rightBCMatrix = Coefficient[#, newYs /. x -> xb]& /@ (rightBCs /. Equal -> Subtract); - leftBCVector = -(leftBCs /. Equal -> Subtract) /. Thread[(newYs /. x -> xa) -> 0]; - rightBCVector = -(rightBCs /. Equal -> Subtract) /. Thread[(newYs /. x -> xb) -> 0]; + + If[Length[leftBCs] > 0, + leftBCMatrix = Coefficient[#, newYs /. x -> xa] & /@ (leftBCs /. Equal -> Subtract); + leftBCVector = -(leftBCs /. Equal -> Subtract) /. Thread[(newYs /. x -> xa) -> 0], + leftBCMatrix = {}; + ]; + + If[Length[rightBCs] > 0, + rightBCMatrix = Coefficient[#, newYs /. x -> xb] & /@ (rightBCs /. Equal -> Subtract); + rightBCVector = -(rightBCs /. Equal -> Subtract) /. Thread[(newYs /. x -> xb) -> 0];, + rightBCMatrix = {}; + ]; (* Return the solutions *) - {AMatrix, {leftBCMatrix, leftBCVector}, {rightBCMatrix, rightBCVector}, {x, xa, xb}} + If[AnyTrue[Join[leftBCVector,rightBCVector],(!PossibleZeroQ[#])&], + Message[ToLinearMatrixForm::inhomogeneousBCs,leftBCVector,rightBCVector]; + {AMatrix, {leftBCMatrix, leftBCVector}, {rightBCMatrix, rightBCVector}, {x, xa, xb}}, + + {AMatrix, leftBCMatrix, rightBCMatrix, {x, xa, xb}} + ] ] +ToLinearMatrixForm[eqns_?ListQ, BCs_?ListQ, {depvarLeft_, depvarRight_}, {x_, xa_, xmatch_, xb_}] := + Module[{leftEqns, rightEqns, leftBCs, rightBCs, FMatrix, GMatrix, leftAMatrix, rightAMatrix, leftBCMatrix, rightBCMatrix, stuff, xLeft, xRight,interfaceBCs,flatBCs}, + flatBCs = Flatten[BCs]; + leftEqns = Select[eqns, ! FreeQ[#, depvarLeft] &]; + rightEqns = Select[eqns, ! FreeQ[#, depvarRight] &]; + leftBCs = Select[flatBCs, ! FreeQ[#, \[FormalA]_[xa] /; ! (\[FormalA] === Derivative), All] &]; + rightBCs = Select[flatBCs, ! FreeQ[#, \[FormalA]_[xb] /; ! (\[FormalA] === Derivative), All] &]; + interfaceBCs = Select[flatBCs, ! FreeQ[#, \[FormalA]_[xmatch] /; ! (\[FormalA] === Derivative), All] &]; + FMatrix = ExtractInterface[interfaceBCs, depvarLeft, {x, xmatch}]; + GMatrix = ExtractInterface[interfaceBCs, depvarRight, {x, xmatch}]; + {leftAMatrix, leftBCMatrix, stuff, xLeft} = ToLinearMatrixForm[leftEqns, leftBCs, depvarLeft, {x, xa, xmatch}]; + {rightAMatrix, stuff, rightBCMatrix, xRight} = ToLinearMatrixForm[rightEqns, rightBCs, depvarRight, {x, xmatch, xb}]; + (*Return the system for using in CMM function*) + {{leftAMatrix, rightAMatrix}, leftBCMatrix,rightBCMatrix, {FMatrix, GMatrix}, {x, xa, xmatch, xb}} + + ] + +ExtractInterface[eqn_, depvars_, {x_,xmatch_}] := + Module[{undifferentiatedVariables = {},neweqn,depVariables,nDepVars,nODEi,eqns,allVariablesInd,originalYVariablesInd,allVariables,originalYVariables,newYs,newYSubs}, + (*Cover for the case of a single variable or multiple:*) + neweqn = eqn /. \[FormalY]_[xmatch] -> \[FormalY][x]; + depVariables = Flatten[{depvars}]; + If[! ListQ[neweqn], eqns = {neweqn}, eqns = neweqn]; + nDepVars = Length[depVariables]; + Table[(*Find highest derivative that occurs for each dependent variable*) + nODEi[i] = Max[0, Cases[eqns, Derivative[m_][depVariables[[i]]][a_] :> m, Infinity]]; + (*Produce a list of all the derivatives for that variable*) + allVariablesInd[i] = Table[D[depVariables[[i]][x], {x, n}], {n, 0, nODEi[i]}]; + (*If no derivatives exist, then that variable is undifferentiated and need to catch it. Else put all in *) + If[Length[allVariablesInd[i]] == 1, + AppendTo[undifferentiatedVariables, allVariablesInd[i][[1]]]; + originalYVariablesInd[i] = {}, + originalYVariablesInd[i] = allVariablesInd[i]], {i, nDepVars}]; + allVariables = Flatten[Table[allVariablesInd[i], {i, 1, nDepVars}]]; + (*Combine the solutions together*) + originalYVariables = Flatten[Table[originalYVariablesInd[i], {i, 1, nDepVars}]]; + + (*Replace all the original variables with a set indexed by Y*) + newYs = Through[Array[\[FormalY], {Length[originalYVariables]}][x]]; + newYSubs = Thread[originalYVariables -> newYs]; + Transpose[Table[Coefficient[neweqn /. newYSubs /. Equal -> Subtract, i], {i, newYs}]] + ] + SelectNegativeEigenvectors[mat_?MatrixQ, x_ /; !NumericQ[x]] := Module[{limitMatrix}, limitMatrix = Limit[mat, x -> -\[Infinity]]; @@ -274,6 +365,154 @@ Extract[#[[2]], Position[#[[1]], a_ /; Re[a] > 0, 1]] &@ Eigensystem[limitMatrix] ] +CompoundMatrixMethod[{\[FormalLambda]_ /; !NumericQ[\[FormalLambda]], \[FormalLambda]0_?NumericQ}, {{ALeftMatrix_?MatrixQ, ARightMatrix_?MatrixQ}, + leftBCMatrix_?MatrixQ, rightBCMatrix_?MatrixQ, {FMatrix_?MatrixQ, GMatrix_?MatrixQ}, {x_ /; ! NumericQ[x], xa_?NumericQ, xm_?NumericQ, xb_?NumericQ}}] := + CompoundMatrixMethod[{\[FormalLambda], \[FormalLambda]0}, {ALeftMatrix, ARightMatrix}, leftBCMatrix, rightBCMatrix, {FMatrix, GMatrix}, {x, xa, xm, xb}] + +CompoundMatrixMethod[{\[FormalLambda]_ /; !NumericQ[\[FormalLambda]], \[FormalLambda]0_? + NumericQ}, {ALeftMatrix_?MatrixQ, ARightMatrix_?MatrixQ}, leftBCMatrix_?MatrixQ, rightBCMatrix_?MatrixQ, + {FMatrix_?MatrixQ, GMatrix_?MatrixQ}, {x_ /; !NumericQ[x], xa_?NumericQ, xm_?NumericQ, xb_?NumericQ}] := + Module[{dettt, len, subsets, newYs, leftYICs, rightYICs, phiLeftVector, phiRightVector, LeftBCSolution, RightBCSolution, yLeft, yRight, + phiLeft, phiRight, LeftPositiveEigenvalues, RightNegativeEigenvalues, phiLeftICs, phiRightICs, QQ, solutionFromRight, + solutionFromLeft, det, matchPoint,lenLeft,lenRight,subsetsLeft,subsetsRight,QLeft,QRight}, + If[(xa <= xm <= xb && NumericQ[xm]), matchPoint = xm]; + (*Check some conditions are true, square matrix with full rank and homogeneous BCs*) + If[Length[ALeftMatrix] != Length[Transpose[ALeftMatrix]], + Message[CompoundMatrixMethod::nonSquareMatrix]; Return[$Failed]]; + If[Length[ARightMatrix] != Length[Transpose[ARightMatrix]], + Message[CompoundMatrixMethod::nonSquareMatrix]; Return[$Failed]]; + (*Check that the eigenvalue appears in the matrix A*) + If[MatrixQ[ALeftMatrix, FreeQ[#, \[FormalLambda]] &] && + MatrixQ[ARightMatrix, FreeQ[#, \[FormalLambda]] &], + Message[CompoundMatrixMethod::noEigenvalue, + {ALeftMatrix,ARightMatrix}, \[FormalLambda]];]; + (*Check that the matrix components are numerical, at least at the endpoints*) + If[! MatrixQ[ALeftMatrix /. \[FormalLambda] -> \[FormalLambda]0 /. x -> xa,NumericQ], + Message[CompoundMatrixMethod::nonNumericalMatrix, ALeftMatrix, xa]; Return[$Failed]]; + If[! MatrixQ[ARightMatrix /. \[FormalLambda] -> \[FormalLambda]0 /. x -> xa,NumericQ], + Message[CompoundMatrixMethod::nonNumericalMatrix, ARightMatrix, xa]; Return[$Failed]]; + If[! MatrixQ[ALeftMatrix /. \[FormalLambda] -> \[FormalLambda]0 /. x -> xb,NumericQ], + Message[CompoundMatrixMethod::nonNumericalMatrix, ALeftMatrix, xb]; Return[$Failed]]; + If[! MatrixQ[ARightMatrix /. \[FormalLambda] -> \[FormalLambda]0 /. x -> xb,NumericQ], + Message[CompoundMatrixMethod::nonNumericalMatrix, ARightMatrix, xb]; Return[$Failed]]; + + + If[Length[ARightMatrix] != Length[ALeftMatrix], + Message[CompoundMatrixMethod::MatrixSizesDiffer, ALeftMatrix, ARightMatrix];Return[$Failed]]; + + len = Length[ARightMatrix]; + + If[len>10, Message[CompoundMatrixMethod::InterfaceTooBig, len];Return[$Failed]]; + + + newYs = Through[Array[\[FormalY], {len}][x]]; + + (*Initial conditions for shooting from the LHS*) + LeftBCSolution = + Quiet@Solve[leftBCMatrix.newYs == 0, newYs]; + leftYICs = + NullSpace[leftBCMatrix /. x -> xa, Method -> "DivisionFreeRowReduction"]; + lenLeft = Length[leftYICs]; + subsetsLeft = Subsets[Range[len], {lenLeft}]; + (*Initial conditions for shooting from the RHS*) + RightBCSolution = + Quiet@Solve[rightBCMatrix.newYs == 0, newYs]; + rightYICs = + NullSpace[rightBCMatrix /. x -> xb, Method -> "DivisionFreeRowReduction"]; + lenRight = Length[rightYICs]; + subsetsRight = Subsets[Range[len], {lenRight}]; + (*Check the initial conditions for each side are enough*) + If[Length[LeftBCSolution] != 1, + Message[CompoundMatrixMethod::boundarySolutionFailed, xa]; + Return[$Failed]]; + If[Length[RightBCSolution] != 1, + Message[CompoundMatrixMethod::boundarySolutionFailed, xb]; + Return[$Failed]]; + If[Length[leftYICs] + Length[rightYICs] != len, + Message[CompoundMatrixMethod::boundaryConditionRank]; + Return[$Failed]]; + (*Generate two sets of Phi vaiables,these will be the matrix minors*) + + phiLeftVector = + Table[\[FormalPhi]L[i][x], {i, 1, Length[subsetsLeft]}]; + phiRightVector = + Table[\[FormalPhi]R[i][x], {i, 1, Length[subsetsRight]}]; + (*Full set of Initial Conditions for the left and right sides, with the BCs incorporated*) + yLeft = + Transpose[ + leftYICs + Table[newYs, {lenLeft}] /. LeftBCSolution[[1]] /. + Thread[newYs -> 0]]; + yRight = + Transpose[ + rightYICs + Table[newYs, {lenRight}] /. RightBCSolution[[1]] /. + Thread[newYs -> 0]]; + (*Use the initial conditions on the Y vectors to generate initial conditions for the minors phi*) + phiLeft = (Det[(yLeft /. + x -> xa /. \[FormalLambda] -> \[FormalLambda]0)[[#]]] & /@ + subsetsLeft); + phiRight = (Det[(yRight /. + x -> xb /. \[FormalLambda] -> \[FormalLambda]0)[[#]]] & /@ + subsetsRight); + (*Find the exponentially growing modes from each side, positive or negative eigenvalues depending on which side*) + LeftPositiveEigenvalues = + Select[Eigenvalues[ + ALeftMatrix /. x -> xa /. \[FormalLambda] -> \[FormalLambda]0], + Re[#] > 0 &]; + RightNegativeEigenvalues = + Select[Eigenvalues[ + ARightMatrix /. x -> xb /. \[FormalLambda] -> \[FormalLambda]0], + Re[#] < 0 &]; + (*Apply the initial conditions for the left and right solutions*) + phiLeftICs = + Thread[Through[Array[\[FormalPhi]L, {Length[subsetsLeft]}][xa]] == + phiLeft]; + phiRightICs = + Thread[Through[Array[\[FormalPhi]R, {Length[subsetsRight]}][xb]] == + phiRight]; + (*Calculate the Q matrix (phi'=Q phi) for each side*) + QLeft = + qMatrix[len, lenLeft] /. \[FormalCapitalA][i_, j_] :> + ALeftMatrix[[i, j]] /. \[FormalLambda] -> \[FormalLambda]0; + QRight = + qMatrix[len, lenRight] /. \[FormalCapitalA][i_, j_] :> + ARightMatrix[[i, j]] /. \[FormalLambda] -> \[FormalLambda]0; + (*Solve for integrating from the left and right*) + solutionFromLeft = + NDSolve[{Thread[ + D[phiLeftVector, + x] == (QLeft - + Total[Re@LeftPositiveEigenvalues] IdentityMatrix[ + Length[QLeft]]).phiLeftVector], phiLeftICs}, + Array[\[FormalPhi]L, {Length[subsetsLeft]}], {x, xa, xm}, + MaxStepFraction -> 0.01][[1]]; + solutionFromRight = + NDSolve[{Thread[ + D[phiRightVector, + x] == (QRight - + Total[Re@RightNegativeEigenvalues] IdentityMatrix[ + Length[QRight]]).phiRightVector], phiRightICs}, + Array[\[FormalPhi]R, {Length[subsetsRight]}], {x, xm, xb}, + MaxStepFraction -> 0.01][[1]]; + + (* Now we need to account for the jump conditions at the interface, so instead of the normal determinant it needs + modifying by multiplication by the matrices F and G. *) + + det = Total@Table[\[FormalPhi]\[FormalPhi]L[i][x] + \[FormalPhi]\[FormalPhi]R[Complement[Range[len], i]][x] (-1)^(Total[Range[lenLeft] + i]), {i, subsetsLeft}]; + + dettt = + det /. rulesFG[len]/.reprules2 /.Thread[subsetsLeft -> Range[Length[subsetsLeft]]] + /.Thread[subsetsRight -> Range[Length[subsetsRight]]]; + + Exp[-Integrate[ + Tr[ALeftMatrix /. \[FormalLambda] -> \[FormalLambda]0], {x, + xa, xm}]] dettt /. {F[i_, j_] :> FMatrix[[i, j]], + G[i_, j_] :> GMatrix[[i, j]]} /. + x -> xm /. \[FormalLambda] -> \[FormalLambda]0 /. solutionFromRight /. solutionFromLeft + ] + + + End[]; (* End Private Context *) EndPackage[]; diff --git a/CompoundMatrixMethod/PacletInfo.m b/CompoundMatrixMethod/PacletInfo.m index 6877353..1cc88e0 100644 --- a/CompoundMatrixMethod/PacletInfo.m +++ b/CompoundMatrixMethod/PacletInfo.m @@ -3,7 +3,7 @@ (* ::Input::Initialization:: *) Paclet[ Name->"CompoundMatrixMethod", -Version->"0.4", +Version->"0.5", MathematicaVersion->"9+", Description->"Solve Eigenvalue Boundary Value Problems using the Compound Matrix Method.", Creator->"Simon Pearce",