Abstract
Perturbation theory is systematically used to generate root finding algorithms with fourth order derivatives. Depending on the number of correction terms in the perturbation expansion and the number of Taylor expansion terms, different root finding formulas can be generated. Expanding Taylor series up to fourth order derivatives and taking two, three and four correction terms in the perturbation expansions, three different root finding algorithms are derived. The algorithms are contrasted numerically with each other as well as with the Newton-Raphson algorithm. It is found that the quadruple-correction-term algorithm performs better than the others.
Highlights
Perturbation theory is well established and used in search of approximate solutions of algebraic equations, differential equations, integro-differential equations, difference equations etc
In the book by Hinch [2], the perturbation method and iteration method were treated as separate methods
Depending on the number of terms taken in the perturbation expansion, on the number of terms in the Taylor expansion and the way the resulting equations are separated, different iteration formulas which may or may not belong to the mentioned class of iteration formulas can be generated
Summary
Perturbation theory is well established and used in search of approximate solutions of algebraic equations, differential equations, integro-differential equations, difference equations etc. Root finding formulas consisting of up to third order derivatives were derived in that work. With referral to the number of terms taken in the perturbation expansions, formulas were classified as single-correction-term algorithms, doublecorrection-term algorithms, triple-correction-term algorithms. Two correction, three correction and four correction terms are taken each leading to a different root finding algorithm Numerical comparisons of the three formulas derived and Newton-Raphson formula yield that the higher order algorithms perform better than the Newton-Raphson method with the best being quadruple-correction-term algorithm. Based on this work as well as on the previous work [4], one may conclude that the best algorithms can be derived by taking the same number of correction terms both in Taylor expansions and perturbation expansions. The formulas are classified with respect to the number of correction terms in the perturbation expansion
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