$E(i+1)=(I-AT)E(i)+1/2(AT)^2$


How to find the maximum value of $E$ in this expression without using the iterative method? An approximate estimation is also acceptable. Only the $E$ vector is unknown, and the others are known matrix vectors.


It would be better if we could get a formula of $E$ about $T$.



How to find the maximum value of the following difference equation without using iterative method？
Concerning your request "It would be better if we could get a formula of $E$":


By induction on $i$,
$$E(i)=B^i E(0)+\sum\_{j=0}^{i-1} B^jC \tag{1}\label{1}$$
for $i=0,1,\dots$, where $B:=I-AT$ and $C:=1/2(AT)^2$.


If $(I-B)^{-1}$ exists, then $\sum\_{j=0}^{i-1} B^j=(I-B^i)(I-B)^{-1}$ and hence \eqref{1} can be rewritten as
$$E(i)=B^i (E(0)-(I-B)^{-1}C)+(I-B)^{-1}C. \tag{2}\label{2}$$




---


(As for your other request, "How to find the maximum value of $E$", it is incomprehensible to me, as $E(i)$ apparently is a *vector* function of $i$.)


