Newton's method is run on a polynomial with a root at of unknown multiplicity
PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Numerical and computational mathematics · Numerical and computational mathematics
Mathematical Methods for Physics, Ch Root finding, interpolation, numerical linear algebra and quadrature
Problem
Newton's method is run on a polynomial p(x) with a root at x=2 of unknown multiplicity. The printed errors ∣xn−2∣ are 0.100, 0.0750, 0.0563, 0.0422.
((a)) For f(x)=(x−2)m with an integer m≥2, show that Newton's method gives en+1=mm−1en exactly, where en=xn−2, and explain from the error law en+1=2f′(xn)f′′(ξ)en2 why the quadratic convergence of a simple root is lost.
((b)) Read the multiplicity of the root of p from the printed errors, and find how many further steps plain Newton needs to bring the error from 0.0422 below 10−10.
((c)) Show that the modified step xn+1=xn−mf(xn)/f′(xn) lands on the root of (x−2)m in one step.
((d)) A colleague proposes to switch to bisection for its guarantee. Decide whether bisection can be used on this root, and compare its rate with that of plain Newton here.