I apologize if this is not a research level question (already tried asking https://math.stackexchange.com/questions/1303288/relation-between-parallel-transport-and-jacobi-field-iion stack exchange with no response), but it's not a homework question either: an applied mathematics/medical imaging paper I was reading tried to numerically approximate parallel transport at time step $i$ using Jacobi fields, and when I ran their algorithm and checked the norms of the parallel transport, it's blowing up (literally) exponentially as a function of the time steps $i$ I used to divide $[0,1]$, which tells me it might not be correct.

Here's the the theoretical assumption/question regarding their algorithm:

FIX $t_0>0$.

Let $t>0$ and $W(t_0)\in T_{c(t_0)}M$ be the parallel transport of $W\in T_{c(0)}M$. and $J_{t_0,W(t_0)}$ be the Jacobi field along a geodesic $c$ such that $J_{t_0,W(t_0)}(t_0)=0, J_{t_0,W(t_0)}'(t_0)=W(t_0)$. Let $P_{t_0,t_0+t}$ denote the parallel transport from time $t_0$ to $t_0+t$ for a FIXED $t_0\in \mathbb{R}$

By now, one can prove that:(I agree and I've the proof)

$$\left|\left|\frac{P_{t_0,t_0+t}(W(t_0))-\frac{J_{t_0,W(t_0)}\ \ \ (t_0+t)}{t}}{t}\right| \right|\to0$$ as $t\to 0$.

Now, let $t_0$ VARY, and indeed assume that $t_0=t$. My question is: is the same formula going to be true? That is, do we also have:

$$\left| \left|\frac{P_{t,2t}(W(t))-\frac{J_{t,W(t)}\ \ \ (2t)}{t}}{t}\right| \right|\to0$$ as $t\to 0$? THIS IS MY MAIN QUESTION. The paper seemingly used this. A definite yes or no answer is okay with me for the moment. Proofs very much appreciated.

In the above, $J_{t,W(t)}$ is the unique Jacobi field along $c$ so that $J(t)=0, J'(t)=W(t)$.

In that paper I mentioned, they assume $t=\frac{1}{N}$and $P_{t,2t}(W(t))=P_{0,2t}(W)$ is approximately $\frac{J_{t,W(t)}(2t)}{t}$, and use it to repeatedly calculate Jacobi and parallel transport. But the norms of $P_{t,2t}(W(t))$ is showing huge errors and $P_{t,kt}(W(t))$ is blowing up with $k$.