An Analytical Study on Two High-Order Hybrid Methods to Solve Systems of Nonlinear Equations

<table class="table-group" id="tab4"><tr><td><table class="table"><tr><td class="thead-hr" colspan="3"><hr/></td></tr><tr class="thead"><td class="align_left">Methods</td><td class="align_center">Error</td><td class="align_center">CPU time (s)</td></tr><tr><td class="thead-hr" colspan="3"><hr/></td></tr><tr><td class="align_left">SORCL1</td><td class="align_center">8.38<i>e − </i>13</td><td class="align_center">601.4073</td></tr><tr><td class="align_left">SORCL2</td><td class="align_center">8.40<i>e − </i>13</td><td class="align_center">664.9404</td></tr><tr><td class="align_left">ode15s</td><td class="align_center">2.48<i>E − </i>13</td><td class="align_center">910.9302</td></tr><tr><td class="align_left">Newton</td><td class="align_center">—</td><td class="align_center">—</td></tr><tr><td class="align_left">VIM</td><td class="align_center">—</td><td class="align_center">—</td></tr><tr><td class="align_left">MVIM</td><td class="align_center">—</td><td class="align_center">—</td></tr><tr><td class="align_left">LADM</td><td class="align_center">—</td><td class="align_center">—</td></tr><tr><td class="align_left">LADM Padè</td><td class="align_center">—</td><td class="align_center">—</td></tr><tr class="table-tr"><td colspan="3"><hr class="tbody-hr"/></td></tr></table></td></tr></table>

<div>Numerical comparison different methods, with <svg height="8.55521pt" id="M452" style="vertical-align:-0.2063904pt" version="1.1" viewbox="-0.0498162 -8.34882 38.8499 8.55521" width="38.8499pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M615 290C615 392 563 448 529 448C511 448 491 441 484 434L483 425C518 400 546 349 546 273C546 158 504 46 417 46C381 46 356 77 356 142C356 184 380 293 392 334L387 340L307 323C305 288 296 228 287 188C265 94 221 52 180 52C144 52 112 89 112 171C112 278 164 372 241 430L226 454C106 394 23 288 23 155C23 40 79 -12 139 -12C184 -12 245 32 285 85C292 31 322 -12 376 -12C423 -12 470 18 501 45C570 104 615 184 615 290Z"></path></g><g transform="matrix(.013,0,0,-0.013,11.927,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,23.19,0)"><path d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z"></path></g><g transform="matrix(.013,0,0,-0.013,29.43,0)"><path d="M113 -12C146 -12 170 11 170 46C170 78 146 103 114 103S58 78 58 46C58 11 82 -12 113 -12Z"></path></g><g transform="matrix(.013,0,0,-0.013,32.394,0)"><path d="M244 635C114 635 38 519 38 422C38 317 111 240 217 240C236 240 255 244 277 256L345 292C311 140 203 39 59 15L64 -15C89 -15 150 -5 204 17C339 72 440 202 440 386C440 521 368 635 244 635ZM228 602C326 602 352 479 352 390C352 370 351 347 348 324C327 308 293 296 258 296C174 296 124 369 124 458C124 517 152 602 228 602Z"></path></g></svg> and <svg height="11.439pt" id="M453" style="vertical-align:-2.15067pt" version="1.1" viewbox="-0.0498162 -9.28833 39.4241 11.439" width="39.4241pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M475 507C475 612 440 712 326 712C139 712 23 420 23 215C23 96 58 -12 180 -12C369 -12 475 293 475 507ZM391 522C391 486 387 448 379 394H126C155 538 222 677 310 677C386 677 391 571 391 522ZM373 346C344 193 283 22 189 22C126 22 106 114 106 196C106 243 111 293 118 346H373Z"></path></g><g transform="matrix(.013,0,0,-0.013,10.107,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,21.37,0)"><path d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z"></path></g><g transform="matrix(.013,0,0,-0.013,27.61,0)"><path d="M368 703H309L44 -163H104L368 703Z"></path></g><g transform="matrix(.013,0,0,-0.013,32.971,0)"><path d="M412 140C382 77 369 73 315 73H129L270 222C362 320 402 379 402 466C402 571 322 635 234 635C177 635 130 609 99 576L42 495L64 475C90 514 133 568 201 568C274 568 318 519 318 435C318 349 255 267 193 193C144 135 87 78 32 23V0H405C417 45 427 89 440 131L412 140Z"></path></g></svg> for SORCL1 and SORCL2 to solve Problem <a href="https://static.hindawi.com/articles/jmath/volume-2023/9917774/figures/#problem1" target="_blank">1</a>.</div>

Journal of Mathematics

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Table 4

Table 4: An Analytical Study on Two High-Order Hybrid Methods to Solve Systems of Nonlinear Equations 