ξ = AdjDforwardLogs(M,x,ν)

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

Ouput

• ξ – resulting tangent vector in $T_x\mathcal M$ representing the adjoint differentials of the logs.
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AdjDxExp(M,x,ξ,η)

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

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AdjDxGeo(M,x,y,t,η)

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

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AdjDxLog(M,x,y,η)

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

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AdjDyGeo(M,x,y,t,η)

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

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AdjDyLog(M,x,y,η)

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

AdjDξExp(M,x,ξ,η)
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.