New Highly Efficient Families of Higher-Order Methods for Simple Roots, Permitting <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.60088pt;width:92.712502px;" id="M1" height="29.1625" version="1.1" viewBox="0 0 92.712502 29.1625" width="92.712502">
	
		
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<table>The basins of attraction for <svg height="19.924999" id="M561" style="vertical-align:-4.22832pt" version="1.1" viewbox="0 0 32.912498 19.924999" width="32.912498" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,15.775)"><path d="M614 175l29 -10q-33 -109 -57 -154q-121 -26 -184 -26q-90 0 -160.5 29t-112.5 77t-63.5 105.5t-21.5 119.5q0 157 108 253t277 96q36 0 71.5 -5t69 -13.5t36.5 -8.5q15 -102 20 -150l-29 -8q-20 79 -66.5 114t-128.5 35q-119 0 -187.5 -86t-68.5 -207
q0 -140 73.5 -227.5t188.5 -87.5q73 0 119.5 37.5t86.5 116.5z" id="x43"></path></g><g transform="matrix(.017,-0,0,-.017,11.367,15.775)"><path d="M861 0h-263v28q61 5 73 22t12 97l-4 408h-4l-244 -548h-20l-224 534h-2l-15 -285q-5 -98 -3 -166q1 -34 19.5 -45t72.5 -17v-28h-233v28q45 5 62.5 17t22.5 43q11 65 19 183l14 208q3 36 3.5 55t-2.5 37.5t-6.5 25t-16.5 13t-25.5 8t-39.5 4.5v28h164l228 -501l232 501
h170v-28q-68 -6 -81 -21.5t-11 -91.5l9 -362q2 -80 15 -97t78 -22v-28z" id="x4D"></path></g>
<g transform="matrix(.012,-0,0,-.012,26.513,7.613)"><path d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" id="x31"></path></g><g transform="matrix(.012,-0,0,-.012,26.513,19.863)"><path d="M456 178h-96v-72q0 -51 12.5 -62.5t72.5 -16.5v-27h-256v27q65 5 78 17t13 62v72h-260v28q182 271 300 426h40v-407h96v-47zM280 225v295h-2q-107 -148 -196 -295h198z" id="x34"></path></g>
</svg> (a), <svg height="15.25" id="M562" style="vertical-align:-3.13504pt" version="1.1" viewbox="0 0 33.049999 15.25" width="33.049999" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.112)"><path d="M667 0h-153l-248 291q-28 32 -54 27v-190q0 -65 14.5 -80t77.5 -20v-28h-260v28q58 5 71 20t13 80v392q0 48 -6 66.5t-22 25.5t-59 10v28h258v-28q-59 -5 -73 -20.5t-14 -81.5v-175q29 6 70 45q101 97 167 178q23 27 20 37t-36 14l-23 3v28h240v-28q-57 -5 -86 -18.5
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<g transform="matrix(.012,-0,0,-.012,26.65,15.187)"><path d="M456 178h-96v-72q0 -51 12.5 -62.5t72.5 -16.5v-27h-256v27q65 5 78 17t13 62v72h-260v28q182 271 300 426h40v-407h96v-47zM280 225v295h-2q-107 -148 -196 -295h198z" id="x34"></path></g>
</svg> (b), and <svg height="15.25" id="M563" style="vertical-align:-3.13504pt" version="1.1" viewbox="0 0 48.1875 15.25" width="48.1875" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.112)"><path d="M861 0h-263v28q61 5 73 22t12 97l-4 408h-4l-244 -548h-20l-224 534h-2l-15 -285q-5 -98 -3 -166q1 -34 19.5 -45t72.5 -17v-28h-233v28q45 5 62.5 17t22.5 43q11 65 19 183l14 208q3 36 3.5 55t-2.5 37.5t-6.5 25t-16.5 13t-25.5 8t-39.5 4.5v28h164l228 -501l232 501
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<g transform="matrix(.012,-0,0,-.012,41.8,15.187)"><path d="M456 178h-96v-72q0 -51 12.5 -62.5t72.5 -16.5v-27h-256v27q65 5 78 17t13 62v72h-260v28q182 271 300 426h40v-407h96v-47zM280 225v295h-2q-107 -148 -196 -295h198z" id="x34"></path></g>
</svg> (c) in problem 3.</table>

International Journal of Mathematics and Mathematical Sciences

fig5

Figure 5

Figure 5: New Highly Efficient Families of Higher-Order Methods for Simple Roots, Permitting <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.60088pt;width:92.712502px;" id="M1" height="29.1625" version="1.1" viewBox="0 0 92.712502 29.1625" width="92.712502">
	
		
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