Fix a few typos

Author: kellertuer

src/documentation_glossary.jl

@@ -120,7 +120,7 @@ define!(:LaTeX, :transp, raw"\mathrm{T}")
 define!(:LaTeX, :vdots, raw"\vdots")
 define!(:LaTeX, :vert, raw"\vert")
 define!(:LaTeX, :widehat, (letter) -> raw"\widehat{" * "$letter" * "}")
-define!(:LaTeX, :widetilde, raw"\widetilde")
+define!(:LaTeX, :widetilde, (letter) -> raw"\widetilde{" * "$letter" * "}")
 _tex(args...; kwargs...) = glossary(:LaTeX, args...; kwargs...)
 #
 # ---

src/plans/quasi_newton_plan.jl

@@ -136,16 +136,16 @@ struct InverseDFP <: AbstractQuasiNewtonUpdateRule end
 
 indicates in [`AbstractQuasiNewtonDirectionUpdate`](@ref) that the Riemannian SR1 update is used in the Riemannian quasi-Newton method.
 
-Denote by ``$(_tex(:widetilde, "H"))_k^$(_tex(:rm, "SR1"))`` the operator concatenated with a vector transport and its inverse before and after to act on ``x_{k+1} = R_{x_k}(α_k η_k)``.
+Denote by ``$(_tex(:widetilde, "H"))_k^{$(_tex(:rm, "SR1"))}`` the operator concatenated with a vector transport and its inverse before and after to act on ``x_{k+1} = R_{x_k}(α_k η_k)``.
 Then the update formula reads
 
 ```math
-H^$(_tex(:rm, "SR1"))_{k+1} = $(_tex(:widetilde, "H"))^$(_tex(:rm, "SR1"))_k
+H^{$(_tex(:rm, "SR1"))}_{k+1} = $(_tex(:widetilde, "H"))^{$(_tex(:rm, "SR1"))}_k
 + $(
     _tex(
         :frac,
-        "(y_k - $(_tex(:widetilde, "H"))^$(_tex(:rm, "SR1"))_k s_k) (y_k - $(_tex(:widetilde, "H"))^$(_tex(:rm, "SR1"))_k s_k)^{$(_tex(:rm, "T"))}",
-        "(y_k - $(_tex(:widetilde, "H"))^$(_tex(:rm, "SR1"))_k s_k)^{$(_tex(:rm, "T"))} s_k",
+        "(y_k - $(_tex(:widetilde, "H"))^{$(_tex(:rm, "SR1"))}_k s_k) (y_k - $(_tex(:widetilde, "H"))^{$(_tex(:rm, "SR1"))}_k s_k)^{$(_tex(:rm, "T"))}",
+        "(y_k - $(_tex(:widetilde, "H"))^{$(_tex(:rm, "SR1"))}_k s_k)^{$(_tex(:rm, "T"))} s_k",
     )
 )
 ```
@@ -160,7 +160,7 @@ $(_tex(:grad))f(x_{k+1}) - T^{S}_{x_k, α_k η_k}($(_tex(:grad))f(x_k)) ∈ T_{x
 respectively.
 
 This method can be stabilized by only performing the update if denominator is larger than
-``r$(_tex(:norm, "s_k"; index = "x_{k+1}"))$(_tex(:norm, "y_k - $(_tex(:widetilde, "H"))^$(_tex(:rm, "SR1"))_k s_k"; index = "x_{k+1}"))``
+``r$(_tex(:norm, "s_k"; index = "x_{k+1}"))$(_tex(:norm, "y_k - $(_tex(:widetilde, "H"))^{$(_tex(:rm, "SR1"))}_k s_k"; index = "x_{k+1}"))``
 for some ``r>0``. For more details, see Section 6.2 in [NocedalWright:2006](@cite).
 
 # Constructor
@@ -178,16 +178,18 @@ end
 
 indicates in [`AbstractQuasiNewtonDirectionUpdate`](@ref) that the inverse Riemannian SR1 update is used in the Riemannian quasi-Newton method.
 
-Denote by ``$(_tex(:widetilde, "B"))_k^$(_tex(:rm, "SR1"))`` the operator concatenated with a vector transport and its inverse before and after to act on ``x_{k+1} = R_{x_k}(α_k η_k)``.
+Denote by ``$(_tex(:widetilde, "B"))_k^{$(_tex(:rm, "SR1"))}`` the operator concatenated with a vector transport and its inverse before and after to act on ``x_{k+1} = R_{x_k}(α_k η_k)``.
 Then the update formula reads
 
 ```math
-B^$(_tex(:rm, "SR1"))_{k+1} = $(_tex(:widetilde, "B"))^$(_tex(:rm, "SR1"))_k
-+ \frac{
-  (s_k - $(_tex(:widetilde, "B"))^$(_tex(:rm, "SR1"))_k y_k) (s_k - $(_tex(:widetilde, "B"))^$(_tex(:rm, "SR1"))_k y_k)^{$(_tex(:rm, "T"))}
-}{
-  (s_k - $(_tex(:widetilde, "B"))^$(_tex(:rm, "SR1"))_k y_k)^{$(_tex(:rm, "T"))} y_k
-}
+B^{$(_tex(:rm, "SR1"))}_{k+1} = $(_tex(:widetilde, "B"))^{$(_tex(:rm, "SR1"))}_k
++ $(
+    _tex(
+        :frac,
+        "(s_k - $(_tex(:widetilde, "B"))^{$(_tex(:rm, "SR1"))}_k y_k) (s_k - $(_tex(:widetilde, "B"))^{$(_tex(:rm, "SR1"))}_k y_k)^{$(_tex(:rm, "T"))}",
+        "(s_k - $(_tex(:widetilde, "B"))^{$(_tex(:rm, "SR1"))}_k y_k)^{$(_tex(:rm, "T"))} y_k"
+    )
+)
 ```
 
 where ``s_k`` and ``y_k`` are the coordinate vectors with respect to the current basis (from [`QuasiNewtonState`](@ref)) of
@@ -200,7 +202,7 @@ $(_tex(:grad))f(x_{k+1}) - T^{S}_{x_k, α_k η_k}($(_tex(:grad))f(x_k)) ∈ T_{x
 respectively.
 
 This method can be stabilized by only performing the update if denominator is larger than
-``r$(_tex(:norm, "y_k"; index = "x_{k+1}"))$(_tex(:norm, "s_k - $(_tex(:widetilde, "H"))^$(_tex(:rm, "SR1"))_k y_k"; index = "x_{k+1}"))``
+``r$(_tex(:norm, "y_k"; index = "x_{k+1}"))$(_tex(:norm, "s_k - $(_tex(:widetilde, "H"))^{$(_tex(:rm, "SR1"))}_k y_k"; index = "x_{k+1}"))``
 for some ``r>0``. For more details, see Section 6.2 in [NocedalWright:2006](@cite).
 
 # Constructor
@@ -222,14 +224,20 @@ Denote by ``$(_tex(:widetilde, "H"))_k^$(_tex(:rm, "Br"))`` the operator concate
 Then the update formula reads
 
 ```math
-H^$(_tex(:rm, "Br"))_{k+1} = $(_tex(:widetilde, "H"))^$(_tex(:rm, "Br"))_k
-  - $(_tex(:frac, "$(_tex(:widetilde, "H"))^$(_tex(:rm, "Br"))_k s_k s^{$(_tex(:rm, "T"))}_k $(_tex(:widetilde, "H"))^$(_tex(:rm, "Br"))_k}{s^{$(_tex(:rm, "T"))}_k $(_tex(:widetilde, "H"))^$(_tex(:rm, "Br"))_k s_k} + \frac{y_k y^{$(_tex(:rm, "T"))}_k", "s^{$(_tex(:rm, "T"))}_k y_k"))
-  + φ_k s^{$(_tex(:rm, "T"))}_k $(_tex(:widetilde, "H"))^$(_tex(:rm, "Br"))_k s_k
-  $(_tex(:Bigl))(
-        $(_tex(:frac, "y_k", "s^{$(_tex(:rm, "T"))}_k y_k")) - $(_tex(:frac, "$(_tex(:widetilde, "H"))^$(_tex(:rm, "Br"))_k s_k", "s^{$(_tex(:rm, "T"))}_k $(_tex(:widetilde, "H"))^$(_tex(:rm, "Br"))_k s_k"))
+H^{$(_tex(:rm, "Br"))}_{k+1}
+=   $(_tex(:widetilde, "H"))^{$(_tex(:rm, "Br"))}_k - $(
+    _tex(
+        :frac,
+        "$(_tex(:widetilde, "H"))^{$(_tex(:rm, "Br"))}_k s_k s^{$(_tex(:rm, "T"))}_k $(_tex(:widetilde, "H"))^{$(_tex(:rm, "Br"))}_k",
+        "s^{$(_tex(:rm, "T"))}_k $(_tex(:widetilde, "H"))^{$(_tex(:rm, "Br"))}_k s_k"
+    )
+) + $(_tex(:frac, "y_k y^{$(_tex(:rm, "T"))}_k", "s^{$(_tex(:rm, "T"))}_k y_k"))
+    + φ_k s^{$(_tex(:rm, "T"))}_k $(_tex(:widetilde, "H"))^{$(_tex(:rm, "Br"))}_k s_k
+    $(_tex(:Bigl))(
+        $(_tex(:frac, "y_k", "s^{$(_tex(:rm, "T"))}_k y_k")) - $(_tex(:frac, "$(_tex(:widetilde, "H"))^{$(_tex(:rm, "Br"))}_k s_k", "s^{$(_tex(:rm, "T"))}_k $(_tex(:widetilde, "H"))^{$(_tex(:rm, "Br"))}_k s_k"))
   $(_tex(:Bigr)))
   $(_tex(:Bigl))(
-        $(_tex(:frac, "y_k", "s^{$(_tex(:rm, "T"))}_k y_k")) - $(_tex(:frac, "$(_tex(:widetilde, "H"))^$(_tex(:rm, "Br"))_k s_k", "s^{$(_tex(:rm, "T"))}_k $(_tex(:widetilde, "H"))^$(_tex(:rm, "Br"))_k s_k"))
+        $(_tex(:frac, "y_k", "s^{$(_tex(:rm, "T"))}_k y_k")) - $(_tex(:frac, "$(_tex(:widetilde, "H"))^{$(_tex(:rm, "Br"))}_k s_k", "s^{$(_tex(:rm, "T"))}_k $(_tex(:widetilde, "H"))^{$(_tex(:rm, "Br"))}_k s_k"))
   $(_tex(:Bigr)))^{$(_tex(:rm, "T"))}
 ```
 
@@ -263,24 +271,24 @@ and its inverse before and after to act on ``x_{k+1} = R_{x_k}(α_k η_k)``.
 Then the update formula reads
 
 ```math
-B^$(_tex(:rm, "Br"))_{k+1}
-= $(_tex(:widetilde, "B"))^$(_tex(:rm, "Br"))_k
+B^{$(_tex(:rm, "Br"))}_{k+1}
+= $(_tex(:widetilde, "B"))^{$(_tex(:rm, "Br"))}_k
    - $(
     _tex(
         :frac,
-        "$(_tex(:widetilde, "B"))^$(_tex(:rm, "Br"))_k y_k y^{$(_tex(:rm, "T"))}_k $(_tex(:widetilde, "B"))^$(_tex(:rm, "Br"))_k",
-        "y^{$(_tex(:rm, "T"))}_k $(_tex(:widetilde, "B"))^$(_tex(:rm, "Br"))_k y_k"
+        "$(_tex(:widetilde, "B"))^{$(_tex(:rm, "Br"))}_k y_k y^{$(_tex(:rm, "T"))}_k $(_tex(:widetilde, "B"))^{$(_tex(:rm, "Br"))}_k",
+        "y^{$(_tex(:rm, "T"))}_k $(_tex(:widetilde, "B"))^{$(_tex(:rm, "Br"))}_k y_k"
     )
 )
     + $(_tex(:frac, "s_k s^{$(_tex(:rm, "T"))}_k", "s^{$(_tex(:rm, "T"))}_k y_k"))
-    + φ_k y^{$(_tex(:rm, "T"))}_k $(_tex(:widetilde, "B"))^$(_tex(:rm, "Br"))_k y_k
+    + φ_k y^{$(_tex(:rm, "T"))}_k $(_tex(:widetilde, "B"))^{$(_tex(:rm, "Br"))}_k y_k
     $(_tex(:Bigl))(
         $(_tex(:frac, "s_k", "s^{$(_tex(:rm, "T"))}_k y_k"))
-        - $(_tex(:frac, "$(_tex(:widetilde, "B"))^$(_tex(:rm, "Br"))_k y_k", "y^{$(_tex(:rm, "T"))}_k $(_tex(:widetilde, "B"))^$(_tex(:rm, "Br"))_k y_k"))
+        - $(_tex(:frac, "$(_tex(:widetilde, "B"))^{$(_tex(:rm, "Br"))}_k y_k", "y^{$(_tex(:rm, "T"))}_k $(_tex(:widetilde, "B"))^{$(_tex(:rm, "Br"))}_k y_k"))
     $(_tex(:Bigr)))
     $(_tex(:Bigl))(
         $(_tex(:frac, "s_k", "s^{$(_tex(:rm, "T"))}_k y_k"))
-        - $(_tex(:frac, "$(_tex(:widetilde, "B"))^$(_tex(:rm, "Br"))_k y_k", "y^{$(_tex(:rm, "T"))}_k $(_tex(:widetilde, "B"))^$(_tex(:rm, "Br"))_k y_k"))
+        - $(_tex(:frac, "$(_tex(:widetilde, "B"))^{$(_tex(:rm, "Br"))}_k y_k", "y^{$(_tex(:rm, "T"))}_k $(_tex(:widetilde, "B"))^{$(_tex(:rm, "Br"))}_k y_k"))
      $(_tex(:Bigr)))^{$(_tex(:rm, "T"))}
 ```
 
@@ -507,7 +515,7 @@ end
 _doc_QN_B = """
 ```math
 $(_tex(:Cal, "B"))_k^{(0)}[⋅]
-= $(_tex(:frac, "$(_tex(:inner, "s_{k-1}", "y_{k-1}"; index = "p_k"))", "$(_tex(:inner, "y_{k-1}", "y_{k-1}"; index = "p_k"))"))$(_tex(:Id))_{$(_math(:TpM))"[⋅]
+= $(_tex(:frac, "$(_tex(:inner, "s_{k-1}", "y_{k-1}"; index = "p_k"))", "$(_tex(:inner, "y_{k-1}", "y_{k-1}"; index = "p_k"))"))$(_tex(:Id))_{$(_math(:TpM))}[⋅]
 ```
 """
 
@@ -516,19 +524,19 @@ $(_tex(:Cal, "B"))_k^{(0)}[⋅]
 
 This [`AbstractQuasiNewtonDirectionUpdate`](@ref) represents the limited-memory Riemannian BFGS update,
 where the approximating operator is represented by ``m`` stored pairs of tangent
-vectors ``$(_math(:Sequence, _tex(:widehat, "s"), "i", "k-m", "k-1"))`` and ``$(_math(:Sequence, _tex(:widehat, "y"), "i", "k-m", "k-1"))`` in the ``k``-th iteration.
-For the calculation of the search direction ``X_k``, the generalisation of the two-loop recursion
-is used (see [HuangGallivanAbsil:2015](@cite)),
+vectors ``$(_math(:Sequence, "$(_tex(:widetilde, "s"))", "i", "k-m", "k-1"))`` and ``$(_math(:Sequence, "$(_tex(:widetilde, "y"))", "i", "k-m", "k-1"))``
+in the ``k``-th iteration. For the calculation of the search direction ``X_k``, the generalisation
+of the two-loop recursion is used (see [HuangGallivanAbsil:2015](@cite)),
 since it only requires inner products and linear combinations of tangent vectors in ``$(_math(:TpM; p = "p_k"))``.
-For that the stored pairs of tangent vectors ``$(_tex(:widehat, "s"))_i,  $(_tex(:widehat, "y"))_i``,
+For that the stored pairs of tangent vectors ``s_i, y_i``,
 the gradient ``$(_tex(:grad)) f(p_k)`` of the objective function ``f`` in ``p_k``
 and the positive definite self-adjoint operator
 
 $(_doc_QN_B)
 
 are used. The two-loop recursion can be understood as that the [`InverseBFGS`](@ref) update
 is executed ``m`` times in a row on ``$(_tex(:Cal, "B"))^{(0)}_k[⋅]`` using the tangent vectors ``$(_tex(:widehat, "s"))_i,$(_tex(:widehat, "y"))_i``,
-and in the same time the resulting operator ``$(_tex(:Cal, "B"))^{LRBFGS}_k [⋅]`` is directly applied on ``$(_tex(:grad))f(x_k)``.
+and in the same time the resulting operator ``$(_tex(:Cal, "B"))^{$(_tex(:rm, "LRBFGS"))}_k [⋅]`` is directly applied on ``$(_tex(:grad))f(x_k)``.
 When updating there are two cases: if there is still free memory, ``k < m``, the previously
 stored vector pairs ``$(_tex(:widehat, "s"))_i,$(_tex(:widehat, "y"))_i`` have to be
 transported into the upcoming tangent space ``$(_math(:TpM; p = "p_{k+1}"))``.