= Solution
For $[a,b]=[1,2]$, $z(0)=-3$ and $\cosh\theta=3$ gives $e^\theta=3+2\sqrt2$. The normalization factors cancel, so
$$
\boxed{\|x-y^{(k)}\|_2\leq\frac{2}{(3+2\sqrt2)^k+(3-2\sqrt2)^k}\|x-x^{(0)}\|_2
\leq2(3-2\sqrt2)^k\|x-x^{(0)}\|_2.}
$$
If the <eigenvalues> are known to be exactly $1,3/2,2$, use the more targeted <annihilating polynomial>
$$
\boxed{p_3(r)=(1-r)(1-2r/3)(1-r/2).}
$$
It equals one at zero and vanishes on the entire spectrum, so the <spectral theorem> gives $p_3(A)=0$. Thus $y^{(3)}=x$ in exact arithmetic. Degree three is the least degree that can vanish at all three distinct <eigenvalues>; a particular initial error supported on fewer eigenspaces can terminate earlier.
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