Past exam of the mathematics course of the University of Oxford / 2021 / Prelims / Mathematics IV / 7 / c / Solution

For multivariable Newton iteration, use the Jacobian matrixAt ${\mathbf{x}}_0=(1/2,-1/2,1/2)$, direct evaluation givesWith ${\mathbf{s}}_0={\mathbf{x}}_1-{\mathbf{x}}_0$, the required linear system is thereforeAfter solving this linear system, form $\boxed{{\mathbf{x}}_1={\mathbf{x}}_0+{\mathbf{s}}_0}$. To get the second iterate, evaluate both the Jacobian matrix and the residual anew at ${\mathbf{x}}_1$, solveand set $\boxed{{\mathbf{x}}_2={\mathbf{x}}_1+{\mathbf{s}}_1}$. Reusing the initial Jacobian matrix would define a different iteration. The requested linear system has deliberately been left unsolved.