Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 40A / a / i / Solution

Writing the error as $e_k=x_k-x^{\ast }$ givesprovided the fixed-point consistency conditionholds. Convergence for every starting vector is therefore equivalent to $H^k\to 0$, which in finite dimensions is equivalent toThese two conditions are necessary and sufficient.