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7 months ago in Computational Methods , Numerical Analysis By Shraddha

How can the first approximation for the root of x³ - 3x + 1 = 0 be found using Newton-Raphson when f’(1) = 0?

I'm using the Newton-Raphson method to find roots of a cubic polynomial. My initial guess near x=1 causes the algorithm to fail because the derivative there is zero. I need a reliable strategy to obtain a first approximation and proceed, understanding why the standard form fails and what modifications are valid.

RootFinding Newton-Raphson Method SecantMethod

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By Nisha Ali Answered 4 months ago

This is a classic pitfall I've encountered in practice. The standard Newton-Raphson iteration blows up because it divides by the derivative. Your immediate fix is to simply choose a different initial guess, like x? = 0.5 or 1.5, away from the critical point. For a more robust, general-purpose solution, I would recommend switching to the Secant Method, which approximates the derivative and avoids this division entirely. Alternatively, you could use a modified Newton method that adds a small regularization constant to the denominator when the derivative is near zero.

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How can the first approximation for the root of x³ - 3x + 1 = 0 be found using Newton-Raphson when f’(1) = 0?

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