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I'm currently trying to implement some code to compute the Frechet derivative of a system of PDEs, and the associated formal adjoint. The code I'm using to compute derivative is:

This works an absolute charm. Then to compute the associated adjoint, I use:Code (Text):FrechetD[support_List, dependVar_List,

independVar_List, testfunction_List] :=

Block[{indep, frechet, deriv, \[Epsilon], r0, x1, x2},

r0 = Function[indep, x1 + \[Epsilon] x2];

frechet = {}; Do[deriv = {};

Do[AppendTo[deriv, \!\(

\*SubscriptBox[\(\[PartialD]\), \(\[Epsilon]\)]\ \((support[[

j]] /. \[IndentingNewLine]dependVar[[

i]] -> \((r0 /. \[IndentingNewLine]{indep ->

independVar, \[IndentingNewLine]x1 ->

dependVar[[i]] @@ independVar, \[IndentingNewLine]x2 ->

testfunction[[i]] @@

independVar})\))\)\) /. \[Epsilon] -> 0],

{i, 1, Length[support]}];

AppendTo[frechet, deriv],

{j, 1, Length[support]}];

frechet]

This doesn't work, sadly. All it seems to do is give me the transpose of the original Frechet derivative matrix. The code originally came from a book I've looked at, so it can't be too far wrong. Sadly, my mathematica skills are pretty much non-existent, so I haven't a clue what the problem is! If anyone could give me some pointers it would be much appreciated!Code (Text):AdjointFrechetD[support_List, dependVar_List,

independVar_List, testfunction_List] :=

Block[{subrule, $testf, frechet, n, b},

subrule = b_. ($testf^(n__)) @@ independVar :>

(-1)^Plus @@ {n} \!\(

\*SubscriptBox[\(\[PartialD]\), \(Delete[Thread[{independVar, {n}}],

0]\)]\((b\ $testf @@ independVar)\)\);

frechet = FrechetD[support, dependVar,

independVar , testfunction];

Do[frechet =

frechet /.

(subrule /. $testf -> testfunction[[i]]),

{i, 1, Length[testfunction]}];

frechet = Transpose[frechet]]

Many thanks,

Ant

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