Past exam of the numerical linear algebra and approximation course of the University of Oxford / 2008 / 3 / c / Solution

For $[a,b]=[1,2]$, $z(0)=-3$ and $\cosh \theta =3$ gives $e^{\theta }=3+2\sqrt{2}$. The normalization factors cancel, soIf the eigenvalues are known to be exactly $1,3/2,2$, use the more targeted annihilating polynomialIt 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.