Six short, independent…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Linear algebra and vector spaces · Numerical and computational mathematics

Error, Chance and the Machine, Ch Iteration: nonlinear systems and eigenvalues

Problem

Six short, independent checks. ((a)) The iteration xk+1=13(xk2+2)x_{k+1} = \tfrac13(x_k^2+2) has a fixed point at x∗=1x^*=1. Find g′(1)g'(1) and state whether the iteration contracts near x∗=1x^*=1. ((b)) An iteration produces the errors e0=0.2e_0=0.2, e1=0.02e_1=0.02, e2=2.0×10−4e_2=2.0\times10^{-4}, e3=2.0×10−8e_3=2.0\times10^{-8}. Estimate the order pp from the ratios ln⁡(ek+1/ek)/ln⁡(ek/ek−1)\ln(e_{k+1}/e_k)/\ln(e_k/e_{k-1}). ((c)) A circuit with a nonlinear (square-law) element and a linear resistor obeys F1(V,I)=I−kV2=0F_1(V,I)=I-kV^2=0 and F2(V,I)=V+IR−E=0F_2(V,I)=V+IR-E=0, with k=0.002 A V−2k=0.002\ \mathrm{A\,V^{-2}}, R=100 ΩR=100\ \Omega, E=5 VE=5\ \mathrm{V}. Starting from the guess (V0,I0)=(2 V,0.03 A)(V_0,I_0)=(2\ \mathrm{V},0.03\ \mathrm{A}), take one Newton step and report the improved (V1,I1)(V_1,I_1). ((d)) For A=(3113)\mathsf{A}=\begin{pmatrix}3&1\\1&3\end{pmatrix} and x0=(1,0)T\mathbf{x}_0=(1,0)^{\rm T}, carry out three power-iteration steps, normalising after each multiplication by dividing by the largest-magnitude component. Give the vector x3\mathbf{x}_3 and the Rayleigh quotient estimate of the dominant eigenvalue at that vector. ((e)) For A=(2112)\mathsf{A}=\begin{pmatrix}2&1\\1&2\end{pmatrix}, form the QR factorisation by Gram-Schmidt and compute one QR step A1=RQ\mathsf{A}_1=\mathsf{R}\mathsf{Q}. Report the entries of A1\mathsf{A}_1 and check that its trace and determinant match those of A\mathsf{A}. ((f)) For M=(61010101−4)\mathsf{M}=\begin{pmatrix}6&1&0\\1&0&1\\0&1&-4\end{pmatrix}, write down the three Gershgorin discs and state how many eigenvalues are guaranteed to be negative.

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