Numerical Methods Madasmaths Apr 2026

(a) Show that the Newton-Raphson iterative formula for this root is [ x_n+1 = x_n - \frac\ln(x_n+2) - x_n\frac1x_n+2 - 1. ]

Consider this gem from a past MADASMATHS worksheet: "The equation ( e^x - 3x = 0 ) has a root in ( [0, 0.5] ). Perform one bisection iteration. What is the maximum possible error in your approximation after this iteration?" The answer: half the interval width (0.25). But the follow-up asks: "How many iterations are needed to guarantee an error less than ( 10^-6 )? Write your answer as an inequality." numerical methods madasmaths

(b) Starting with ( x_0 = 0.5 ), find ( x_2 ) correct to 5 decimal places. (a) Show that the Newton-Raphson iterative formula for

In the polished world of pure mathematics, answers are often exact: ( \sqrt2 ), ( \pi ), or a neatly factored root of a polynomial. But the real world—physics, engineering, finance—rarely offers such tidy solutions. It demands approximations. It demands numerical methods. What is the maximum possible error in your

(c) Without performing further iterations, state the order of convergence of Newton-Raphson for this root. Give a reason for your answer.

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