We present a new fourth order method for finding simple roots of a nonlinear equation <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.42947pt" id="M1" height="12.7178pt" version="1.1" viewBox="-0.0498162 -9.28833 45.9876 12.7178" width="45.9876pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-103" d="M619 670C619 686 593 712 555 712S459 686 410 634S335 504 320 430H250L219 400L222 388H312L258 73C223 -133 201 -166 187 -180C175 -191 158 -199 140 -199C123 -199 88 -188 74 -172C68 -166 63 -164 54 -171C38 -185 23 -201 23 -215C23 -236 60 -261 93 -261C122 -261 161 -247 207 -200C268 -138 300 -71 337 94C365 220 376 277 394 387L501 399L521 430H401C432 623 464 665 501 665C524 665 544 651 567 627C577 617 583 618 592 625C601 631 619 651 619 670Z"/><glyph.data ascent="3473" descent="-2876" horiz-adv-x="642" vert-adv-y="642"/></g><g transform="matrix(.013,0,0,-0.013,8.352,0)"><path id="g113-41" d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"/><glyph.data ascent="3473" descent="-2876" horiz-adv-x="346" vert-adv-y="346"/></g><g transform="matrix(.013,0,0,-0.013,12.85,0)"><path id="g113-121" d="M536 404C536 423 520 448 491 448C445 448 398 404 308 283L286 338C255 416 242 448 217 448C182 448 138 402 92 341L111 321C149 368 169 378 178 378C188 378 198 363 214 320L254 212C185 117 138 65 107 65C98 65 85 69 82 75C79 82 73 86 65 86C44 86 23 60 23 39C23 7 44 -12 71 -12C119 -12 168 33 265 177L306 61C321 17 347 -12 373 -12C413 -12 465 33 507 96L491 119C459 84 432 60 413 60C395 60 378 92 358 148L321 250C341 279 369 310 389 332C417 363 439 382 456 382C466 382 475 376 481 368C486 361 492 358 496 358C513 358 536 381 536 404Z"/><glyph.data ascent="3473" descent="-2876" horiz-adv-x="559" vert-adv-y="559"/></g><g transform="matrix(.013,0,0,-0.013,20.117,0)"><path id="g113-42" d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"/><glyph.data ascent="3473" descent="-2876" horiz-adv-x="346" vert-adv-y="346"/></g><g transform="matrix(.013,0,0,-0.013,28.247,0)"><path id="g117-34" d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"/><glyph.data ascent="3473" descent="-2876" horiz-adv-x="587" vert-adv-y="587"/></g><g transform="matrix(.013,0,0,-0.013,39.51,0)"><path id="g113-49" d="M241 635C89 635 35 457 35 312C35 153 89 -12 240 -12C390 -12 443 166 443 312C443 466 390 635 241 635ZM238 602C329 602 354 454 354 312C354 172 330 22 240 22C152 22 124 173 124 313S148 602 238 602Z"/><glyph.data ascent="3473" descent="-2876" horiz-adv-x="480" vert-adv-y="480"/></g></svg>. In terms of computational cost, per iteration the method uses one evaluation of the function and two evaluations of its first derivative. Therefore, the method has optimal order with efficiency index 1.587 which is better than efficiency index 1.414 of Newton method and the same with Jarratt method and King’s family. Numerical examples are given to support that the method thus obtained is competitive with other similar robust methods.  The conjugacy maps and extraneous fixed points of the presented method and other existing fourth order methods are discussed, and their basins of attraction are also given to demonstrate their dynamical behavior in the complex plane.

Rajni Sharma

Ashu Bahl

An Optimal Fourth Order Iterative Method for Solving Nonlinear Equations and Its Dynamics

Nevanlinna theory provides us with many tools applicable to the study of value distribution of meromorphic solutions of differential equations. Analogues of some of these tools have been recently developed for difference, <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.32506pt" id="M1" height="9.32511pt" version="1.1" viewBox="-0.0498162 -6.00005 6.61095 9.32511" width="6.61095pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g50-114" d="M481 430L460 436C440 442 393 451 366 451C348 451 325 448 313 446C267 437 195 409 147 367C79 308 24 211 24 105C24 37 56 -12 93 -12C121 -12 156 1 199 32C246 66 313 122 349 168H351L286 -147C274 -205 261 -215 210 -224L199 -226L193 -255L438 -243L442 -214L417 -212C364 -208 359 -200 370 -140L433 204C455 324 469 389 481 430ZM391 385C382 336 366 260 358 238C321 178 209 59 143 59C129 59 115 83 115 131C115 207 152 317 220 373C243 392 282 403 314 403C348 403 374 395 391 385Z"/></g></svg>-difference, and ultradiscrete equations. In many cases, the methodologies used in the study of meromorphic solutions of differential, difference, and <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.32506pt" id="M2" height="9.32511pt" version="1.1" viewBox="-0.0498162 -6.00005 6.61095 9.32511" width="6.61095pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g50-114"/></g></svg>-difference equations are largely similar. The purpose of this paper is to collect some of these tools in a common toolbox for the study of general classes of functional equations by introducing notion of a good linear operator, which satisfies certain regularity conditions in terms of value distribution theory. As an example case, we apply our methods to study the growth of meromorphic solutions of the functional equation <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.40337pt" id="M3" height="12.6917pt" version="1.1" viewBox="-0.0498162 -9.28833 132.453 12.6917" width="132.453pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g50-78" d="M1006 650H829L492 147L357 650H174L166 618C270 609 275 601 247 502L194 315C158 189 136 127 122 96C104 56 80 37 18 32L12 0H256L263 32C181 40 176 51 183 98C188 132 197 179 226 281L299 541H301L433 6H458L809 551H811L753 141C739 41 734 39 650 32L642 0H917L923 32C836 39 827 41 839 142L887 535C896 610 901 612 997 618L1006 650Z"/></g><g transform="matrix(.013,0,0,-0.013,12.937,0)"><path id="g50-41" d="M309 -142C211 -61 151 99 151 271S210 601 309 683L288 710C141 611 75 451 75 272V271C75 90 141 -72 288 -169L309 -142Z"/></g><g transform="matrix(.013,0,0,-0.013,17.591,0)"><path id="g50-123" d="M481 445L456 468C426 437 403 427 388 427C372 427 356 428 302 438C256 447 221 451 187 451C159 451 130 428 105 384C93 363 80 340 65 305L96 290C122 342 149 381 193 381C216 381 267 377 294 374S346 372 365 376L75 105C37 70 24 35 24 16C24 2 30 -5 35 -8C42 -8 46 -6 48 -2C58 14 75 35 90 41C102 46 116 46 147 32C200 8 267 -21 298 -21C347 -21 390 12 448 126L420 151C369 77 338 56 311 56C287 56 249 68 212 81C180 92 149 100 127 98C212 172 353 308 413 372L481 445Z"/></g><g transform="matrix(.013,0,0,-0.013,24.157,0)"><path id="g50-45" d="M98 134C72 134 46 117 46 90C46 73 55 65 60 64C95 55 124 32 124 -4C124 -42 95 -68 44 -89L57 -123C122 -104 194 -60 194 22C194 94 136 134 98 134Z"/></g><g transform="matrix(.013,0,0,-0.013,29.444,0)"><path id="g50-103" d="M620 667C620 684 593 710 552 710C515 710 459 687 409 634C360 582 334 506 320 433H247L215 400L220 388H310L259 80C224 -129 199 -161 185 -174C175 -183 160 -192 141 -192C123 -192 90 -181 75 -165C69 -158 64 -156 54 -164C39 -178 24 -195 24 -208C24 -231 61 -257 94 -257C124 -257 162 -243 209 -194C267 -134 303 -66 343 107C369 220 378 279 393 385L503 397L523 433H404C432 619 464 660 500 660C522 660 542 646 566 621C573 614 583 615 593 621C603 628 620 648 620 667Z"/></g><g transform="matrix(.013,0,0,-0.013,37.822,0)"><path id="g50-42" d="M283 271C283 451 220 610 70 710L48 683C146 603 207 443 207 271S149 -59 48 -142L70 -169C220 -67 283 90 283 271Z"/></g><g transform="matrix(.013,0,0,-0.013,45.416,0)"><path id="g54-36" d="M556 236V289H337V504H275V289H56V236H275V-4H337V236H556Z"/></g><g transform="matrix(.013,0,0,-0.013,56.299,0)"><path id="g50-81" d="M622 479C622 589 545 650 400 650H146L140 618C229 609 230 602 215 527L135 132C117 43 112 41 27 32L18 0H305L313 32C213 40 205 44 222 132L251 284L324 262H365C484 269 622 333 622 479ZM528 477C528 356 420 305 344 305C300 305 276 314 259 321L303 554C311 597 313 600 331 606C345 611 361 614 379 614C463 614 528 580 528 477Z"/></g><g transform="matrix(.013,0,0,-0.013,64.39,0)"><use xlink:href="#g50-41"/></g><g transform="matrix(.013,0,0,-0.013,69.044,0)"><use xlink:href="#g50-123"/></g><g transform="matrix(.013,0,0,-0.013,75.611,0)"><use xlink:href="#g50-45"/></g><g transform="matrix(.013,0,0,-0.013,80.897,0)"><use xlink:href="#g50-103"/></g><g transform="matrix(.013,0,0,-0.013,89.275,0)"><use xlink:href="#g50-42"/></g><g transform="matrix(.013,0,0,-0.013,97.604,0)"><path id="g54-34" d="M556 331V384H55V331H556ZM556 140V193H55V140H556Z"/></g><g transform="matrix(.013,0,0,-0.013,109.223,0)"><path id="g50-105" d="M509 90L493 118C465 91 427 70 418 70S409 77 415 103L469 327C494 429 466 451 436 451C412 451 387 444 356 427C307 400 222 342 160 264H158L246 667C253 701 252 710 243 710C228 710 169 680 87 673L83 644H117C155 644 160 640 149 588L24 -4L32 -12C57 -4 86 4 108 9L143 187C198 275 324 386 375 386C394 386 399 367 385 311L331 87C315 22 322 -12 351 -12C387 -12 452 24 509 90Z"/></g><g transform="matrix(.013,0,0,-0.013,116.074,0)"><use xlink:href="#g50-41"/></g><g transform="matrix(.013,0,0,-0.013,120.728,0)"><use xlink:href="#g50-123"/></g><g transform="matrix(.013,0,0,-0.013,127.294,0)"><use xlink:href="#g50-42"/></g></svg>, where <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.40337pt" id="M4" height="12.6917pt" version="1.1" viewBox="-0.0498162 -9.28833 42.7168 12.6917" width="42.7168pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g50-78"/></g><g transform="matrix(.013,0,0,-0.013,12.937,0)"><use xlink:href="#g50-41"/></g><g transform="matrix(.013,0,0,-0.013,17.591,0)"><use xlink:href="#g50-123"/></g><g transform="matrix(.013,0,0,-0.013,24.157,0)"><use xlink:href="#g50-45"/></g><g transform="matrix(.013,0,0,-0.013,29.444,0)"><use xlink:href="#g50-103"/></g><g transform="matrix(.013,0,0,-0.013,37.822,0)"><use xlink:href="#g50-42"/></g></svg> is a linear polynomial in <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.40337pt" id="M5" height="12.6917pt" version="1.1" viewBox="-0.0498162 -9.28833 8.50303 12.6917" width="8.50303pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g50-103"/></g></svg> and <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.40337pt" id="M6" height="12.6917pt" version="1.1" viewBox="-0.0498162 -9.28833 25.1663 12.6917" width="25.1663pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g50-77" d="M567 167L534 181C505 122 482 94 454 70C428 46 385 42 330 42C288 42 248 44 235 55C216 70 213 91 220 130L295 528C310 609 314 611 401 618L409 650H140L131 618C222 611 224 609 209 528L133 126C118 47 108 41 25 33L18 0H497C514 33 554 126 567 167Z"/></g><g transform="matrix(.013,0,0,-0.013,7.289,0)"><use xlink:href="#g50-41"/></g><g transform="matrix(.013,0,0,-0.013,11.943,0)"><use xlink:href="#g50-103"/></g><g transform="matrix(.013,0,0,-0.013,20.321,0)"><use xlink:href="#g50-42"/></g></svg>, where <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.04981995pt" id="M7" height="8.64656pt" version="1.1" viewBox="-0.0498162 -8.59674 7.41997 8.64656" width="7.41997pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g50-77"/></g></svg> is good linear operator, <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.40337pt" id="M8" height="12.6917pt" version="1.1" viewBox="-0.0498162 -9.28833 37.8496 12.6917" width="37.8496pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g50-81"/></g><g transform="matrix(.013,0,0,-0.013,8.091,0)"><use xlink:href="#g50-41"/></g><g transform="matrix(.013,0,0,-0.013,12.745,0)"><use xlink:href="#g50-123"/></g><g transform="matrix(.013,0,0,-0.013,19.311,0)"><use xlink:href="#g50-45"/></g><g transform="matrix(.013,0,0,-0.013,24.598,0)"><use xlink:href="#g50-103"/></g><g transform="matrix(.013,0,0,-0.013,32.976,0)"><use xlink:href="#g50-42"/></g></svg> is a polynomial in <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.40337pt" id="M9" height="12.6917pt" version="1.1" viewBox="-0.0498162 -9.28833 8.50303 12.6917" width="8.50303pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g50-103"/></g></svg> with degree deg <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-1.15896pt" id="M10" height="9.7557pt" version="1.1" viewBox="-0.0498162 -8.59674 29.8942 9.7557" width="29.8942pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g50-81"/></g><g transform="matrix(.013,0,0,-0.013,11.766,0)"><path id="g54-94" d="M552 293V344L59 566V507L489 319L59 131V72L552 293ZM552 -41V12H59V-41H552Z"/></g><g transform="matrix(.013,0,0,-0.013,23.385,0)"><path id="g50-51" d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z"/></g></svg>, both with small meromorphic coefficients, and <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.18977pt" id="M11" height="11.4781pt" version="1.1" viewBox="-0.0498162 -9.28833 22.9088 11.4781" width="22.9088pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g50-105"/></g><g transform="matrix(.013,0,0,-0.013,6.851,0)"><use xlink:href="#g50-41"/></g><g transform="matrix(.013,0,0,-0.013,11.505,0)"><use xlink:href="#g50-123"/></g><g transform="matrix(.013,0,0,-0.013,18.072,0)"><use xlink:href="#g50-42"/></g></svg> is a meromorphic function.

Nan Li

Risto Korhonen

Lianzhong Yang

Good Linear Operators and Meromorphic Solutions of Functional Equations

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Table of Contents

Table of Contents: 2015

An Optimal Fourth Order Iterative Method for Solving Nonlinear Equations and Its Dynamics

Good Linear Operators and Meromorphic Solutions of Functional Equations