Rearrange code to avoid redefining functions.

src/manifolds/MultinomialSymmetric.jl

@@ -1,7 +1,7 @@
 @doc raw"""
     MultinomialSymmetric{T} <: AbstractMultinomialDoublyStochastic{N}
 
-The multinomial symmetric matrices manifold consists of all symmetric $n×n$ matrices with
+The multinomial symmetric matrices manifold consists of all symmetric ``n×n`` matrices with
 positive entries such that each column sums to one, i.e.
 
 ````math
@@ -13,7 +13,7 @@ positive entries such that each column sums to one, i.e.
 \end{aligned}
 ````
 
-where $\mathbf{1}_n$ is the vector of length $n$ containing ones.
+where ``\mathbf{1}_n`` is the vector of length ``n`` containing ones.
 
 It is modeled as [`IsIsometricEmbeddedManifold`](https://juliamanifolds.github.io/ManifoldsBase.jl/stable/decorator.html#ManifoldsBase.IsIsometricEmbeddedManifold).
 via the [`AbstractMultinomialDoublyStochastic`](@ref) type, since it shares a few functions
@@ -29,15 +29,15 @@ X\mathbf{1}_n = \mathbf{0}_n
 \bigr\},
 ````
 
-where $\mathbf{0}_n$ is the vector of length $n$ containing zeros.
+where ``\mathbf{0}_n`` is the vector of length ``n`` containing zeros.
 
 More details can be found in Section IV [DouikHassibi:2019](@cite).
 
 # Constructor
 
     MultinomialSymmetric(n)
 
-Generate the manifold of matrices $\mathbb R^{n×n}$ that are doubly stochastic and symmetric.
+Generate the manifold of matrices ``\mathbb R^{n×n}`` that are doubly stochastic and symmetric.
 """
 struct MultinomialSymmetric{T} <: AbstractMultinomialDoublyStochastic
     size::T
@@ -110,12 +110,12 @@ The formula reads
 ````math
     \operatorname{proj}_p(Y) = Y - (α\mathbf{1}_n^{\mathrm{T}} + \mathbf{1}_n α^{\mathrm{T}}) ⊙ p,
 ````
-where $⊙$ denotes the Hadamard or elementwise product and $\mathbb{1}_n$ is the vector of length $n$ containing ones.
-The two vector $α ∈ ℝ^{n×n}$ is given by solving
+where ``⊙`` denotes the Hadamard or elementwise product and ``\mathbb{1}_n`` is the vector of length ``n`` containing ones.
+The two vector ``α ∈ ℝ^{n×n}`` is given by solving
 ````math
     (I_n+p)α =  Y\mathbf{1},
 ````
-where $I_n$ is teh $n×n$ unit matrix and $\mathbf{1}_n$ is the vector of length $n$ containing ones.
+where ``I_n`` is teh ``n×n`` unit matrix and ``\mathbf{1}_n`` is the vector of length ``n`` containing ones.
 
 """
 project(::MultinomialSymmetric, ::Any, ::Any)
@@ -133,8 +133,8 @@ end
 @doc raw"""
     retract(M::MultinomialSymmetric, p, X, ::ProjectionRetraction)
 
-compute a projection based retraction by projecting $p\odot\exp(X⨸p)$ back onto the manifold,
-where $⊙,⨸$ are elementwise multiplication and division, respectively. Similarly, $\exp$
+compute a projection based retraction by projecting ``p⊙\exp(X⨸p)`` back onto the manifold,
+where ``⊙,⨸`` are elementwise multiplication and division, respectively. Similarly, ``\exp``
 refers to the elementwise exponentiation.
 """
 retract(::MultinomialSymmetric, ::Any, ::Any, ::ProjectionRetraction)

src/manifolds/Oblique.jl

@@ -1,10 +1,10 @@
 @doc raw"""
     Oblique{T,𝔽,S} <: AbstractPowerManifold{𝔽}
 
-The oblique manifold $\mathcal{OB}(n,m)$ is the set of 𝔽-valued matrices with unit norm
+The oblique manifold ``\mathcal{OB}(n,m)`` is the set of 𝔽-valued matrices with unit norm
 column endowed with the metric from the embedding. This yields exactly the same metric as
 considering the product metric of the unit norm vectors, i.e. [`PowerManifold`](https://juliamanifolds.github.io/ManifoldsBase.jl/stable/manifolds.html#ManifoldsBase.PowerManifold) of the
-$(n-1)$-dimensional [`Sphere`](@ref).
+``(n-1)``-dimensional [`Sphere`](@ref).
 
 The [`Sphere`](@ref) is stored internally within `M.manifold`, such that all functions of
 [`AbstractPowerManifold`](https://juliamanifolds.github.io/ManifoldsBase.jl/stable/manifolds.html#ManifoldsBase.AbstractPowerManifold)  can be used directly.
@@ -13,7 +13,7 @@ The [`Sphere`](@ref) is stored internally within `M.manifold`, such that all fun
 
     Oblique(n::Int, m::Int, field::AbstractNumbers=ℝ; parameter::Symbol=:type)
 
-Generate the manifold of matrices $\mathbb R^{n×m}$ such that the $m$ columns are unit
+Generate the manifold of matrices ``\mathbb R^{n×m}`` such that the ``m`` columns are unit
 vectors, i.e. from the [`Sphere`](@ref)`(n-1)`.
 """
 struct Oblique{T,𝔽,S} <: AbstractPowerManifold{𝔽,Sphere{S,𝔽},ArrayPowerRepresentation}
@@ -39,7 +39,7 @@ end
     check_point(M::Oblique, p)
 
 Checks whether `p` is a valid point on the [`Oblique`](@ref)`{m,n}` `M`, i.e. is a matrix
-of `m` unit columns from $\mathbb R^{n}$, i.e. each column is a point from
+of `m` unit columns from ``\mathbb R^{n}``, i.e. each column is a point from
 [`Sphere`](@ref)`(n-1)`.
 """
 check_point(::Oblique, ::Any)

src/manifolds/SphereSymmetricMatrices.jl

@@ -1,13 +1,13 @@
 @doc raw"""
     SphereSymmetricMatrices{T,𝔽} <: AbstractEmbeddedManifold{ℝ,TransparentIsometricEmbedding}
 
-The [`AbstractManifold`](https://juliamanifolds.github.io/ManifoldsBase.jl/stable/types.html#ManifoldsBase.AbstractManifold)  consisting of the $n×n$ symmetric matrices
+The [`AbstractManifold`](https://juliamanifolds.github.io/ManifoldsBase.jl/stable/types.html#ManifoldsBase.AbstractManifold)  consisting of the ``n×n`` symmetric matrices
 of unit Frobenius norm, i.e.
 ````math
 \mathcal{S}_{\text{sym}} :=\bigl\{p  ∈ 𝔽^{n×n}\ \big|\ p^{\mathrm{H}} = p, \lVert p \rVert = 1 \bigr\},
 ````
-where $⋅^{\mathrm{H}}$ denotes the Hermitian, i.e. complex conjugate transpose,
-and the field $𝔽 ∈ \{ ℝ, ℂ\}$.
+where ``⋅^{\mathrm{H}}`` denotes the Hermitian, i.e. complex conjugate transpose,
+and the field ``𝔽 ∈ \{ ℝ, ℂ\}``.
 
 # Constructor
     SphereSymmetricMatrices(n[, field=ℝ])
@@ -119,7 +119,7 @@ Projects `p` from the embedding onto the [`SphereSymmetricMatrices`](@ref) `M`,
 ````math
 \operatorname{proj}_{\mathcal{S}_{\text{sym}}}(p) = \frac{1}{2} \bigl( p + p^{\mathrm{H}} \bigr),
 ````
-where $⋅^{\mathrm{H}}$ denotes the Hermitian, i.e. complex conjugate transposed.
+where ``⋅^{\mathrm{H}}`` denotes the Hermitian, i.e. complex conjugate transposed.
 """
 project(::SphereSymmetricMatrices, ::Any)
 
@@ -135,7 +135,7 @@ Project the matrix `X` onto the tangent space at `p` on the [`SphereSymmetricMat
 ````math
 \operatorname{proj}_p(X) = \frac{X + X^{\mathrm{H}}}{2} - ⟨p, \frac{X + X^{\mathrm{H}}}{2}⟩p,
 ````
-where $⋅^{\mathrm{H}}$ denotes the Hermitian, i.e. complex conjugate transposed.
+where ``⋅^{\mathrm{H}}`` denotes the Hermitian, i.e. complex conjugate transposed.
 """
 project(::SphereSymmetricMatrices, ::Any, ::Any)
 

src/manifolds/StiefelEuclideanMetric.jl

@@ -17,9 +17,9 @@ emanating from `p` in tangent direction `X`.
 \begin{pmatrix}  \exp( -p^{\mathrm{H}}X) \\ 0_n\end{pmatrix},
 ````
 
-where $\operatorname{Exp}$ denotes matrix exponential,
-$⋅^{\mathrm{H}}$ denotes the complex conjugate transpose or Hermitian, and $I_k$ and
-$0_k$ are the identity matrix and the zero matrix of dimension $k×k$, respectively.
+where ``\operatorname{Exp}`` denotes matrix exponential,
+``⋅^{\mathrm{H}}`` denotes the complex conjugate transpose or Hermitian, and ``I_k`` and
+``0_k`` are the identity matrix and the zero matrix of dimension ``k×k``, respectively.
 """
 exp(::Stiefel, ::Any...)
 
@@ -38,25 +38,25 @@ end
 @doc raw"""
     get_basis(M::Stiefel{<:Any,ℝ}, p, B::DefaultOrthonormalBasis)
 
-Create the default basis using the parametrization for any $X ∈ T_p\mathcal M$.
-Set $p_\bot \in ℝ^{n×(n-k)}$ the matrix such that the $n×n$ matrix of the common
-columns $[p\ p_\bot]$ is an ONB.
-For any skew symmetric matrix $a ∈ ℝ^{k×k}$ and any $b ∈ ℝ^{(n-k)×k}$ the matrix
+Create the default basis using the parametrization for any ``X ∈ T_p\mathcal M``.
+Set ``p_\bot \in ℝ^{n×(n-k)}`` the matrix such that the ``n×n`` matrix of the common
+columns ``[p\ p_\bot]`` is an ONB.
+For any skew symmetric matrix ``a ∈ ℝ^{k×k}`` and any ``b ∈ ℝ^{(n-k)×k}`` the matrix
 
 ````math
 X = pa + p_\bot b ∈ T_p\mathcal M
 ````
 
-and we can use the $\frac{1}{2}k(k-1) + (n-k)k = nk-\frac{1}{2}k(k+1)$ entries
-of $a$ and $b$ to specify a basis for the tangent space.
+and we can use the ``\frac{1}{2}k(k-1) + (n-k)k = nk-\frac{1}{2}k(k+1)`` entries
+of ``a`` and ``b`` to specify a basis for the tangent space.
 using unit vectors for constructing both
-the upper matrix of $a$ to build a skew symmetric matrix and the matrix b, the default
+the upper matrix of ``a`` to build a skew symmetric matrix and the matrix b, the default
 basis is constructed.
 
-Since $[p\ p_\bot]$ is an automorphism on $ℝ^{n×p}$ the elements of $a$ and $b$ are
+Since ``[p\ p_⊥]`` is an automorphism on ``ℝ^{n×p}`` the elements of ``a`` and ``b`` are
 orthonormal coordinates for the tangent space. To be precise exactly one element in the upper
-trangular entries of $a$ is set to $1$ its symmetric entry to $-1$ and we normalize with
-the factor $\frac{1}{\sqrt{2}}$ and for $b$ one can just use unit vectors reshaped to a matrix
+trangular entries of ``a`` is set to ``1`` its symmetric entry to ``-1`` and we normalize with
+the factor ``\frac{1}{\sqrt{2}}`` and for ``b`` one can just use unit vectors reshaped to a matrix
 to obtain orthonormal set of parameters.
 """
 get_basis(M::Stiefel{<:Any,ℝ}, p, B::DefaultOrthonormalBasis{ℝ,TangentSpaceType})
@@ -128,7 +128,7 @@ end
     project(M::Stiefel,p)
 
 Projects `p` from the embedding onto the [`Stiefel`](@ref) `M`, i.e. compute `q`
-as the polar decomposition of $p$ such that ``q^{\mathrm{H}}q`` is the identity,
+as the polar decomposition of ``p`` such that ``q^{\mathrm{H}}q`` is the identity,
 where ``⋅^{\mathrm{H}}`` denotes the hermitian, i.e. complex conjugate transposed.
 """
 project(::Stiefel, ::Any, ::Any)
@@ -149,8 +149,8 @@ The formula reads
 \operatorname{proj}_{T_p\mathcal M}(X) = X - p \operatorname{Sym}(p^{\mathrm{H}}X),
 ````
 
-where $\operatorname{Sym}(q)$ is the symmetrization of $q$, e.g. by
-$\operatorname{Sym}(q) = \frac{q^{\mathrm{H}}+q}{2}$.
+where ``\operatorname{Sym}(q)`` is the symmetrization of ``q``, e.g. by
+``\operatorname{Sym}(q) = \frac{q^{\mathrm{H}}+q}{2}``.
 """
 project(::Stiefel, ::Any...)
 

src/manifolds/SymmetricPositiveDefinite.jl

@@ -24,7 +24,7 @@ i.e. the set of symmetric matrices,
 
     SymmetricPositiveDefinite(n; parameter::Symbol=:type)
 
-generates the manifold $\mathcal P(n) \subset ℝ^{n×n}$
+generates the manifold ``\mathcal P(n) \subset ℝ^{n×n}``
 """
 struct SymmetricPositiveDefinite{T} <: AbstractDecoratorManifold{ℝ}
     size::T
@@ -243,7 +243,7 @@ end
 
 Return the injectivity radius of the [`SymmetricPositiveDefinite`](@ref).
 Since `M` is a Hadamard manifold with respect to the [`AffineInvariantMetric`](@ref) and the
-[`LogCholeskyMetric`](@ref), the injectivity radius is globally $∞$.
+[`LogCholeskyMetric`](@ref), the injectivity radius is globally ``∞``.
 """
 injectivity_radius(::SymmetricPositiveDefinite) = Inf
 injectivity_radius(::SymmetricPositiveDefinite, p) = Inf
@@ -265,7 +265,7 @@ is_flat(M::SymmetricPositiveDefinite) = false
     manifold_dimension(M::SymmetricPositiveDefinite)
 
 returns the dimension of
-[`SymmetricPositiveDefinite`](@ref) `M`$=\mathcal P(n), n ∈ ℕ$, i.e.
+[`SymmetricPositiveDefinite`](@ref) `M` ``=\mathcal P(n), n ∈ ℕ``, i.e.
 ````math
 \dim \mathcal P(n) = \frac{n(n+1)}{2}.
 ````
@@ -466,8 +466,8 @@ end
     representation_size(M::SymmetricPositiveDefinite)
 
 Return the size of an array representing an element on the
-[`SymmetricPositiveDefinite`](@ref) manifold `M`, i.e. $n×n$, the size of such a
-symmetric positive definite matrix on $\mathcal M = \mathcal P(n)$.
+[`SymmetricPositiveDefinite`](@ref) manifold `M`, i.e. ``n×n``, the size of such a
+symmetric positive definite matrix on ``\mathcal M = \mathcal P(n)``.
 """
 function representation_size(M::SymmetricPositiveDefinite)
     N = get_parameter(M.size)[1]

src/manifolds/SymmetricPositiveSemidefiniteFixedRank.jl

@@ -1,27 +1,27 @@
 @doc raw"""
     SymmetricPositiveSemidefiniteFixedRank{T,𝔽} <: AbstractDecoratorManifold{𝔽}
 
-The [`AbstractManifold`](https://juliamanifolds.github.io/ManifoldsBase.jl/stable/types.html#ManifoldsBase.AbstractManifold)  $ \operatorname{SPS}_k(n)$ consisting of the real- or complex-valued
-symmetric positive semidefinite matrices of size $n×n$ and rank $k$, i.e. the set
+The [`AbstractManifold`](https://juliamanifolds.github.io/ManifoldsBase.jl/stable/types.html#ManifoldsBase.AbstractManifold)  `` \operatorname{SPS}_k(n)`` consisting of the real- or complex-valued
+symmetric positive semidefinite matrices of size ``n×n`` and rank ``k``, i.e. the set
 
 ````math
 \operatorname{SPS}_k(n) = \bigl\{
 p  ∈ 𝔽^{n×n}\ \big|\ p^{\mathrm{H}} = p,
 apa^{\mathrm{H}} \geq 0 \text{ for all } a ∈ 𝔽
 \text{ and } \operatorname{rank}(p) = k\bigr\},
 ````
-where $⋅^{\mathrm{H}}$ denotes the Hermitian, i.e. complex conjugate transpose,
-and the field $𝔽 ∈ \{ ℝ, ℂ\}$.
-We sometimes $\operatorname{SPS}_{k,𝔽}(n)$, when distinguishing the real- and complex-valued
+where ``⋅^{\mathrm{H}}`` denotes the Hermitian, i.e. complex conjugate transpose,
+and the field ``𝔽 ∈ \{ ℝ, ℂ\}``.
+We sometimes ``\operatorname{SPS}_{k,𝔽}(n)``, when distinguishing the real- and complex-valued
 manifold is important.
 
-An element is represented by $q ∈ 𝔽^{n×k}$ from the factorization $p = qq^{\mathrm{H}}$.
-Note that since for any unitary (orthogonal) $A ∈ 𝔽^{n×n}$ we have
-$(Aq)(Aq)^{\mathrm{H}} = qq^{\mathrm{H}} = p$, the representation is not unique, or in
-other words, the manifold is a quotient manifold of $𝔽^{n×k}$.
+An element is represented by ``q ∈ 𝔽^{n×k}`` from the factorization ``p = qq^{\mathrm{H}}``.
+Note that since for any unitary (orthogonal) ``A ∈ 𝔽^{n×n}`` we have
+``(Aq)(Aq)^{\mathrm{H}} = qq^{\mathrm{H}} = p``, the representation is not unique, or in
+other words, the manifold is a quotient manifold of ``𝔽^{n×k}``.
 
-The tangent space at $p$, $T_p\operatorname{SPS}_k(n)$, is also represented
-by matrices $Y ∈ 𝔽^{n×k}$ and reads as
+The tangent space at ``p``, ``T_p\operatorname{SPS}_k(n)``, is also represented
+by matrices ``Y ∈ 𝔽^{n×k}`` and reads as
 
 ````math
 T_p\operatorname{SPS}_k(n) = \bigl\{
@@ -37,7 +37,7 @@ The metric was used in [JourneeBachAbsilSepulchre:2010](@cite)[MassartAbsil:2020
 
     SymmetricPositiveSemidefiniteFixedRank(n::Int, k::Int, field::AbstractNumbers=ℝ; parameter::Symbol=:type)
 
-Generate the manifold of $n×n$ symmetric positive semidefinite matrices of rank $k$
+Generate the manifold of ``n×n`` symmetric positive semidefinite matrices of rank ``k``
 over the `field` of real numbers `ℝ` or complex numbers `ℂ`.
 """
 struct SymmetricPositiveSemidefiniteFixedRank{T,𝔽} <: AbstractDecoratorManifold{𝔽}
@@ -107,8 +107,8 @@ end
     distance(M::SymmetricPositiveSemidefiniteFixedRank, p, q)
 
 Compute the distance between two points `p`, `q` on the
-[`SymmetricPositiveSemidefiniteFixedRank`](@ref), which is the Frobenius norm of $Y$ which
-minimizes $\lVert p - qY\rVert$ with respect to $Y$.
+[`SymmetricPositiveSemidefiniteFixedRank`](@ref), which is the Frobenius norm of ``Y`` which
+minimizes ``\lVert p - qY\rVert`` with respect to ``Y``.
 """
 distance(M::SymmetricPositiveSemidefiniteFixedRank, p, q) = norm(M, p, log(M, p, q))
 
@@ -129,8 +129,8 @@ which just reads
     ````math
         q_2 = \exp_p(\log_pq)
     ````
-    might yield a matrix $q_2\neq q$, but they represent the same point on the quotient
-    manifold, i.e. $d_{\operatorname{SPS}_k(n)}(q_2,q) = 0$.
+    might yield a matrix ``q_2\neq q``, but they represent the same point on the quotient
+    manifold, i.e. ``d_{\operatorname{SPS}_k(n)}(q_2,q) = 0``.
 """
 exp(::SymmetricPositiveSemidefiniteFixedRank, ::Any, ::Any)
 
@@ -144,7 +144,7 @@ end
 
 test, whether two points `p`, `q` are (approximately) nearly the same.
 Since this is a quotient manifold in the embedding, the test is performed by checking
-their distance, if they are not the same, i.e. that $d_{\mathcal M}(p,q) \approx 0$, where
+their distance, if they are not the same, i.e. that ``d_{\mathcal M}(p,q) \approx 0``, where
 the comparison is performed with the classical `isapprox`.
 The `kwargs...` are passed on to this accordingly.
 """
@@ -170,7 +170,7 @@ is_flat(M::SymmetricPositiveSemidefiniteFixedRank) = false
     log(M::SymmetricPositiveSemidefiniteFixedRank, q, p)
 
 Compute the logarithmic map on the [`SymmetricPositiveSemidefiniteFixedRank`](@ref) manifold
-by minimizing $\lVert p - qY\rVert$ with respect to $Y$.
+by minimizing ``\lVert p - qY\rVert`` with respect to ``Y``.
 
 !!! note
 
@@ -180,8 +180,8 @@ by minimizing $\lVert p - qY\rVert$ with respect to $Y$.
     ````math
         q_2 = \exp_p(\log_pq)
     ````
-    might yield a matrix $q_2\neq q$, but they represent the same point on the quotient
-    manifold, i.e. $d_{\operatorname{SPS}_k(n)}(q_2,q) = 0$.
+    might yield a matrix ``q_2≠q``, but they represent the same point on the quotient
+    manifold, i.e. ``d_{\operatorname{SPS}_k(n)}(q_2,q) = 0``.
 """
 log(::SymmetricPositiveSemidefiniteFixedRank, q, p)
 
@@ -203,7 +203,7 @@ Return the dimension of the [`SymmetricPositiveSemidefiniteFixedRank`](@ref) mat
 \end{aligned}
 ````
 
-where the last $k^2$ is due to the zero imaginary part for Hermitian matrices diagonal
+where the last ``k^2`` is due to the zero imaginary part for Hermitian matrices diagonal
 """
 manifold_dimension(::SymmetricPositiveSemidefiniteFixedRank)
 

test/ambiguities.jl

@@ -1,3 +1,14 @@
+"""
+    has_type_in_signature(sig, T)
+    Test whether the signature `sig` has an argument of type `T` as one of its paramaters
+"""
+function has_type_in_signature(sig, T::Type)
+    return any(map(Base.unwrap_unionall(sig.sig).parameters) do x
+        xw = Base.rewrap_unionall(x, sig.sig)
+        return (xw isa Type ? xw : xw.T) <: T
+    end)
+end
+
 @testset "Ambiguities" begin
     if VERSION.prerelease == () && !Sys.iswindows() && VERSION < v"1.10.0"
         mbs = Test.detect_ambiguities(ManifoldsBase)

test/approx_inverse_retraction.jl

@@ -1,7 +1,7 @@
 using NLsolve
 using LinearAlgebra
 
-include("utils.jl")
+include("header.jl")
 
 Random.seed!(10)
 

test/differentiation.jl

@@ -1,6 +1,6 @@
 # A bit of duplication of tests in ManifoldDiff.jl but it ensures that it works here too.
 
-include("utils.jl")
+include("header.jl")
 using Manifolds:
     default_differential_backend,
     _derivative,

test/groups/circle_group.jl

@@ -1,4 +1,4 @@
-include("../utils.jl")
+include("../header.jl")
 include("group_utils.jl")
 
 using Manifolds:

test/groups/connections.jl

@@ -1,4 +1,4 @@
-include("../utils.jl")
+include("../header.jl")
 include("group_utils.jl")
 
 using Manifolds: connection

test/groups/general_linear.jl

@@ -1,4 +1,4 @@
-include("../utils.jl")
+include("../header.jl")
 include("group_utils.jl")
 using NLsolve
 

test/groups/general_unitary_groups.jl

@@ -1,4 +1,4 @@
-include("../utils.jl")
+include("../header.jl")
 include("group_utils.jl")
 
 @testset "General Unitary Groups" begin