naa a22 4500 510774113 CHVBK 20180411083224.0 cr unu---uuuuu 180411e20131201xx s 000 0 eng 10.1007/s40295-015-0071-z doi (NATIONALLICENCE)springer-10.1007/s40295-015-0071-z Generalized Least Squares and Newton's Method Algorithms for Nonlinear Root-Solving Applications [Elektronische Daten] [Ahmad Younes, James Turner] Many problems in science and engineering must solve nonlinear necessary conditions. For example, a standard problem in optimization involves solving for the roots of nonlinear functions defined by f(x) = 0, where x is the unknown variable. Classically one develops a first-order Taylor series model that defines the necessary condition that must be iteratively refined. The standard assumption is that the correction terms are small. Two classes of problems arise: (1) non-square systems that lead to least-squares solutions, and (2) square systems that are often handled by Newton-like methods. The accuracy of the starting guess impacts the number of iteration cycles required. To handle more nonlinear problems, both approaches are generalized to account for first- through fourth-order approximations. Computational differentiation tools are used for automatically formulating and numerically computing the partial derivatives. Two solution approaches are presented for inverting the tensor-valued necessary condition: (1) an integrated Legendre transformation, homotopy method, and high-order vector reversion of series algorithm; and (2) a computational differentiation-based generalized linear algebra approach. Several numerical examples are presented to demonstrate generalized multilinear Least-Squares and Newton-Raphson Methods. Accelerated convergence rates are demonstrated for scalar and vector root-solving problems. The integration of generalized algorithms and automatic differentiation is expected to have broad potential for impacting the design and use of mathematical programming tools for knowledge discovery applications in science and engineering. American Astronautical Society, 2015 Least squares nationallicence Newton's method nationallicence Computational differentiation nationallicence OCEA nationallicence Younes Ahmad Aerospace Engineering, Khalifa University, Abu Dhabi, United Arab Emirates aut Turner James Aerospace Engineering, Khalifa University, Abu Dhabi, United Arab Emirates aut The Journal of the Astronautical Sciences Springer US; http://www.springer-ny.com 60/3-4(2013-12-01), 517-540 0021-9142 60:3-4<517 2013 60 40295 https://doi.org/10.1007/s40295-015-0071-z text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s40295-015-0071-z text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Younes Ahmad Aerospace Engineering, Khalifa University, Abu Dhabi, United Arab Emirates aut NATIONALLICENCE 700 1- Turner James Aerospace Engineering, Khalifa University, Abu Dhabi, United Arab Emirates aut NATIONALLICENCE 773 0- The Journal of the Astronautical Sciences Springer US; http://www.springer-ny.com 60/3-4(2013-12-01), 517-540 0021-9142 60:3-4<517 2013 60 40295 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer