{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,18]],"date-time":"2026-05-18T21:49:08Z","timestamp":1779140948685,"version":"3.51.4"},"reference-count":74,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2020,4,13]],"date-time":"2020-04-13T00:00:00Z","timestamp":1586736000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2020,4,13]],"date-time":"2020-04-13T00:00:00Z","timestamp":1586736000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2020,11]]},"DOI":"10.1007\/s11075-019-00836-1","type":"journal-article","created":{"date-parts":[[2020,4,13]],"date-time":"2020-04-13T05:02:43Z","timestamp":1586754163000},"page":"763-785","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":36,"title":["Descent Perry conjugate gradient methods for systems of monotone nonlinear equations"],"prefix":"10.1007","volume":"85","author":[{"given":"Mohammed Yusuf","family":"Waziri","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Kabiru Ahmed","family":"Hungu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jamilu","family":"Sabi\u2019u","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2020,4,13]]},"reference":[{"issue":"2","key":"836_CR1","first-page":"319","volume":"34","author":"N Andrei","year":"2011","unstructured":"Andrei, N.: Open problems in conjugate gradient algorithms for unconstrained optimization. Bull. Malays. Math. Sci. Soc. 34(2), 319\u2013330 (2011)","journal-title":"Bull. Malays. Math. Sci. Soc."},{"key":"836_CR2","first-page":"325","volume":"149-164","author":"N Andrei","year":"2017","unstructured":"Andrei, N.: Accelerated adaptive Perry conjugate gradient algorithms based on the self-scaling BFGS update. J. Comput. Appl. Math. 149-164, 325 (2017)","journal-title":"J. Comput. Appl. Math."},{"issue":"72","key":"836_CR3","doi-asserted-by":"crossref","first-page":"361","DOI":"10.3336\/gm.52.2.12","volume":"52","author":"MR Arazm","year":"2017","unstructured":"Arazm, M.R., Babaie-Kafaki, S., Ghanbari, R.: An extended Dai-Liao conjugate gradient method with global convergence for nonconvex functions. Glasnik Matematic 52(72), 361\u2013375 (2017)","journal-title":"Glasnik Matematic"},{"issue":"5","key":"836_CR4","doi-asserted-by":"crossref","first-page":"1374","DOI":"10.1016\/j.cam.2010.01.052","volume":"234","author":"S Babaie-Kafaki","year":"2010","unstructured":"Babaie-Kafaki, S., Ghanbari, R., Mahdavi-Amiri, N.: Two new conjugate gradient methods based on modified secant equations. J. Comput. Appl. Math. 234 (5), 1374\u20131386 (2010)","journal-title":"J. Comput. Appl. Math."},{"issue":"3","key":"836_CR5","doi-asserted-by":"crossref","first-page":"583","DOI":"10.1080\/10556788.2013.833199","volume":"29","author":"S Babaie-Kafaki","year":"2013","unstructured":"Babaie-Kafaki, S., Ghanbari, R.: A descent family of Dai-Liao conjugate gradient methods. Optim. Methods Softw. 29(3), 583\u2013591 (2013)","journal-title":"Optim. Methods Softw."},{"key":"836_CR6","doi-asserted-by":"crossref","first-page":"625","DOI":"10.1016\/j.ejor.2013.11.012","volume":"234","author":"S Babaie-Kafaki","year":"2014","unstructured":"Babaie-Kafaki, S., Ghanbari, R.: The Dai-Liao nonlinear conjugate gradient method with optimal parameter choices. Eur. J. Oper. Res. 234, 625\u2013630 (2014)","journal-title":"Eur. J. Oper. Res."},{"key":"836_CR7","doi-asserted-by":"crossref","first-page":"2277","DOI":"10.1080\/02331934.2014.938072","volume":"64","author":"S Babaie-Kafaki","year":"2015","unstructured":"Babaie-Kafaki, S., Ghanbari, R.: Two optimal Dai-Liao conjugate gradient methods. Optimization. 64, 2277\u20132287 (2015)","journal-title":"Optimization."},{"issue":"2014","key":"836_CR8","doi-asserted-by":"crossref","first-page":"2005","DOI":"10.1016\/j.camwa.2014.09.019","volume":"68","author":"S Babaie-Kafaki","year":"2014","unstructured":"Babaie-Kafaki, S., Ghanbari, R.: A descent extension of of the Polak-Ribier\u00e9-Polyak conjugate gradient method. Comput. Math. Appl. 68(2014), 2005\u20132011 (2014)","journal-title":"Comput. Math. Appl."},{"key":"836_CR9","first-page":"82","volume":"377-400","author":"A Bouaricha","year":"1998","unstructured":"Bouaricha, A., Schnabel, R.B.: Tensor methods for large sparse systems of nonlinear equations. Math. Program. 377-400, 82 (1998)","journal-title":"Math. Program."},{"key":"836_CR10","first-page":"19","volume":"577-593","author":"CG Broyden","year":"1965","unstructured":"Broyden, C.G.: A class of methods for solving nonlinear simultaneous equations. Math. Comput. 577-593, 19 (1965)","journal-title":"Math. Comput."},{"key":"836_CR11","first-page":"50","volume":"15-20","author":"W Cheng","year":"2009","unstructured":"Cheng, W.: A PRP type method for systems of monotone equations. Math. Comput. Modell. 15-20, 50 (2009)","journal-title":"Math. Comput. Modell."},{"issue":"1","key":"836_CR12","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1007\/s002450010019","volume":"43","author":"YH Dai","year":"2001","unstructured":"Dai, Y.H., Liao, L.Z.: New conjugacy conditions and related nonlinear conjugate gradient methods. Appl. Math. Optim. 43(1), 87\u2013101 (2001)","journal-title":"Appl. Math. Optim."},{"key":"836_CR13","volume-title":"Nonlinear Conjugate Gradient Methods","author":"YH Dai","year":"2000","unstructured":"Dai, Y.H., Yuan, Y.X.: Nonlinear Conjugate Gradient Methods. Shanghai Scientific and Technical Publishers, Shanghai (2000)"},{"key":"836_CR14","first-page":"378","volume":"270","author":"Z Dai","year":"2015","unstructured":"Dai, Z., Chen, X., Wen, F.: A modified Perrys conjugate gradient method-based derivative-free method for solving large-scale nonlinear monotone equation. Appl. Math. Comput. 270, 378\u2013386 (2015)","journal-title":"Appl. Math. Comput."},{"issue":"1","key":"836_CR15","doi-asserted-by":"crossref","first-page":"117","DOI":"10.11113\/mjfas.v15n2019.1085","volume":"15","author":"MK Dauda","year":"2019","unstructured":"Dauda, M.K., Mamat, M., Mohamed, M.A., Waziri, M.Y.: Improved quasi-Newton method via SR1 update for solving symmetric systems of nonlinear equations. Malayan J. Fund. Appl. Sci. 15(1), 117\u2013120 (2019)","journal-title":"Malayan J. Fund. Appl. Sci."},{"issue":"2, Ser. A","key":"836_CR16","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1007\/s101070100263","volume":"91","author":"ED Dolan","year":"2002","unstructured":"Dolan, E.D., Mor\u00e9, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91(2, Ser. A), 201\u20132013 (2002)","journal-title":"Math. Program."},{"key":"836_CR17","first-page":"34","volume":"343-358","author":"G Fasano","year":"2006","unstructured":"Fasano, G., Lampariello, F., Sciandrone, M.: A truncated nonmonotone Gauss-Newton method for large-scale nonlinear least-squares problems. Comput. Optim. Appl. 343-358, 34 (2006)","journal-title":"Comput. Optim. Appl."},{"issue":"2","key":"836_CR18","doi-asserted-by":"crossref","first-page":"587","DOI":"10.1007\/s10957-015-0786-9","volume":"169","author":"M Fatemi","year":"2016","unstructured":"Fatemi, M.: An optimal parameter for Dai\u2013Liao family of conjugate gradient methods. J. Optim. Theory Appl. 169(2), 587\u2013605 (2016)","journal-title":"J. Optim. Theory Appl."},{"issue":"13","key":"836_CR19","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1016\/0377-0427(94)90309-3","volume":"50","author":"JA Ford","year":"1994","unstructured":"Ford, J.A., Moghrabi, I.A.: Multi-step quasi-Newton methods for optimization. J. Comput. Appl. Math. 50(13), 305\u2013323 (1994). 66(12), 201\u2013211(1996)","journal-title":"J. Comput. Appl. Math."},{"issue":"2","key":"836_CR20","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1007\/s10589-007-9087-z","volume":"40","author":"JA Ford","year":"2008","unstructured":"Ford, J.A., Narushima, Y., Yabe, H.: Multi-step nonlinear conjugate gradient methods for unconstrained minimization. Comput. Optim. Appl. 40(2), 191\u2013216 (2008)","journal-title":"Comput. Optim. Appl."},{"key":"836_CR21","first-page":"23","volume":"707-716","author":"L Grippo","year":"1986","unstructured":"Grippo, L., Lampariello, F., Lucidi, S.: A nonmonotone linesearch technique for Newtons method. SIAM J. Numer. Anal. 707-716, 23 (1986)","journal-title":"SIAM J. Numer. Anal."},{"issue":"1","key":"836_CR22","first-page":"35","volume":"2","author":"WW Hager","year":"2006","unstructured":"Hager, W.W., Zhang, H.: A survey of nonlinear conjugate gradient methods. Pac. J. Optim. 2(1), 35\u201358 (2006)","journal-title":"Pac. J. Optim."},{"key":"836_CR23","doi-asserted-by":"crossref","first-page":"409","DOI":"10.6028\/jres.049.044","volume":"49","author":"MR Hestenes","year":"1952","unstructured":"Hestenes, M.R., Stiefel, E.L.: Methods of conjugate gradients for solving linear systems. J. Res. Nat. Bur. Stand. 49, 409\u2013436 (1952)","journal-title":"J. Res. Nat. Bur. Stand."},{"key":"836_CR24","doi-asserted-by":"crossref","first-page":"375","DOI":"10.1016\/j.cam.2004.02.013","volume":"172","author":"C Kanzow","year":"2004","unstructured":"Kanzow, C., Yamashita, N., Fukushima, M.: Levenberg-Marquardt methods for constrained nonlinear equations with strong local convergence properties. J. Comput. Appl. Math. 172, 375\u2013397 (2004)","journal-title":"J. Comput. Appl. Math."},{"key":"836_CR25","doi-asserted-by":"publisher","unstructured":"Khoshgam, Z., Ashrafi, A.: A new modidified scaled conjugate gradient method for large-scale unconstrained optimization with non-convex objective function. Optimization Methods and Software, https:\/\/doi.org\/10.1080\/10556788.2018.1457152 (2018)","DOI":"10.1080\/10556788.2018.1457152"},{"key":"836_CR26","volume-title":"Numerical Analysis","author":"D Kincaid","year":"1991","unstructured":"Kincaid, D., Cheney, W.: Numerical Analysis. Brooks\/Cole Publishing Company, California (1991)"},{"issue":"3","key":"836_CR27","doi-asserted-by":"crossref","first-page":"755","DOI":"10.1007\/s41980-018-0163-1","volume":"45","author":"M Koorapetse","year":"2018","unstructured":"Koorapetse, M., Kaelo, P., Offen, E.R.: A scaled derivative-free projection method for solving nonlinear monotone equations. Bullet. Iran. Math. Soc. 45(3), 755\u2013770 (2018)","journal-title":"Bullet. Iran. Math. Soc."},{"key":"836_CR28","first-page":"2","volume":"164-166","author":"K Levenberg","year":"1944","unstructured":"Levenberg, K.: A method for the solution of certain non-linear problems in least squares. Q. Appl. Math. 164-166, 2 (1944)","journal-title":"Q. Appl. Math."},{"issue":"1","key":"836_CR29","first-page":"152","volume":"37","author":"DH Li","year":"2000","unstructured":"Li, D.H., Fukushima, M.: A globally and superlinearly convergent Gauss-Newton-based BFGS method for symmetric nonlinear equations. SIAM J. Numer. Anal. 37(1), 152\u2013172 (2000)","journal-title":"SIAM J. Numer. Anal."},{"key":"836_CR30","first-page":"13","volume":"583-599","author":"DH Li","year":"2000","unstructured":"Li, D.H., Fukushima, M.: A derivative-free linesearch and global convergence of Broyden-like method for nonlinear equations. Optim. Methods Softw. 583-599, 13 (2000)","journal-title":"Optim. Methods Softw."},{"issue":"2","key":"836_CR31","doi-asserted-by":"crossref","first-page":"523","DOI":"10.1016\/j.cam.2006.03.005","volume":"202","author":"G Li","year":"2007","unstructured":"Li, G., Tang, C., Wei, Z.: New conjugacy condition and related new conjugate gradient methods for unconstrained optimization. J. Comput. Appl. Math. 202(2), 523\u2013539 (2007)","journal-title":"J. Comput. Appl. Math."},{"key":"836_CR32","doi-asserted-by":"crossref","unstructured":"Liu, D.Y., Shang, Y.F.: A new Perry conjugate gradient method with the generalized conjugacy condition. In: 2010 International Conference on Issue Computational Intelligence and Software Engineering (CiSE) (2010)","DOI":"10.1109\/CISE.2010.5677114"},{"key":"836_CR33","doi-asserted-by":"crossref","unstructured":"Liu, D.Y., Xu, G.Q.: A Perry descent conjugate gradient method with restricted spectrum. Optimization Online, Nonlinear Optimization (unconstrained optimization), pp. 1\u201319 (2011)","DOI":"10.1186\/1029-242X-2011-57"},{"key":"836_CR34","doi-asserted-by":"publisher","unstructured":"Liu, J. K., Feng, Y. M.: A norm descent derivative-free algorithm for solving large-scale nonlinear symmetric equations. https:\/\/doi.org\/10.1016\/j.cam.2018.05.006(2017)","DOI":"10.1016\/j.cam.2018.05.006"},{"issue":"10","key":"836_CR35","doi-asserted-by":"crossref","first-page":"2442","DOI":"10.1016\/j.camwa.2015.09.014","volume":"70","author":"JK Liu","year":"2015","unstructured":"Liu, J.K., Li, S.J.: A projection method for convex constrained monotone nonlinear equationswith applications. Comput. Math. Appl. 70(10), 2442\u20132453 (2015)","journal-title":"Comput. Math. Appl."},{"key":"836_CR36","first-page":"9197","volume":"218","author":"IE Livieris","year":"2012","unstructured":"Livieris, I.E., Pintelas, P.: Globally convergent modified Perrys conjugate gradient method. Appl. Math. Comput. 218, 9197\u20139207 (2012)","journal-title":"Appl. Math. Comput."},{"key":"836_CR37","doi-asserted-by":"crossref","first-page":"396","DOI":"10.1016\/j.cam.2012.09.007","volume":"239","author":"IE Livieris","year":"2013","unstructured":"Livieris, I.E., Pintelas, P.: A new class of spectral conjugate gradient methods based on a modified secant equation for unconstrained optimization. J. Comput. Appl. Math. 239, 396\u2013405 (2013)","journal-title":"J. Comput. Appl. Math."},{"key":"836_CR38","doi-asserted-by":"crossref","unstructured":"Livieris, I.E., Pintelas, P.: A descent Dai-Liao conjugate gradient method based on a modified secant equation and its global convergence. ISRN Computational Mathematics (2012)","DOI":"10.5402\/2012\/435495"},{"issue":"2013","key":"836_CR39","first-page":"491","volume":"221","author":"IE Livieris","year":"2013","unstructured":"Livieris, I.E., Pintelas, P.: A new conjugate gradient algorithm for training neural networks based on a modified secant equation. Appl. Math. Comput. 221 (2013), 491\u2013502 (2013)","journal-title":"Appl. Math. Comput."},{"key":"836_CR40","doi-asserted-by":"crossref","first-page":"431","DOI":"10.1137\/0111030","volume":"11","author":"DW Marquardt","year":"1963","unstructured":"Marquardt, D.W.: An algorithm for least-squares estimation of nonlinear parameters. SIAM J. Appl. Math. 11, 431\u2013441 (1963)","journal-title":"SIAM J. Appl. Math."},{"issue":"4","key":"836_CR41","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1007\/s40065-018-0206-8","volume":"7","author":"S Mompati","year":"2018","unstructured":"Mompati, S., Koorapetse, M., Kaelo, P.: Globally convergent three-term conjugate gradient projection methods for solving nonlinear monotone equations. Arabian J. Math. 7(4), 289\u2013301 (2018)","journal-title":"Arabian J. Math."},{"key":"836_CR42","doi-asserted-by":"crossref","DOI":"10.1007\/b98874","volume-title":"Numerical Optimization","author":"J Nocedal","year":"1999","unstructured":"Nocedal, J., Wright, S. J.: Numerical Optimization. Springer, New York (1999)"},{"key":"836_CR43","volume-title":"Iterative Solution of Nonlinear Equations in Several Variables","author":"JM Ortega","year":"1970","unstructured":"Ortega, J. M., Rheinboldt, W. C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York (1970)"},{"key":"836_CR44","doi-asserted-by":"crossref","unstructured":"Peiting, G., Chuanjiang, H.: A derivative-free three-term projection algorithm involving spectral quotient for solving nonlinear monotone equations. Optimization. A Journal of Mathematical Programming and Operations Research, pp. 1\u201318 (2018)","DOI":"10.1080\/02331934.2018.1482490"},{"issue":"6","key":"836_CR45","doi-asserted-by":"crossref","first-page":"1073","DOI":"10.1287\/opre.26.6.1073","volume":"26","author":"A Perry","year":"1978","unstructured":"Perry, A.: A modified conjugate gradient algorithm. Oper. Res. Tech. Notes 26(6), 1073\u20131078 (1978)","journal-title":"Oper. Res. Tech. Notes"},{"key":"836_CR46","first-page":"4","volume":"94-112","author":"BT Polak","year":"1969","unstructured":"Polak, B.T.: The conjugate gradient method in extreme problems. USSR Comput. Math. Math. Phys. 94-112, 4 (1969)","journal-title":"USSR Comput. Math. Math. Phys."},{"key":"836_CR47","first-page":"35","volume":"16","author":"E Polak","year":"1969","unstructured":"Polak, E., Ribire, G.: Note sur la convergence de mthodes de directions conjugues. Rev. Fr. Inform. Rech. Oper. 16, 35\u201343 (1969)","journal-title":"Rev. Fr. Inform. Rech. Oper."},{"issue":"3","key":"836_CR48","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1080\/02331939708844339","volume":"41","author":"VM Solodov","year":"1997","unstructured":"Solodov, V.M., Iusem, A.N.: Newton-type methods with generalized distances for constrained optimization. Optimization 41(3), 257\u201327 (1997)","journal-title":"Optimization"},{"key":"836_CR49","doi-asserted-by":"crossref","unstructured":"Solodov, M.V., Svaiter, B.F.: A globally convergent inexact Newton method for systems of monotone equations. In: Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, pp. 355\u2013369. Springer (1998)","DOI":"10.1007\/978-1-4757-6388-1_18"},{"key":"836_CR50","first-page":"1","volume":"236","author":"M Sun","year":"2017","unstructured":"Sun, M., Wang, X., Feng, D.: A family of conjugate gradient methods for large-scale nonlinear equations. J. Inequal. Appl. 236, 1\u20138 (2017)","journal-title":"J. Inequal. Appl."},{"key":"836_CR51","volume-title":"Optimization Theory and Methods: Nonlinear Programming","author":"W Sun","year":"2006","unstructured":"Sun, W., Yuan, Y. X.: Optimization Theory and Methods: Nonlinear Programming. Springer, NewYork (2006)"},{"issue":"1","key":"836_CR52","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1023\/B:JOTA.0000043997.42194.dc","volume":"123","author":"XJ Tong","year":"2004","unstructured":"Tong, X.J., Qi, L.: On the convergence of a trust-region method for solving constrained nonlinear equations with degenerate solutions. J. Optim. Theory Appl. 123(1), 187\u2013211 (2004)","journal-title":"J. Optim. Theory Appl."},{"key":"836_CR53","doi-asserted-by":"publisher","unstructured":"Waziri, M.Y., Ahmed, K., Sabiu, J.: A Dai-Liao conjugate gradient method via modified secant equation for system of nonlinear equations. Arabian Journal of Mathematics. https:\/\/doi.org\/10.1007\/s40065-019-0264-6, 1\u201315 (2019)","DOI":"10.1007\/s40065-019-0264-6"},{"key":"836_CR54","first-page":"645","volume":"361","author":"MY Waziri","year":"2019","unstructured":"Waziri, M. Y., Ahmed, K., Sabiu, J.: A family of Hager-Zhang conjugate gradient methods for system of monotone nonlinear equations. Appl. Math. Comput. 361, 645\u2013660 (2019)","journal-title":"Appl. Math. Comput."},{"issue":"2","key":"836_CR55","first-page":"241","volume":"5","author":"MY Waziri","year":"2011","unstructured":"Waziri, M.Y., Leong, W.J., Hassan, M.A.: Jacobian free-diagonal Newton\u2019s method for nonlinear systems with singular Jacobian. Malaysian J. Math. Sci. 5(2), 241\u2013255 (2011)","journal-title":"Malaysian J. Math. Sci."},{"key":"836_CR56","doi-asserted-by":"crossref","unstructured":"Waziri, M.Y., Sabiu, J.: A Derivative-free conjugate gradient method and its global convergence for solving symmetric nonlinear equations. Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences, Vol. 2015, pp. 8","DOI":"10.1155\/2015\/961487"},{"key":"836_CR57","doi-asserted-by":"crossref","unstructured":"Wei, Z., Li, G., Qi, L.: New quasi-Newton methods for unconstrained optimization problems, vol. 175 (2006)","DOI":"10.1016\/j.amc.2005.08.027"},{"key":"836_CR58","doi-asserted-by":"crossref","first-page":"310","DOI":"10.1016\/j.jmaa.2013.04.017","volume":"405","author":"Y Xiao","year":"2013","unstructured":"Xiao, Y., Zhu, H.: A conjugate gradient method to solve convex constrained monotone equations with applications in compressive sensing. J. Math. Anal. Appl. 405, 310\u2013319 (2013)","journal-title":"J. Math. Anal. Appl."},{"issue":"2","key":"836_CR59","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1023\/B:COAP.0000026885.81997.88","volume":"28","author":"H Yabe","year":"2004","unstructured":"Yabe, H., Takano, M.: Global convergence properties of nonlinear conjugate gradient methods with modified secant condition. Comput. Optim. Appl. 28(2), 203\u2013225 (2004)","journal-title":"Comput. Optim. Appl."},{"key":"836_CR60","doi-asserted-by":"crossref","first-page":"649","DOI":"10.1016\/j.cam.2010.01.001","volume":"234","author":"QR Yan","year":"2010","unstructured":"Yan, Q.R., Peng, X.Z., Li, D.H.: A globally convergent derivative-free method for solving large-scale nonlinear monotone equations. J. Comput. Appl. Math. 234, 649\u2013657 (2010)","journal-title":"J. Comput. Appl. Math."},{"key":"836_CR61","doi-asserted-by":"crossref","first-page":"149","DOI":"10.3934\/jimo.2010.6.149","volume":"6","author":"G Yu","year":"2010","unstructured":"Yu, G.: A derivative-free method for solving large-scale nonlinear systems of equations. J. Ind. Manag. Optim. 6, 149\u2013160 (2010)","journal-title":"J. Ind. Manag. Optim."},{"key":"836_CR62","first-page":"387","volume":"7","author":"G Yu","year":"2011","unstructured":"Yu, G.: Nonmonotone spectral gradient-type methods for large-scale unconstrained optimization and nonlinearsystems of equations. Pac. J. Optim. 7, 387\u2013404 (2011)","journal-title":"Pac. J. Optim."},{"key":"836_CR63","doi-asserted-by":"crossref","first-page":"325","DOI":"10.1093\/imanum\/11.3.325","volume":"11","author":"YX Yuan","year":"1991","unstructured":"Yuan, Y.X.: A modified BFGS algorithm for unconstrained optimization. IMA J. Numer. Anal. 11, 325\u2013332 (1991)","journal-title":"IMA J. Numer. Anal."},{"key":"836_CR64","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1007\/s11081-008-9064-0","volume":"10","author":"Y Yuan","year":"2009","unstructured":"Yuan, Y.: Subspace methods for large scale nonlinear equations and nonlinear least squares. Optim. Eng. 10, 207\u2013218 (2009)","journal-title":"Optim. Eng."},{"issue":"4","key":"836_CR65","doi-asserted-by":"crossref","first-page":"317","DOI":"10.1007\/s00607-011-0146-z","volume":"92","author":"GL Yuan","year":"2011","unstructured":"Yuan, G.L., Wei, Z.X., Lu, X.W.: A BFGS trust-region method for nonlinear equations. Computing 92(4), 317\u2013333 (2011)","journal-title":"Computing"},{"key":"836_CR66","doi-asserted-by":"crossref","first-page":"186","DOI":"10.1016\/j.cam.2015.03.014","volume":"286","author":"G Yuan","year":"2015","unstructured":"Yuan, G., Zhang, M.: A three term Polak-Ribiere-Polyak conjugate gradient algorithm for large-scale nonlinear equations. J. Comput. Appl. Math. 286, 186\u2013195 (2015)","journal-title":"J. Comput. Appl. Math."},{"issue":"1","key":"836_CR67","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1023\/A:1021898630001","volume":"102","author":"JZ Zhang","year":"1999","unstructured":"Zhang, J.Z., Deng, N.Y., Chen, L.H.: New quasi-Newton equation and related methods for unconstrained optimization. J. Optim. Theory Appl. 102(1), 147\u2013157 (1999)","journal-title":"J. Optim. Theory Appl."},{"issue":"2","key":"836_CR68","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1016\/S0377-0427(00)00713-5","volume":"137","author":"J Zhang","year":"2001","unstructured":"Zhang, J., Xu, C.: Properties and numerical performance of quasi-Newton methods with modified quasi-Newton equations. J. Comput. Appl. Math. 137(2), 269\u2013278 (2001)","journal-title":"J. Comput. Appl. Math."},{"key":"836_CR69","first-page":"58","volume":"283-298","author":"J Zhang","year":"2003","unstructured":"Zhang, J., Wang, Y.: A new trust region method for nonlinear equations. Math. Methods Oper. Res. 283-298, 58 (2003)","journal-title":"Math. Methods Oper. Res."},{"key":"836_CR70","doi-asserted-by":"crossref","first-page":"962","DOI":"10.1137\/S1052623499357957","volume":"11","author":"YB Zhao","year":"2001","unstructured":"Zhao, Y.B., Li, D.: Monotonicity of fixed point and normal mappings associated with variational inequality and its application. SIAM J. Optim. 11, 962\u2013973 (2001)","journal-title":"SIAM J. Optim."},{"issue":"1","key":"836_CR71","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1007\/s10957-014-0547-1","volume":"164","author":"W Zhou","year":"2015","unstructured":"Zhou, W., Shen, D.: Convergence properties of an iterative method for solving symmetric non-linear equations. J. Optim. Theory Appl. 164(1), 277\u2013289 (2015)","journal-title":"J. Optim. Theory Appl."},{"issue":"1","key":"836_CR72","first-page":"89","volume":"25","author":"W Zhou","year":"2007","unstructured":"Zhou, W., Li, D.: Limited memory bfgs method for nonlinear monotone equations. J. Comput. Math. 25(1), 89\u201396 (2007)","journal-title":"J. Comput. Math."},{"key":"836_CR73","first-page":"1","volume":"261","author":"W Zhou","year":"2015","unstructured":"Zhou, W., Wang, F.: A PRP-based residual method for large-scale monotone nonlinear equations. Appl. Math. Comput. 261, 1\u20137 (2015)","journal-title":"Appl. Math. Comput."},{"issue":"5","key":"836_CR74","doi-asserted-by":"crossref","first-page":"707","DOI":"10.1080\/10556780500137041","volume":"21","author":"W Zhou","year":"2006","unstructured":"Zhou, W., Zhang, L.: A nonlinear conjugate gradient method based on the MBFGS secant condition. Optim. Methods Softw. 21(5), 707\u2013714 (2006)","journal-title":"Optim. Methods Softw."}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-019-00836-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11075-019-00836-1\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-019-00836-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,10,21]],"date-time":"2022-10-21T03:44:20Z","timestamp":1666323860000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11075-019-00836-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,4,13]]},"references-count":74,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2020,11]]}},"alternative-id":["836"],"URL":"https:\/\/doi.org\/10.1007\/s11075-019-00836-1","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,4,13]]},"assertion":[{"value":"9 April 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"22 October 2019","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 April 2020","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Compliance with ethical standards"}},{"value":"The authors declare that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"<!--Emphasis Type='Bold' removed-->Conflict of interest"}}]}}