| Summary: | In the paper, we iteratively solve a scalar nonlinear equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>∈</mo><mi mathvariant="script">C</mi><mo>(</mo><mi>I</mi><mo>,</mo><mi mathvariant="double-struck">R</mi><mo>)</mo><mo>,</mo><mspace width="3.33333pt"></mspace><mi>x</mi><mo>∈</mo><mi>I</mi><mo>⊂</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula>, and <i>I</i> includes at least one real root <i>r</i>. Three novel two-step iterative schemes equipped with memory updating methods are developed; they are variants of the fixed-point Newton method. A triple data interpolation is carried out by the two-degree Newton polynomial, which is used to update the values of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>f</mi><mo>′</mo></msup><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>f</mi><mrow><mo>″</mo></mrow></msup><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. The relaxation factor in the supplementary variable is accelerated by imposing an extra condition on the interpolant. The new memory method (NMM) can raise the efficiency index (E.I.) significantly. We apply the NMM to five existing fourth-order iterative methods, and the computed order of convergence (COC) and E.I. are evaluated by numerical tests. When the relaxation factor acceleration technique is combined with the modified D<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mi mathvariant="normal">z</mi><mo>ˇ</mo></mover></semantics></math></inline-formula>uni<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mi mathvariant="normal">c</mi><mo>´</mo></mover></semantics></math></inline-formula>’s memory method, the value of E.I. is much larger than that predicted by the paper [Kung, H.T.; Traub, J.F. <i>J. Assoc. Comput. Machinery</i> <b>1974</b>, <i>21</i>]. for the iterative method without memory.
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