Iterative solution of nonlinear equations in several variables

2014

4,184 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Numerical Analysis

Citations in the corpus

• Group Knowledge Transfer: Collaborative Training of Large CNNs on the Edge (2020, background): “The essence of GKT is that it reformulates FL as an alternating minimization (AM) approach [14, 15, 16, 17, 18, 19], which optimizes two random variables (the edge model and the server model) by alternatively fixing one and optimizing another.”
• New Insights and Perspectives on the Natural Gradient Method (2014, methodology): “Second, as we establish in Section 9, it is very often equivalent to the so-called “Generalized Gauss-Newton matrix” (GGN) (discussed in Section 8), a generalization of the classical Gauss-Newton matrix (e.g. Dennis Jr and Schnabel, 1996; Ortega and Rheinboldt, 2000; Nocedal and Wright, 2006), which”
• Coordinate descent algorithms (2015, methodology): “The 1970 text of Ortega and Rheinboldt [40, Section 14.6] included a comprehensive discussion of “univariate relaxation,” and such optimization specialists as Luo and Tseng [30,31], Tseng [55], and Bertsekas and Tsitsiklis [5] made important contributions to understanding the convergence properties…”
• A New Conjugate Gradient Method with Guaranteed Descent and an Efficient Line Search (2005, neutral): “Since the root convergence order [29] of the secant method is (1 + √ 5)/2, the order of convergence for a double secant step is (1 + √ 5)(2)/4.”
• Some derivative free quadratic and cubic convergence iterative formulas for solving nonlinear equations (2010, background): “1 Introduction A large number of papers have been written about iterative methods for the solution of the the nonlinear equations [3, 7, 8, 9, 10, 12, 13].”
• GEE analysis of clustered binary data with diverging number of covariates (2011, methodology): “…(1984), this technique was applied to establish the existence and consistency of an M-estimator for i.i.d. k k This approach follows from Theorem 6.3.4 of Ortega and Rheinboldt (1970). data; in a different setting, it was used by Wang et al. (2010) to study a partial linear single-index model.”
• Recurrence Relations for Chebyshev-Type Methods (2000, methodology): “Basic results concerning convergence of the Newton method, existence and uniqueness of solutions, and error estimates are given by Kantorovich-type theorems (see [5], [6], [8], and [9]).”
• Quasi-Newton Methods, Motivation and Theory (1974, neutral): “This update was proposed by Greenstadt (1970), but it has not received any more attention in the literature since it does not perform as well as the PSB update (7.13). It is interesting that the underlying single rank method was obtained by Broyden (1965), but that this update has also been neglecte”
• Introduction to Numerical Continuation Methods (1987, methodology): “Full theoretical discussions of Newton’s method (3.2.6) using the MoorePenrose inverse can be found in several text books e.g. Ortega & Rheinboldt (1970) or Ben-Israel & Greville (1974).”
• A distributed block coordinate descent method for training $l_1$ regularized linear classifiers (2014, background): “See chapter 9 of [12] for definitions of Q-linear and R-linear convergence.”
• On the Convergence of Alternating Minimization for Convex Programming with Applications to Iteratively Reweighted Least Squares and Decomposition Schemes (2015, methodology): “The AM method in its general sense, that is, when the decision variables vector is decomposed into p subblocks (p being an integer greater than one), is a rather old and fundamental algorithm [6, 24].”
• ON ITERATIVE IMPES FORMULATION FOR TWO-PHASE FLOW WITH CAPILLARITY IN HETEROGENEOUS POROUS MEDIA (2010, positive): “It is obtained from (50) that [57] ‖Φc(Sw)− Φc(Sw)− Φ′c(Sw)(Sw − Sw)‖ ≤ 1 2 Cpc‖Sw − Sw‖(2) ≤ 1 2 Cpc‖Sw − Sw‖.”
• Finite element algorithms for thermoelastic wear problems (2002, methodology): “Instead of solving (36) directly using Algorithm MBN, one can adopt an iterative strategy, which from an algorithmic point of view is a method of Gauss–Seidel type, see, e.g., (Ortega and Reinboldt, 1970).”
• An Inexact Cayley Transform Method For Inverse Eigenvalue Problems (2004, methodology): “In the following, we show that the root-convergence rate of our method is at least β. Here, we recall the definition of root-convergence, see [ 29, chapter 9 ].”
• Coverage Optimization and Spatial Load Balancing by Robotic Sensor Networks (2010, background): “root-convergence factor [24], convergence is guaranteed if changes in occur between increasingly longer time instants.”
• The random-utility-based multiregional input-output model: solution existence and uniqueness (2004, methodology): “5 According to the Norm-Equivalence Theorem (see Ortege and Rheinboldt, 1970), all norms on R n are equivalent.”