Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 4 / 40E (Numerical Analysis)

Let $Mx=(A-B)x$ and suppose $Hx=\lambda x$, so $-Bx=\lambda Mx$. Put $a=x^{\ast }Ax>0$ and $m=x^{\ast }Mx$. Since $A=M-(-B)$, $a=(1-\lambda )m$. AlsoPositive definiteness makes this positive, hence $|\lambda |<1$ for every eigenvalue and $\rho (H)<1$.

After multiplying the five-point equations by $-1$ if necessary, their matrix $A$ is symmetric positive definite. For any ordering write $A=D+L+L^T$, where $D=4I$. Gauss-Seidel method uses $M=D+L$ and has iteration matrix $-M^{-1}{L}^T$. In part (a) take $B=L^T$; then $A-B=M$ andwhich is positive definite. Therefore the iteration matrix has spectral radius below one for every ordering, proving convergence.