Cofactor Matrix #218

termi-official · 2024-05-23T08:52:16Z

This is commonly used to compute the push-forward of surface areas from reference to current configuration in continuum mechanics (Nansons formula).

using Tensors, BenchmarkTools

F = Tensor{2,3,Float64}((1.1,-0.1,0.1, -0.1,0.9,0.0 ,0.25,0.0,1.2))

# Master version
cof2(F) = det(F)*transpose(inv(F))
@btime cof2($F); # 16.583 ns
@btime gradient(cof2, $F); # 233.892 ns
@btime gradient(cof2, $F, :all); # 236.603 ns

# This PR
@btime cof($F); # 5.890 ns
@btime gradient(cof, $F); # 78.131 ns
@btime gradient(cof, $F, :all); # 92.096 ns

termi-official · 2024-05-23T10:03:22Z

Nightly failure is unrelated to the PR. The behavior for the built-in eigenvalue reconstruction seems to have changed.

fredrikekre · 2024-05-23T11:02:46Z

Can cof not be re-used in inv? Also, can you add a note in the changelog for 1.17.0 (and update the version number in the project file).

fredrikekre · 2024-05-23T11:03:18Z

src/Tensors.jl

 using StaticArrays
 # re-exports from LinearAlgebra
-export ⋅, ×, dot, diagm, tr, det, norm, eigvals, eigvecs, eigen
+export ⋅, ×, dot, diagm, tr, det, cof, norm, eigvals, eigvecs, eigen


I don't think this come from LinearAlgebra so maybe list separately. Is cof a standard name for this operation btw?

Oh, right. In the continuum mechanics books I read this operation is usually abbreviated with cof (Holzapfel, Wriggers). German Wikipedia also uses it.

https://de.wikipedia.org/wiki/Minor_(Lineare_Algebra)#Eigenschaften

But possible that this abbreviation is a German thing. If you want something different then we can also use it.

I don't mind cof, just wondering if it was called that, or something else, in other software. I can't find anything from a quick serach though.

My only worry here is that some standard library might include cofactors in the future. :D

termi-official · 2024-05-23T11:12:22Z

Can cof not be re-used in inv? Also, can you add a note in the changelog for 1.17.0 (and update the version number in the project file).

We could do, but I am not sure about the performance implications. Will add a benchmark.

termi-official · 2024-05-23T11:31:09Z

If I understood you correctly, then you propose something along this line:

@generated function inv_new(t::Tensor{2, dim}) where {dim}
    Tt = get_base(t)
    idx(i,j) = compute_index(Tt, i, j)
    ex = quote
        transpose(cof(t))*dinv
    end
    return quote
        $(Expr(:meta, :inline))
        dinv = 1 / det(t)
        @inbounds return $ex
    end
end

Seems like this is slightly faster on my machine in 3D.

julia> F3 = Tensor{2,3,Float64}((1.1,-0.1,0.1, -0.1,0.9,0.0 ,0.25,0.0,1.2))
3×3 Tensor{2, 3, Float64, 9}:
  1.1  -0.1  0.25
 -0.1   0.9  0.0
  0.1   0.0  1.2

julia> F2 = Tensor{2,2,Float64}((1.1,-0.1, -0.1,0.9))
2×2 Tensor{2, 2, Float64, 4}:
  1.1  -0.1
 -0.1   0.9

julia> @btime Tensors.inv_new($F2)
  6.240 ns (0 allocations: 0 bytes)
2×2 Tensor{2, 2, Float64, 4}:
 0.918367  0.102041
 0.102041  1.12245

julia> @btime inv_new($F3)
  11.331 ns (0 allocations: 0 bytes)
3×3 Tensor{2, 3, Float64, 9}:
  0.936281    0.104031    -0.195059
  0.104031    1.12267     -0.0216732
 -0.0780234  -0.00866927   0.849588

julia> @btime Tensors.inv($F2)
  6.240 ns (0 allocations: 0 bytes)
2×2 Tensor{2, 2, Float64, 4}:
 0.918367  0.102041
 0.102041  1.12245

julia> @btime Tensors.inv($F3)
  11.612 ns (0 allocations: 0 bytes)
3×3 Tensor{2, 3, Float64, 9}:
  0.936281    0.104031    -0.195059
  0.104031    1.12267     -0.0216732
 -0.0780234  -0.00866927   0.849588

Reproducer

using Tensors, BenchmarkTools

F2 = Tensor{2,2,Float64}((1.1,-0.1, -0.1,0.9))
F3 = Tensor{2,3,Float64}((1.1,-0.1,0.1, -0.1,0.9,0.0 ,0.25,0.0,1.2))

@btime Tensors.inv_new($F2)
@btime Tensors.inv_new($F3)
@btime Tensors.inv($F2)
@btime Tensors.inv($F3)

Should I include the change?

KnutAM · 2024-05-23T11:39:33Z

I'm getting

julia> @btime inv_new($F2);
  2.100 ns (0 allocations: 0 bytes)

julia> @btime inv_new($F3);
  6.800 ns (0 allocations: 0 bytes)

julia> @btime Tensors.inv($F2);
  1.900 ns (0 allocations: 0 bytes)

julia> @btime Tensors.inv($F3);
  4.300 ns (0 allocations: 0 bytes)

julia> @btime myinv($F2);
  1.900 ns (0 allocations: 0 bytes)

julia> @btime myinv($F3);
  7.307 ns (0 allocations: 0 bytes)

with

@generated function inv_new(t::Tensor{2, dim}) where {dim}
           Tt = Tensors.get_base(t)
           idx(i,j) = Tensors.compute_index(Tt, i, j)
           ex = quote
               transpose(cof(t))*dinv
           end
           return quote
               $(Expr(:meta, :inline))
               dinv = 1 / det(t)
               @inbounds return $ex
           end
       end
end

myinv(F) = cof(F)'/det(F);

termi-official · 2024-05-23T11:41:59Z

I guess the optimizer has some trouble with the transpose statement here. It is also questionable how reliable the measurements are within this regime.

KnutAM · 2024-05-23T11:51:37Z

Could do

cof(t::SecondOrderTensor) = _cof(t, Val(false))
inv(t::SecondOrderTensor) = _cof(t, Val(true)) / det(t)
@generated function _cof(t::Tensor{2, dim, T}, ::Val{Transpose}) where {dim, T, Transpose}
    Tt = get_base(t)
    idx(i,j) = Transpose ? compute_index(Tt, j, i) : compute_index(Tt, i, j)
    ...

but not sure if that is worth the added complexity compared to just duplicating the code...

KnutAM · 2025-11-24T16:47:09Z

src/math_ops.jl

+@generated function cof(t::Tensor{2, dim, T}) where {dim, T}
+    Tt = get_base(t)
+    idx(i,j) = compute_index(Tt, i, j)
+    if dim == 1
+        ex = :($Tt((T(1.0), )))
+    elseif dim == 2
+        ex = quote
+            v = get_data(t)
+            $Tt((v[$(idx(2,2))], -v[$(idx(1,2))],
+                -v[$(idx(2,1))],  v[$(idx(1,1))]))
+        end
+    else # dim == 3
+        ex = quote
+            v = get_data(t)
+            $Tt(((v[$(idx(2,2))]*v[$(idx(3,3))] - v[$(idx(2,3))]*v[$(idx(3,2))]), #11
+                -(v[$(idx(1,2))]*v[$(idx(3,3))] - v[$(idx(1,3))]*v[$(idx(3,2))]), #21
+                 (v[$(idx(1,2))]*v[$(idx(2,3))] - v[$(idx(1,3))]*v[$(idx(2,2))]), #31
+
+                -(v[$(idx(2,1))]*v[$(idx(3,3))] - v[$(idx(2,3))]*v[$(idx(3,1))]), #12
+                 (v[$(idx(1,1))]*v[$(idx(3,3))] - v[$(idx(1,3))]*v[$(idx(3,1))]), #22
+                -(v[$(idx(1,1))]*v[$(idx(2,3))] - v[$(idx(1,3))]*v[$(idx(2,1))]), #32
+
+                 (v[$(idx(2,1))]*v[$(idx(3,2))] - v[$(idx(2,2))]*v[$(idx(3,1))]), #13
+                -(v[$(idx(1,1))]*v[$(idx(3,2))] - v[$(idx(1,2))]*v[$(idx(3,1))]), #23
+                 (v[$(idx(1,1))]*v[$(idx(2,2))] - v[$(idx(1,2))]*v[$(idx(2,1))])))#33
+        end
+    end
+    return quote
+        $(Expr(:meta, :inline))
+        @inbounds return $ex
+    end
+end


Why is a generated function required? This seems easier (and equivalent for symmetric tensor)? I get similar or slightly better performance with this.

Suggested change

@generated function cof(t::Tensor{2, dim, T}) where {dim, T}

Tt = get_base(t)

idx(i,j) = compute_index(Tt, i, j)

if dim == 1

ex = :($Tt((T(1.0), )))

elseif dim == 2

ex = quote

v = get_data(t)

$Tt((v[$(idx(2,2))], -v[$(idx(1,2))],

-v[$(idx(2,1))], v[$(idx(1,1))]))

end

else # dim == 3

ex = quote

v = get_data(t)

$Tt(((v[$(idx(2,2))]*v[$(idx(3,3))] - v[$(idx(2,3))]*v[$(idx(3,2))]), #11

-(v[$(idx(1,2))]*v[$(idx(3,3))] - v[$(idx(1,3))]*v[$(idx(3,2))]), #21

(v[$(idx(1,2))]*v[$(idx(2,3))] - v[$(idx(1,3))]*v[$(idx(2,2))]), #31

-(v[$(idx(2,1))]*v[$(idx(3,3))] - v[$(idx(2,3))]*v[$(idx(3,1))]), #12

(v[$(idx(1,1))]*v[$(idx(3,3))] - v[$(idx(1,3))]*v[$(idx(3,1))]), #22

-(v[$(idx(1,1))]*v[$(idx(2,3))] - v[$(idx(1,3))]*v[$(idx(2,1))]), #32

(v[$(idx(2,1))]*v[$(idx(3,2))] - v[$(idx(2,2))]*v[$(idx(3,1))]), #13

-(v[$(idx(1,1))]*v[$(idx(3,2))] - v[$(idx(1,2))]*v[$(idx(3,1))]), #23

(v[$(idx(1,1))]*v[$(idx(2,2))] - v[$(idx(1,2))]*v[$(idx(2,1))])))#33

end

end

return quote

$(Expr(:meta, :inline))

@inbounds return $ex

end

end

@inline cof(t::AbstractTensor{2, 1}) = one(t)

@inline function cof(t::Tensor{2, 2})

t11, t21, t12, t22 = get_data(t)

return Tensor{2,2}((t22, -t21, -t12, t11))

end

@inline function cof(t::Tensor{2, 3})

t11, t21, t31, t12, t22, t32, t13, t23, t33 = get_data(t)

c11 = t22 * t33 - t23 * t32

c21 = t13 * t32 - t12 * t33

c31 = t12 * t23 - t13 * t22

c12 = t23 * t31 - t21 * t33

c22 = t11 * t33 - t13 * t31

c32 = t13 * t21 - t11 * t23

c13 = t21 * t32 - t22 * t31

c23 = t12 * t31 - t11 * t23

c33 = t11 * t22 - t12 * t21

return Tensor{2, 3}((c11, c21, c31, c12, c22, c32, c13, c23, c33))

end

Tensor{2,3} benchmark tested, hopefully no typos when transferring here :)

I used generate, because the other variants above were generated. I do not have a strong opinion here about either.

Was there any reason why, @KristofferC or @fredrikekre?
(IMO we should avoid @generated when possible)

termi-official added 2 commits May 23, 2024 10:50

Add direct computation of cofactor matrix.

8dd6c9f

Add to docs.

4b85c2e

fredrikekre reviewed May 23, 2024

View reviewed changes

termi-official added 2 commits November 24, 2025 14:07

Merge branch 'master' into do/cofactor-matrix

0aaeb98

Update to new main and add changelog

68db706

KnutAM reviewed Nov 24, 2025

View reviewed changes

Cofactor Matrix #218

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Cofactor Matrix #218

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