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I've been trying to learn about root finding using Newton's method, which uses a quadratic interpolating polynomial. I found this text, Numerical methods for roots of polynomials, PART 2: McNamee & Pan, which has relevant theory, but there seems to be some error in the text.

Book section describing the quadratic part of Newton's method

While calculating $x_{i+1}$, shouldn't the second term in the RHS of that equation be reciprocal of what it is, or am I wrong? I don't understand what am I missing here.

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No, the book is right. I know that the second term on the right looks superficially like the standard solution of a quadratic equation, upside down, but look more closely: the denominator $2f(x_i)$ is $2c$, not the $2a$ which you might have expected.

Muller's method uses a less common expression for the roots of a quadratic equation: $$ x = \frac{-2c}{b\pm\sqrt{b^2-4ac}} $$ which is mentioned on the Wikipedia page for quadratic equations. They even mention there that it is used in Muller's method. And on the Wikipedia page for Muller's method they say that the standard formula for a quadratic equation is not used because it may lead to a loss of significance.

I'm not sure that Muller's method is the go-to method for this kind of root finding. Inverse quadratic interpolation is used within the standard Brent method for locating roots. I don't know enough about the topic to explain in more detail, however.

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