MultiLinear and Linear function

• To: mathgroup at smc.vnet.net
• Subject: [mg25540] MultiLinear and Linear function
• From: "Arturas Acus" <acus at itpa.lt>
• Date: Sat, 7 Oct 2000 03:35:43 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```Dear group,

Here was a question how to make a function linear. Below is the
solution
from (I think Noncommutative Algebra package, but not sure):

SetLinear[a__] := (Function[x, x[0] := 0;
x[y_ + z_] := x[y] + x[z];
x[c_ y_] := c x[y] /; NumberQ[c];
x[y_/c_] := (1/c) x[y] /; NumberQ[c];
x[c_] := c x[1] /; NumberQ[c] && Not[TrueQ[c == 1]];
x[-y_] := -x[y];
LinearQ[x] = True;] /@ {a});

It works fine, but is only applied to function of multiple variables.
Bolow is my solution for MultiLinear function. It works, but is not
so
elegant as example bekow. Probably anybody knows a better approach?

makePattern[x_, y_, {n_Integer, {m_Integer}}] :=
Module[{z, p1, p2}, p1 = Table[Unique[], {n - 1}];
p2 = Table[Unique[], {z, n + 1, m}];
ReplaceAll[
Function[y,
x[Sequence @@ Flatten[Pattern[#, Blank[]] & /@ p1], y,
Sequence @@ Flatten[Pattern[#, Blank[]] & /@ p2]]],
Pattern -> myPattern]]

SetMultiLinear[
s__Symbol, {i_Integer, {l_Integer}}] := (Function[x,
Module[{pureF, locvar},
pureF = makePattern[x, locvar, {i, {l}}];
SetDelayed @@ {ReplaceAll[
Evaluate[pureF[y_ + z_], myPattern -> Pattern]],
ReplaceAll[Evaluate[pureF[y] + pureF[z]],
myPattern[a_, Blank[]] :> a]};

SetDelayed @@ {ReplaceAll[
Evaluate[pureF[c_*y_], myPattern -> Pattern]],
ReplaceAll[Evaluate[c*pureF[y]],
myPattern[a_, Blank[]] :> a] /; NumberQ[c]};

SetDelayed @@ {ReplaceAll[
Evaluate[pureF[y_/c_], myPattern -> Pattern]],
ReplaceAll[Evaluate[(1/c)*pureF[y]],
myPattern[a_, Blank[]] :> a] /; NumberQ[c]};

SetDelayed @@ {ReplaceAll[
Evaluate[pureF[c_], myPattern -> Pattern]],
ReplaceAll[Evaluate[c*pureF[1]],
myPattern[a_, Blank[]] :> a] /;
NumberQ[c] && Not[TrueQ[c === 1]]};
LinearQ[x, {i, {l}}] = True;
];
] /@ {s}) /; Greater[l, 1]

SetMultiLinear[s__Symbol, {All, {l_Integer}}] :=
Map[SetMultiLinear[s, {#, {l}}] &, Range[l]]

Example of usage: Make   Commutator and AntiCommutator linear
functions of both variables:
SetMultiLinear[Commutator,AntiCommutator,{All,{2}}].

Make f be linear in 3-th its argument of total 5:
SetMultiLinear[f,{3,{5}}].

Better solution welcome

Dr. Arturas Acus
Institute of Theoretical
Physics and Astronomy
Gostauto 12, 2600,Vilnius
Lithuania

E-mail: acus at itpa.lt
Fax: 370-2-225361
Tel: 370-2-612906

```

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