
# The adjoint differential of the end point of a geodesic

This function evaluates for $F(y)=\geo{x}{y}(t)$ with fixed $x\in\M$ and $t\in\mathbb R$ the adjoint differenital $D^*_yF(y)[\eta].$

It is calculated a corresponding adjoint Jacobi field AdjJacoiField. Since we can rewrite the problem as computing the differential of the reverted geodesic $\geo{y}{x}(1-t)$, this differential is computed using the adjoint differential of the start point of a geodesic.

### Matlab Documentation

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 % AdjDyGeo(x,y,t,eta) - Adjoint of the Derivative of geo(x,y,t) wrt y. % % For a function f: M \mapsto R and fixed x,t we have for the % gradient of g(y) = f(geo(x,y,t)) that % _y = _g(x,y,t) % hence with the Adjoint we obtain % grad g = AdjDxGeo(x,.,t)(y)[grad f]. % This function hence only requires eta=grad f to computed % the chain rule. % % INPUT % x : start point of a geodesic, g(x,y,0)=x % y : end point of a geodesic, geo(x,y,1) = y % t : [0,1] a point on the geodesic to be evaluated, % may exceed [0,1] to leave the segment between x and y % eta : (in Tg(x,y,t)) direction to take the Adjoint derivative at. % % OUTPUT % xi : ( in TyM ) - the adjoint of DyGeo with respect to eta % --- % MVIRT R. Bergmann, 2017-12-04