compute the adjoibnt differential of forwardLogs $F$ orrucirng, in the power manifold array, the differential of the function

$F_i(x) = \sum_{j\in\mathcal I_i} \log_{x_i} x_j$

where $i$ runs over all indices of the Power manifold M and $\mathcal I_i$ denotes the forward neighbors of $i$ Let $n$ be the number dimensions of the Power manifold (i.e. length(size(x))). Then the input tangent vector lies on the manifold $\mathcal M' = \mathcal M^n$.

Input

• M – a Power manifold
• x – a PowPoint.
• ν – a PowTVector from $T_X\mathcal M'$, where $X = (x,...,x)\in\mathcal M'$ is an $n$-fold copy of $x$ where \mathcal N (x,...,x)N.

Ouput

• ξ – resulting tangent vector in $T_x\mathcal M$ representing the adjoint differentials of the logs.
source

computes the adjoint of $D_x\exp_x\xi[\eta]$.

source

computes the adjoint of $D_xg(t;x,y)[\eta]$.

source

computes the adjoint of $D_xlog_xy[\eta]$.

source

computes the adjoint of $D_yg(t;x,y)[\eta]$.

source

computes the adjoint of $D_ylog_xy[\eta]$.

computes the adjoint of $D_\xi\exp_x\xi[\eta]$. Note that $\xi\in T_\xi(T_x\mathcal M) = T_x\mathcal M$ is still a tangent vector.