(i) Formulate the conjugate gradient method for the solution of a system $A \mathbf{x}=\mathbf{b}$ with $A \in \mathbb{R}^{n \times n}$ and $\mathbf{b} \in \mathbb{R}^{n}, n>0$.

(ii) Prove that if the conjugate gradient method is applied in exact arithmetic then, for any $\mathbf{x}^{(0)} \in \mathbb{R}^{n}$, termination occurs after at most $n$ iterations.

(iii) The polynomial $p(x)=x^{m}+\sum_{i=0}^{m-1} c_{i} x^{i}$ is the minimal polynomial of the $n \times n$ matrix $A$ if it is the polynomial of lowest degree that satisfies $p(A)=0 . \quad[$ Note that $m \leqslant n$.] Give an example of a $3 \times 3$ symmetric positive definite matrix with a quadratic minimal polynomial.

Prove that (in exact arithmetic) the conjugate gradient method requires at most $m$ iterations to calculate the exact solution of $A \mathbf{x}=\mathbf{b}$, where $m$ is the degree of the minimal polynomial of $A$.