| Summary: | We have developed a two-point iterative scheme for multiple roots that achieves fifth order convergence by using two function evaluations and two derivative evaluations each iteration. Weight function approach is utilized to frame the scheme. The weight function named as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><mo>(</mo><msub><mi>υ</mi><mi>t</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula> is used, which is a function of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>υ</mi><mi>t</mi></msub></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>υ</mi><mi>t</mi></msub></semantics></math></inline-formula> is a function of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ω</mi><mi>t</mi></msub></semantics></math></inline-formula>, i.e., <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>υ</mi><mi>t</mi></msub><mo>=</mo><mfrac><msub><mi>ω</mi><mi>t</mi></msub><mrow><mn>1</mn><mo>+</mo><mi>a</mi><msub><mi>ω</mi><mi>t</mi></msub></mrow></mfrac><mo>,</mo></mrow></semantics></math></inline-formula> where <i>a</i> is a real number and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ω</mi><mi>t</mi></msub><mo>=</mo><msup><mfenced separators="" open="(" close=")"><mfrac><mrow><mi>g</mi><mo>(</mo><msub><mi mathvariant="monospace">y</mi><mi>t</mi></msub><mo>)</mo></mrow><mrow><mi>g</mi><mo>(</mo><msub><mi mathvariant="monospace">x</mi><mi>t</mi></msub><mo>)</mo></mrow></mfrac></mfenced><mfrac><mn>1</mn><mover accent="true"><mi>m</mi><mo stretchy="false">˜</mo></mover></mfrac></msup></mrow></semantics></math></inline-formula> is a multi-valued function. The consistency of the newly generated methods is ensured numerically and through the basins of attraction. Four complex functions are considered to compare the new methods with existing schemes via basins of attraction, and all provided basins of attraction possess reflection symmetry. Further, five numerical examples are used to verify the theoretical results and to contrast the presented schemes with some recognized schemes of fifth order. The results obtained have proved that the new schemes are better than the existing schemes of the same nature.
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