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Modified Newton’s method with third-order convergence and multiple roots. (English) Zbl 1030.65044
Some modified Newton methods, with order of convergence three in the case of simple roots, obtained by using interpolatory quadrature formula and not requiring the second or higher derivatives of the function, are studied in the case of multiple roots. A typical result (Theorem 3) reads as follows:
If $$\xi$$ is a root of $$f(x)$$ with multiplicity $$p>1$$ ($$f(\xi)=0,\;f^{\prime }(\xi)=0,\dots ,f^{(p)}(\xi)\neq 0$$) then the modified Newton method obtained by using a quadrature formula of order at least $$p-1$$ and using the corrected Newton’s method has order of convergence two.

##### MSC:
 65H05 Numerical computation of solutions to single equations
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##### References:
 [1] Dennis, J.E.; Schnable, R.B., Numerical methods for unconstrained optimization and nonlinear equations, (1983), Prentice-Hall Englewood Cliffs, NJ [2] Frontini, M.; Sormani, E., Some variants of Newton’s method with third-order convergence, Appl. math. comput., 140, 419-426, (2003) · Zbl 1037.65051 [3] Gautschi, W., Numerical analysis an introduction, (1997), Birkhäuser Basel · Zbl 0877.65001 [4] Weerakoom, S.; Fernando, T.G.I., A variant of Newton’s method with accelerated third-order convergence, Appl. math. lett., 13, 87-93, (2000) · Zbl 0973.65037
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