# How to speed up RotationMatrix?

I frequently run into the situation that I have to apply RotationMatrix to a huge bunch of 3D vectors and angles for numerical computations. On one hand, the syntax of RotationMatrix forces me to perform (several) transpositions, in order to generate data onto which it can be mapped upon (since RotationMatrix is not Listable). On the other hand, the execution is way too slow. What can we do about it?

As an example, let's assume we are given two lists of 1000 3D vectors each, and we seek the rotations that rotate each vector in the first list to the corresponding vector in the second list. We can do that with

n = 1000;
udata = RandomReal[{-1, 1}, {n, 3}];
vdata = RandomReal[{-1, 1}, {n, 3}];
First @ RepeatedTiming[result = RotationMatrix /@ Transpose[{udata, vdata}];]


0.17

but admittedly, 0.17 seconds for only 1000 $3 \times 3$ matrices is pretty slow for a numerical function...

• Can you give a short example to clarify (sample input data)? I just wrote an answer and then realized that it might not be what you wanted. Dec 3, 2017 at 12:48
• @Szabolcs Uh sorry. See my answer below. Dec 3, 2017 at 12:50

Having this problem so often, I also generated some tools to handle it which I'd like to share. This is the code (along with a usage message which is basically a small modification of RotationMatrix::usage. Note that it does not handle exceptions and that it assumes that a C compiler is installed.

Quiet@Block[{angle, v, vv, u, uu, ww, e1, e2, e2prime, e3},
uu = Table[u[[i]], {i, 1, 3}];
vv = Table[v[[i]], {i, 1, 3}];

rotationMatrix2D = Compile[
{{angle, _Real}},
{{Cos[angle], -Sin[angle]}, {Sin[angle], Cos[angle]}},
CompilationTarget -> "C",
RuntimeAttributes -> {Listable},
Parallelization -> True,
RuntimeOptions -> "Speed"
];

With[{code = N[
Simplify[ComplexExpand[RotationMatrix[angle, uu]], u[[1]] \[Element] Reals]
] /. Part -> CompileGetElement},
rotationMatrix3DAngleVector = Compile[
{ {angle, _Real},{u, _Real, 1}},
code,
CompilationTarget -> "C",
RuntimeAttributes -> {Listable},
Parallelization -> True,
RuntimeOptions -> "Speed"
]
];

ww = Cross[uu, vv];
e2 = Cross[ww, uu];
e2prime = Cross[ww, vv];

With[{code = N[
Plus[
KroneckerProduct[vv, uu]/Sqrt[uu.uu]/Sqrt[vv.vv],
KroneckerProduct[e2prime, e2]/Sqrt[e2.e2]/Sqrt[e2prime.e2prime],
KroneckerProduct[ww, ww]/ww.ww
]
] /. Part -> CompileGetElement},
rotationMatrix3DVectorVector = Compile[
{{u, _Real, 1}, {v, _Real, 1}},
code,
CompilationTarget -> "C",
RuntimeAttributes -> {Listable},
Parallelization -> True,
RuntimeOptions -> "Speed"
]
];

e1 = uu/Sqrt[uu.uu];
ww = Cross[uu, vv];
e3 = ww/Sqrt[ww.ww];
e2 = Simplify[Cross[e3, e1]];

With[{code = N[Simplify@Plus[
Cos[angle] Simplify@KroneckerProduct[e1, e1],
Sin[angle] Simplify@KroneckerProduct[e2, e1],
-Sin[angle] Simplify@KroneckerProduct[e1, e2],
Cos[angle] Simplify@KroneckerProduct[e2, e2],
Simplify@KroneckerProduct[e3, e3]
]] /. Part -> CompileGetElement},
rotationMatrix3DAngleVectorVector = Compile[
{{angle, _Real}, {u, _Real, 1}, {v, _Real, 1}},
code,
CompilationTarget -> "C",
RuntimeAttributes -> {Listable},
Parallelization -> True,
RuntimeOptions -> "Speed"
]
];
];

ClearAll[MyRotationMatrix];

MyRotationMatrix[angle_] := rotationMatrix2D[angle];

MyRotationMatrix[angle_, u_] := rotationMatrix3DAngleVector[angle, u];

MyRotationMatrix[{u_, v_}] := rotationMatrix3DVectorVector[u, v];

MyRotationMatrix[angle_, {u_, v_}] := rotationMatrix3DAngleVectorVector[angle, u, v];

MyRotationMatrix::usage =
"\!$$\*RowBox[{\"MyRotationMatrix\", \"[\", StyleBox[\"\[Theta]\", \ \"TR\"], \"]\"}]$$ gives the 2D rotation matrix that rotates 2D \
vectors counterclockwise by \!$$\*StyleBox[\"\[Theta]\", \"TR\"]$$ \
radians.\n\!$$\*RowBox[{\"MyRotationMatrix\", \"[\", \ RowBox[{StyleBox[\"\[Theta]\", \"TR\"], \",\", StyleBox[\"w\", \ \"TI\"]}], \"]\"}]$$ gives the 3D rotation matrix for a \
counterclockwise rotation around the 3D vector \!$$\*StyleBox[\"w\", \ \"TI\"]$$.\n\!$$\*RowBox[{\"MyRotationMatrix\", \"[\", RowBox[{\"{\", \ RowBox[{StyleBox[\"u\", \"TI\"], \",\", StyleBox[\"v\", \"TI\"]}], \ \"}\"}], \"]\"}]$$ gives the 3D matrix that rotates the vector \
\!$$\*StyleBox[\"u\", \"TI\"]$$ to the direction of the vector \
\!$$\*StyleBox[\"v\", \"TI\"]$$.\n\!$$\*RowBox[{\"MyRotationMatrix\", \ \"[\", RowBox[{StyleBox[\"\[Theta]\", \"TR\"], \",\", RowBox[{\"{\", \ RowBox[{StyleBox[\"u\", \"TI\"], \",\", StyleBox[\"v\", \"TI\"]}], \ \"}\"}]}], \"]\"}]$$ gives the matrix that rotates by \!$$\*StyleBox[\ \"\[Theta]\", \"TR\"]$$ radians in the hyperplane spanned by \
\!$$\*StyleBox[\"u\", \"TI\"]$$ and \!$$\*StyleBox[\"v\", \"TI\"]$$.";


And here is a short test suite:

n = 1000;
angledata = RandomReal[{-2 Pi, 2 Pi}, n];
udata = RandomReal[{-1, 1}, {n, 3}];
vdata = RandomReal[{-1, 1}, {n, 3}];

t1 = First@RepeatedTiming[aa = MyRotationMatrix[angledata];];
t2 = First@RepeatedTiming[bb = RotationMatrix /@ angledata;];
Association["MyTime" -> t1, "Time" -> t2, "SpeedUp" -> t2/t1,
"Error" -> Max[Abs[aa - bb]]]

t1 = First@RepeatedTiming[aa = MyRotationMatrix[angledata , vdata];];
t2 = First@ RepeatedTiming[ bb = RotationMatrix @@@ Transpose[{angledata, vdata}];];
Association["MyTime" -> t1, "Time" -> t2, "SpeedUp" -> t2/t1, "Error" -> Max[Abs[aa - bb]]]

t1 = First@RepeatedTiming[aa = MyRotationMatrix[{udata, vdata}];];
t2 = First@ RepeatedTiming[bb = RotationMatrix /@ Transpose[{udata, vdata}];];
Association["MyTime" -> t1, "Time" -> t2, "SpeedUp" -> t2/t1,  "Error" -> Max[Abs[aa - bb]]]

t1 = First@RepeatedTiming[aa = MyRotationMatrix[angledata, {udata, vdata}];];
t2 = First@RepeatedTiming[bb = RotationMatrix @@@Transpose[{angledata, Transpose[{udata, vdata}]}];];
Association["MyTime" -> t1, "Time" -> t2, "SpeedUp" -> t2/t1, "Error" -> Max[Abs[aa - bb]]]


<|"MyTime" -> 0.000067, "Time" -> 0.032, "SpeedUp" -> 4.9*10^2, "Error" -> 1.11022*10^-16|>

<|"MyTime" -> 0.000098, "Time" -> 0.273, "SpeedUp" -> 2.8*10^3, "Error" -> 9.99201*10^-16|>

<|"MyTime" -> 0.00010, "Time" -> 0.17, "SpeedUp" -> 1.7*10^3, "Error" -> 8.88178*10^-16|>

<|"MyTime" -> 0.000096, "Time" -> 0.16, "SpeedUp" -> 1.7*10^3, "Error" -> 2.03171*10^-14|>

Edit

Fixed the argument pattern of the angle + vector case to make it compatible with RotationMatrix.

I have previously used the following routine based on ideas by Möller and Hughes in these previous answers, but it would be good to have it as an explicit answer here:

vectorRotate[vv1_?VectorQ, vv2_?VectorQ] :=
Module[{v1 = Normalize[vv1], v2 = Normalize[vv2], c, d, d1, d2, t1, t2},
d = v1.v2;
If[TrueQ[Chop[1 + d] == 0],
c = UnitVector[3, First[Ordering[Abs[v1], 1]]];
t1 = c - v1; t2 = c - v2; d1 = t1.t1; d2 = t2.t2;
IdentityMatrix[3] - 2 (Outer[Times, t2, t2]/d2 -
2 t2.t1 Outer[Times, t2, t1]/(d2 d1) + Outer[Times, t1, t1]/d1),
c = Cross[v1, v2];
d IdentityMatrix[3] + Outer[Times, c, c]/(1 + d) - LeviCivitaTensor[3].c]]


Using the simple test in the OP:

With[{n = 1000},
udata = RandomReal[{-1, 1}, {n, 3}];
vdata = RandomReal[{-1, 1}, {n, 3}]];

First @ RepeatedTiming[result1 = RotationMatrix /@ Transpose[{udata, vdata}];]
0.272

First @ RepeatedTiming[result2 = MapThread[vectorRotate, {udata, vdata}];]
0.19

Max[Abs[result1 - result2]]
7.99916*10^-14
`
• (As for the axis-angle case, I implemented the Rodrigues rotation formula in this answer.) Mar 4, 2018 at 17:37