Exercise 1.3.12

Answers

Centered difference matrices of even size seem to be invertible. Look at eqns. 1 and 4:

[\begin{matrix} 0 & 1 & 0 & 0 \\ - 1 & 0 & 1 & 0 \\ 0 & - 1 & 0 & 1 \\ 0 & 0 & - 1 & 0 \end{matrix}] [\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{matrix}] = [\begin{matrix} b_{1} \\ b_{2} \\ b_{3} \\ b_{4} \end{matrix}] First solve x_{2} = b_{1} [\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{matrix}] = [\begin{matrix} - b_{2} - b_{4} \\ b_{1} \\ - b_{4} \\ b_{1} + b_{3} \end{matrix}]

mathchakra

2021-11-20 14:16