Two branch currents and in a network containing nonlinear resistors

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

Two branch currents I1I_1 and I2I_2 in a network containing nonlinear resistors, measured in units of a reference current Iref=1 AI_{\rm ref} = 1\ \mathrm{A} so that every term below is dimensionless, satisfy the coupled Kirchhoff equations I12−I2−1=0,I1+I22−3=0.I_1^2 - I_2 - 1 = 0, \qquad I_1 + I_2^2 - 3 = 0. ((a)) Starting from the guess (I1,I2)=(1,1)(I_1,I_2) = (1,1), take one Newton step (solve the linearised system with the exact Jacobian, do not use a difference approximation) to find the improved estimate of (I1,I2)(I_1,I_2). ((b)) Take two more steps (a computer may be used). The root is (1.4925727,1.2277733)(1.4925727, 1.2277733). Find the error ∥ek∥\|\mathbf{e}_k\| after each of the three steps, and show that ∥ek+1∥/∥ek∥2\|\mathbf{e}_{k+1}\|/\|\mathbf{e}_k\|^2 settles to a constant, as quadratic convergence requires.

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