Abstract 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.
Generalized Least Squares and Newton’s Method Algorithms for Nonlinear Root-Solving Applications
The Journal of the Astronautical Sciences ; 60 , 3 ; 517-540
2013-12-01
24 pages
Article (Journal)
Electronic Resource
English
Generalized Least Squares and Newton’s Method Algorithms for Nonlinear Root-Solving Applications
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