Clifford algebra decompositions of conformal orthogonal group elements with Mathematica

G Stacey Staples, November 2014  (Revised May 2015)

Quit[];  (* Quit the kernel to get a fresh start, if needed. *)

<<CliffMath2014a`

SetSignature[2,1]

Generate a random vector in current Clifford algebra with random integer coefficients between - ell and ell.

Intrv[ell_]:=Total[Table[Random[Integer, {-ell,ell}](Subscript[e, {j}]),{j,1,MaxIndex}]]

For example, compute MaxIndex random vectors with integer coefficients in [-5,5].

L=Table[Intrv[5],{t,1,MaxIndex}]

B=L[[1]]⊙L[[2]]//ClExpand

Compute the exterior product of the first five vectors in L.

L=Table[Intrv[5],{t,1,MaxIndex}];
Timing[B=WExpand[WListProd[L[[1;;5]]]]]

The 5-blade computed above.

B

(* Gram-Schmidt for Euclidean signatures *)
GramSchmidt[A_]:=Module[{U,j,ut,ell},
U={Expand[A[[1]]/Sqrt[A[[1]].A[[1]]]]};
For[ell=2,ell≤Length[A], ell++,
ut=Expand[A[[ell]]-Sum[Expand[(U[[j]].A[[ell]])U[[j]]],{j,1,ell-1}]];
U=Append[U,Expand[ut/Sqrt[ut.ut]]]];
Return[U]];

Decomposition algorithms

The following procedure factors versors in Clifford algebras of definite signature.

(* Works in definite signatures.  Decomposition by geometric considerations. *)
VersorFactor[B_]:=Module[{ell,j,X,gb,bi,Bprime,btemp,k,x,z,BNorm},
BNorm=N[CliffordNorm[B,2]];
ell=1;
Bprime=Expand[B/BNorm];
While[Grade[Bprime]>1,
bi=GetIndex[Bprime];
x=N[(Subscript[e, {bi[[1Random[Integer,{1,Length[bi]}]]]}])];

X=Chop[N[ClExpand[CircleDot[Bprime,x,CliffordConjugate[Bprime]]]]];
gb=(((-1))^(Grade[Bprime]));
If[Chop[LeftContract[x,X]≠1],
btemp=Chop[Expand[x-X]];
(Subscript[b, ell])=Chop[Expand[btemp/CliffordNorm[btemp,2]]];

(* If odd grade, pull out one reflection first *)
If[gb<0,
Bprime=Chop[ClExpand[CircleDot[Bprime,(Subscript[b, ell])]]];
ell=ell+1;,
(* For even-grade Bprime, compute bivector *)
z=Chop[Expand[N[ClExpand[CircleDot[Bprime,(Subscript[b, ell]),Reversion[Bprime]]]]]];
If[Chop[LeftContract[(Subscript[b, ell]),z]]≠-1,
(Subscript[b, ell+1])=N[Chop[Expand[((Subscript[b, ell])+z)/CliffordNorm[((Subscript[b, ell])+z),2]]]];
Bprime=Expand[Chop[ClExpand[CircleDot[Bprime,(Subscript[b, ell]),(Subscript[b, ell+1])]]]];
ell=ell+2;,
Bprime=Expand[Chop[ClExpand[CircleDot[Bprime,(Subscript[b, ell])]]]];
ell=ell+1;];];];];
Return[Append[Table[(Subscript[b, j]),{j,ell-1}],Expand[BNorm Bprime]]];];

(* A very fast algorithm dealing strictly with blades *)
FastBladeFactor[B_]:=Module[{ell,α,gb,F,Fi,Bprime,btemp,u},
α=B[[1]]/.{(Subscript[e, _])->1};
ell=1;
Bprime=Expand[B/α];
F=Expand[B[[1]]/α];
Fi=GetIndex[F];
gb=Grade[F];
While[ell≤gb,
btemp=(Subscript[e, Complement[Fi,{Fi[[ell]]}]]);
(Subscript[b, ell])=GradeKPart[Chop[ClExpand[CircleDot[Bprime,ClInverse[btemp]]]],1];
ell=ell+1];
Return[Prepend[Table[(Subscript[b, j]),{j,ell-1}], α]];];

(* True if all terms of u are of even grade or all terms are of odd grade *)
ParityCheck[u_]:=Xor[AllTrue[Grade[DeleteCases[Variables[u],Except[(Subscript[e, _])]]],EvenQ],((u/.{(Subscript[e, _])->0})==0∧NoneTrue[Grade[DeleteCases[Variables[u],Except[(Subscript[e, _])]]],EvenQ])]

(* Decomposability Test. Necessary in indefinite signatures *)
DecomposableQ[u_]:=ParityCheck[u]∧FreeQ[ClExpand[CircleDot[u,Reversion[u]]],e]∧SignedNormSquare[GradeKPart[u,Grade[u]]]≠0

(* Decompose elements in arbitrary signatures *)
CliffordDecomp[B_]:=Module[{X,bi,gb,Bprime,btemp,facts},
Bprime=B;
facts={};
While[Grade[Bprime]>1,
gb=Grade[Bprime];
bi=GetIndex[Bprime];
x=(Subscript[e, {bi[[Random[Integer,{1,Length[bi]}]]]}]);X=Chop[ClExpand[CircleDot[Bprime,x,ConjugatePseudoInverse[Bprime]]]];
btemp=Expand[x- X];
If[Chop[SignedNormSquare[btemp]]!=0,
Bprime=Chop[ClExpand[CircleDot[Bprime, ClInverse[btemp]]]];
facts=Append[facts, btemp];
If[Grade[Bprime]>gb,Bprime=B; facts={};];
];];
Return[Append[facts,Bprime]];];

OrthoSumDecomp[W_]:=Module[{wprime,F,G,out},
wprime=W;
out={};
While[!FreeQ[wprime,(Subscript[e, _])],
F=Chop[FastBladeFactor[wprime]];
out=Append[out, F];
G=Expand[F[[1]] WListProd[Reverse[F[[2;;]]]]];
wprime=Chop[WExpand[wprime-G]];
];
Return[out];
];

Clifford algebra representations of conformal orthogonal matrices

Extending the idea to matrices, allow input of an invertible matrix.  Output the additive representation in Clifford algebra of definite signature.

SetSignature[8,0]

CliffRep[A_]:=Module[{M,α,IM,ClVec, u,ones,U,x,btemp,Aprime,i},
α=Norm[A,2];
Aprime=Chop[A/α];
ClVec=ClKVectors[1];
IM=IdentityMatrix[MaxIndex];
U=1;
While[Chop[Aprime]!=IM,
x=Table[Random[],{i,MaxIndex}];
u=Normalize[Chop[N[x-Aprime.x]]];
M=(Chop[IdentityMatrix[MaxIndex]-2KroneckerProduct[u,u]]);
Aprime=Chop[Aprime.Inverse[M]];
U=Chop[ClExpand[CircleDot[U,u.ClVec]]];
];
Return[{α,U}];]

J=Permutations[Table[i,{i,MaxIndex}]][[Random[Integer,{1,MaxIndex!}]]];
P=Table[If[J[[i]]==j,1,0],{i,1,MaxIndex},{j,1,MaxIndex}];
Pr=CliffRep[P];
Print["P = ",MatrixForm[P],"                              ",GraphPlot[P, DirectedEdges->True]];
Print["u = ",Expand[Pr[[1]]Pr[[2]]]];

Level - k vector representation of Clifford element .  Basis k - blades ordered by Mathematica' s subset ordering.

KVecRep[x_+y_,k_]:=Block[{$RecursionLimit=∞},KVecRep[x,k]+KVecRep[y,k]];
KVecRep[x_.(Subscript[e, a_]),k_]:=Module[{j,S},S=Subsets[Range[MaxIndex],{k}];x Table[If[a==S[[j]],1,0],{j,Binomial[MaxIndex,k]}]/;FreeQ[x,_.(Subscript[e, _])]];
KVecRep[x_,k_]:=(x Table[If[j==1,1,0],{j,Binomial[MaxIndex,k]}])/;FreeQ[x,_.(Subscript[e, _])];

Verify the result by building the matrix representation of x->u x (OverHat[(u^(-1))]) and comparing with matrix P.  This can take some time.

Print[MatrixForm[CP=Pr[[1]] Table[KVecRep[Chop[ClExpand[CircleDot[Pr[[2]],(Subscript[e, {i}]),GradeInvolution[ClInverse[Pr[[2]]]]]]],1],{i,MaxIndex}]]];
Print["Matrix difference: ",MatrixForm[Chop[P-CP]]];

Another example.  This time, begin with a randomly-generated conformal orthogonal matrix.

MO=Table[Random[Integer,{-5,5}],{i,1,MaxIndex},{j,1,MaxIndex}];
K=Random[Integer,{1,5}]Chop[N[GramSchmidt[MO]]];
Ku=CliffRep[K];
Print["K = ",MatrixForm[K],"                              ",GraphPlot[K, DirectedEdges->True]];
Print["u = ",Expand[Ku[[1]]Ku[[2]]]];

Verify the result by building the matrix representation of x->u x (OverHat[(u^(-1))]) and comparing with matrix K.  This can take some time.

Print[MatrixForm[CK=Ku[[1]] Table[KVecRep[Chop[ClExpand[CircleDot[Ku[[2]],(Subscript[e, {i}]),GradeInvolution[ClInverse[Ku[[2]]]]]]],1],{i,MaxIndex}]]];
Print["Matrix difference: ",MatrixForm[Chop[K-CK]]];

Observe that K is a conformal orthogonal matrix by computing (SuperDagger[K])K.

MatrixForm[Chop[Transpose[K].K]]

Decomposition Examples

For example, generate and decompose a grade-5 element of (Subscript[Cl, 3,5]):

SetSignature[4,4];
grade=5;
L=Table[Intrv[5],{t,1,grade}];
B=Chop[Expand[ClListProd[L]]]

Make sure top form of B is not a null blade.

Print["B is ", If[SignedNormSquare[GradeKPart[B,grade]]==0,"null!  Try again.", "okay."]];

DecomposableQ[B]

Decompose the element (exactly).  Computations can also be done numerically, but error propagation can be a problem.

Bf=Timing[CliffordDecomp[B]];
Print["Decomposition time: ",Bf[[1]]," seconds."];
Print["Decomposition: ",Flatten[Bf[[2;;]]]];

Compute the product of the vectors obtained.

B2=Chop[Expand[ClListProd[Reverse[Bf[[2]]]]]]

Verify the result.

CliffordNorm[Chop[Expand[B-B2]],2]

Now for some definite signature examples...

SetSignature[7,0];
grade=3;
L=Table[Intrv[5],{t,1,grade}];
B=Chop[Expand[ClListProd[L]]]

First apply VersorFactor.

Timing[Y=VersorFactor[B]]

Compute the product of the vectors found.

CheckY=Chop[Expand[ClListProd[L]]]

Verify the result.

B-CheckY//Chop

CliffDecomp is faster.

Timing[X=CliffordDecomp[B]]

Compute the product and compare with the original element.

CheckX=Chop[Expand[ClListProd[L]]]

B-CheckX//Chop

Retrieve high-order blade from B and apply FastBladeFactor.

Print[blade=GradeKPart[B,Grade[B]]];
Timing[Blff=FastBladeFactor[blade]]

Compute product of vectors obtained.

Blff2=Chop[Expand[WListProd[Reverse[Blff]]]]

Check the result.

blade-Blff2//Chop

SetSignature[8,0]

Construct a random sum of pairwise-orthogonal blades, and iteratively factor them out with FastBladeFactor.

M=Table[Random[Integer,{-5,5}],{i,1,MaxIndex},{j,1,MaxIndex}];
K=Chop[N[GramSchmidt[M]]];
CliffK=K.Table[(Subscript[e, {i}]),{i,MaxIndex}];
w1=WListProd[CliffK[[1;;2]]];
w2=WListProd[CliffK[[3;;5]]];
w3=WListProd[CliffK[[6;;8]]];
w=Expand[(w1+w2+w3)];
Print["Element to be decomposed: ",w];

Z=OrthoSumDecomp[w]

Check the result.

Print["w is the sum of..."];
For[k=1,k≤Length[Z],k++,
Print["w_",k," = ",WExpand[WListProd[Reverse[Z[[k]]]]]]];

Print["Orthosum = ",dcheck=Sum[WExpand[WListProd[Reverse[Z[[i]]]]],{i,1,Length[Z]}]];
Print["Difference w-Orthosum: ",Chop[w-dcheck]];

Apply CliffordDecomp and FastBladeFactor to elements of particular grade in (Subscript[Cl, 6,0]).  Compile runtimes over MaxTrials trials for analysis and comparison.
Data from 500 trials can be found at the end of this notebook.

SetSignature[6,0];
grade=4;
MaxTrials=50;
coeffrange=5;

correct=0;
trials=0;
Timelist46={};
For[m=1,m≤MaxTrials,m++,
bfacts={};
buildtable=Table[Intrv[coeffrange],{i,1,grade}];
u=Chop[Expand[N[ClListProd[buildtable]]]];
If[Grade[u]==grade,
trials=trials+1;
facts=Timing[bfacts=CliffordDecomp[u]];
prodcheck=Chop[N[ClListProd[Reverse[bfacts]]]];
If[Chop[SignedNormSquare[ClExpand[prodcheck-u]]]==0,
PrintTemporary["Trial ",m," verified!"];
correct=correct+1;,
Print["Trial ", m, "Not verified!  u=",u ];
];
Timelist46=Append[Timelist46,facts[[1]]];
];
];
Print[correct," correct = ",N[correct/(trials)*100],"%"];

SetSignature[6,0];
grade=4;
MaxTrials=50;
coeffrange=5;

correct=0;
trials=0;
Timelist46B={};
For[m=1,m≤MaxTrials,m++,
ClearSystemCache[];
bfacts={};
buildtable=Table[Intrv[coeffrange],{i,1,grade}];u=Expand[N[WListProd[buildtable]]];
If[Grade[u]==grade,
trials=trials+1;
facts=Timing[bfacts=CliffordDecomp[u]];
prodcheck=Chop[N[ClListProd[Reverse[bfacts]]]];
If[Chop[SignedNormSquare[ClExpand[prodcheck-u]]]==0,
PrintTemporary["Trial ",m," verified!"];
correct=correct+1;,
Print["Trial ", m, "Not verified!  u=",u ];
Print["Product check: ",prodcheck]];
Timelist46B=Append[Timelist46B,facts[[1]]];
];
];
Print[correct," correct = ",N[correct/(trials)*100],"%"];

SetSignature[6,0];
grade=4;
MaxTrials=50;
coeffrange=5;

correct=0;
trials=0;
Timelist46FBF={};
For[m=1,m≤MaxTrials,m++,
ClearSystemCache[];
bfacts={};
buildtable=Table[Intrv[coeffrange],{i,1,grade}];
u=Expand[N[WListProd[buildtable]]];
If[Grade[u]==grade,
trials=trials+1;
facts=Timing[bfacts=FastBladeFactor[u]];
prodcheck=Chop[N[WListProd[Reverse[bfacts]]]];
If[Chop[ClExpand[prodcheck-u]]==0,
{PrintTemporary["Trial ",m," verified!"];
correct=correct+1;},
{Print["Trial ", m, "Not verified!  u=",u ];
Print["Product check: ",prodcheck]}];Timelist46FBF=Append[Timelist46FBF,facts[[1]]];
];
];
Print[correct," correct = ",N[correct/(trials)*100],"%"];

ListPlot[{Timelist46,Timelist46B, Timelist46FBF},AxesLabel->{"Trial #","Time(s)"}]

Print["General element min time: ",Min[Timelist46]," seconds."]

Print["General element max time: ",Max[Timelist46]," seconds."]

Print["General element Mean time: ",Mean[Timelist46]," seconds."]

Print["Blade min time: ",Min[Timelist46B]," seconds."]

Print["Blade max time: ",Max[Timelist46B]," seconds."]

Print["Blade mean time: ",Mean[Timelist46B]," seconds."]

Print["Fast Blade Factor min time: ",Min[Timelist46FBF]," seconds."]

Print["Fast Blade Factor max time: ",Max[Timelist46FBF]," seconds."]

Print["Fast Blade Factor mean time: ",Mean[Timelist46FBF]," seconds."]

Apply CliffordDecomp and FastBladeFactor to elements of particular grade in (Subscript[Cl, 7,0]).  Compile runtimes over MaxTrials trials for analysis and comparison.

SetSignature[7,0];
grade=4;
MaxTrials=50;

correct=0;
trials=0;
Timelist47={};
For[m=1,m≤MaxTrials,m++,
ClearSystemCache[];
bfacts={};
buildtable=Table[Intrv[coeffrange],{i,1,grade}];u=Chop[Expand[N[ClListProd[buildtable]]]];
If[Grade[u]==grade,
trials=trials+1;
facts=Timing[bfacts=CliffordDecomp[u]];
prodcheck=Chop[N[ClListProd[Reverse[bfacts]]]];
If[Chop[ClExpand[prodcheck-u]]==0,
{PrintTemporary["Trial ",m," verified!"];
correct=correct+1;},
{Print["Trial ", m, "Not verified!  u=",u ];
Print["Product check: ",prodcheck]}];
Timelist47=Append[Timelist47,facts[[1]]];
];
];
Print[correct," correct = ",N[correct/(trials)*100],"%"];

SetSignature[7,0];
grade=4;
MaxTrials=50;

correct=0;
trials=0;
Timelist47B={};
For[m=1,m≤MaxTrials,m++,
ClearSystemCache[];
bfacts={};
buildtable=Table[Intrv[coeffrange],{i,1,grade}];
u=Expand[N[WListProd[buildtable]]];
If[Grade[u]==grade,
trials=trials+1;
facts=Timing[bfacts=CliffordDecomp[u]];
prodcheck=Chop[N[ClListProd[Reverse[bfacts]]]];
If[Chop[ClExpand[prodcheck-u]]==0,
{PrintTemporary["Trial ",m," verified!"];
correct=correct+1;},
{Print["Trial ", m, "Not verified!  u=",u ];
Print["Product check: ",prodcheck]}];
Timelist47B=Append[Timelist47B,facts[[1]]];
];];
Print[correct," correct = ",N[correct/(trials)*100],"%"];

SetSignature[7,0];
grade=4;
MaxTrials=50;

correct=0;
trials=0;
Timelist47FBF={};
For[m=1,m≤MaxTrials,m++,
ClearSystemCache[];
bfacts={};
buildtable=Table[Intrv[coeffrange],{i,1,grade}];u=Expand[N[WListProd[buildtable]]];
If[Grade[u]==grade,
trials=trials+1;
facts=Timing[bfacts=FastBladeFactor[u]];
prodcheck=Chop[N[WListProd[Reverse[bfacts]]]];
If[Chop[ClExpand[prodcheck-u]]==0,
{PrintTemporary["Trial ",m," verified!"];
correct=correct+1;},
{Print["Trial ", m, "Not verified!  u=",u ];
Print["Product check: ",prodcheck]}];
Timelist47FBF=Append[Timelist47FBF,facts[[1]]];
];];
Print[correct," correct = ",N[correct/(trials)*100],"%"];

ListPlot[{Timelist47,Timelist47B, Timelist47FBF},AxesLabel->{"Trial #","Time(s)"}]

Print["General element min time: ",Min[Timelist47]," seconds."];
Print["General element Mean time: ",Mean[Timelist47]," seconds."];
Print["General element max time: ",Max[Timelist47]," seconds."];

Print["Blade min time: ",Min[Timelist47B]," seconds."];
Print["Blade mean time: ",Mean[Timelist47B]," seconds."];
Print["Blade max time: ",Max[Timelist47B]," seconds."];

Print["Fast Blade Factor min time: ",Min[Timelist47FBF]," seconds."];
Print["Fast Blade Factor mean time: ",Mean[Timelist47FBF]," seconds."];
Print["Fast Blade Factor max time: ",Max[Timelist47FBF]," seconds."];

ListLinePlot[{Timelist46, Timelist46B, Timelist46FBF, Timelist47,Timelist47B, Timelist47FBF},PlotStyle->{RGBColor[0,0,1],RGBColor[1,0,0], RGBColor[0,1,0],RGBColor[.7,.7,0],RGBColor[0,.7,.7], RGBColor[.7,0,.7]},AxesLabel->{"Trial #","Time(s)"},PlotLegends->{"(Subscript[Cl, 6])","(Subscript[Cl, 6])Blade","(Subscript[Cl, 6])FBF","(Subscript[Cl, 7])","(Subscript[Cl, 7])Blade","(Subscript[Cl, 7])FBF"}]

Below are runtimes summarized in the paper Clifford algebra decompositions of conformal orthogonal transformations by Staples and Wylie (preprint, 2014).

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References

L.Dorst, D.Fontijne, Efficient Algorithms for Factorization and Join of Blades, Geometric Algebra Computing, E.Bayro-Corrochano, G. Scheuermann, Eds., Springer, London, 2010, pp. 457--476.

D. Fontijne,  Efficient Implementation of Geometric Algebra, Ph.D. thesis, University of Amsterdam, 2007. 

P. Lounesto, E. Latvamaa, Conformal transformations and Clifford algebras, Proc. Amer. Math. Soc., 79 (1980),  533--538.

C. Perwass, Geometric Algebra with Applications in Engineering, Springer-Verlag, Berlin, 2009.

G.S. Staples, D. Wylie, Clifford algebra decompositions of conformal orthogonal group elements, Preprint, 2014.

D. Wylie,  Factoring blades and versors in Euclidean Clifford algebras, M.S. Thesis, Southern Illinois University Edwardsville, 2014.