zbMATH — the first resource for mathematics

Transformation methods for finding multiple roots of nonlinear equations. (English) Zbl 1205.65177
The problem of finding a root of multiplicity \((m\geq 1)\) of a nonlinear equation on an interval \((a,b)\) is considered. The function \(f(x)\) is transformed to a hyper tangent function combined with a simple difference formula whose value changes from \((-1)\) to 1 as \((x)\) passes through the root of function. Then the so-called numerical integration method is applied to the transformed equation. A Steffensen-type iterative method which does not require any derivatives of \(f(x)\) nor is quite effected by an initial approximation is proposed. It is shown that the convergence order of the proposed method becomes cubic by simultaneous approximation to the root and its multiplicity.

65H05 Numerical computation of solutions to single equations
Full Text: DOI
[1] Atkinson, K.E., An introduction to numerical analysis, (1988), John Wiley & Sons Singapore
[2] Chun, C.; Bae, H.J.; Neta, B., New families of nonlinear third-order solvers for finding multiple roots, Comput. math. appl., 57, 1574-1582, (2009) · Zbl 1186.65060
[3] Chun, C.; Neta, B., A third-order modification of newton’s method for multiple roots, Appl. math. comput., 211, 474-479, (2009) · Zbl 1162.65342
[4] Dong, C., A family of multipoint iterative formula of the higher order for computing multiple roots of an equation, Int. J. comput. math., 21, 363-367, (1987)
[5] King, R.F., A secant method for multiple roots, Bit, 17, 321-328, (1977) · Zbl 0367.65026
[6] Kioustelidis, J.B., A derivative-free transformation preserving the order of convergence of iteration methods in case of multiple zeros, Numer. math., 33, 385-389, (1979) · Zbl 0422.65033
[7] Neta, B., New third order nonlinear solvers for multiple roots, Appl. math. comput., 202, 162-170, (2008) · Zbl 1151.65041
[8] Neta, B.; Johnson, A.N., High order nonlinear solvers for multiple roots, Comput. math. appl., 55, 2012-2017, (2008) · Zbl 1142.65044
[9] Osada, N., An optimal multiple root finding method of order three, J. comput. appl. math., 51, 131-133, (1994) · Zbl 0814.65045
[10] Parida, P.K.; Gupta, D.K., An improved method for finding multiple roots and it’s multiplicity of nonlinear equations in \(\mathbb{R}\), Appl. math. comput., 202, 498-503, (2008) · Zbl 1151.65042
[11] Traub, J.F., Iterative methods for the solution of equations, (1964), Prentice Hall Englewood · Zbl 0121.11204
[12] Wu, X.Y.; Xia, J.L.; Shao, R., Quadratically convergent multiple roots finding method without derivatives, Comput. math. appl., 42, 115-119, (2001) · Zbl 0985.65048
[13] Yun, B.I., A non-iterative method for solving non-linear equations, Appl. math. comput., 198, 691-699, (2008) · Zbl 1138.65035
[14] Steffensen, L.F., Remark on iteration, vol. 16, (1933), Skand Aktuarietidskr · JFM 59.0535.03
[15] Yun, B.I., A derivative free iterative method for finding multiple roots of nonlinear equations, Appl. math. lett., 22, 1859-1863, (2009) · Zbl 1205.65176
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.