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A Class of Methods for Solving Nonlinear Simultaneous Equations

A Class of Methods for Solving Nonlinear Simultaneous Equations,10.2307/2003941,Mathematics of Computation,C. G. Broyden

A Class of Methods for Solving Nonlinear Simultaneous Equations   (Citations: 453)
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1. Introduction. The solution of a set of nonlinear simultaneous equations is often the final step in the solution of practical problems arising in physics and engi- neering. These equations can be expressed as the simultaneous zeroing of a set of functions, where the number of functions to be zeroed is equal to the number of independent variables. If the equations form a sufficiently good description of a physical or engineering system, they will have a solution that corresponds to some state of this system. Although it may be impossible to prove by mathematics alone that a solution exists, this fact may be inferred from the physical analogy. Similarly, although the solution may not be unique, it is hoped that familiarity with the physi- cal analogue will enable a sufficiently good initial estimate to be found so that any iterative process that may be used will in fact converge to the physically significant solution. The functions that require zeroing are real functions of real variables and it will be assumed that they are continuous and differentiable with respect to these varia- bles. In many practical examples they are extremely complicated anld hence la- borious to compute, an-d this fact has two important immediate consequences. The first is that it is impracticable to compute any derivative that may be required by the evaluation of the algebraic expression of this derivative. If derivatives are needed they must be obtained by differencing. The second is that during any iterative solution process the bulk of the computing time will be spent in evaluating the functions. Thus, the most efficient process will tenid to be that which requires the smallest number of function evaluations. This paper discusses certain modificatioins to Newton's method designed to re- duce the number of function evaluations required. Results of various numerical experiments are given and conditions under which the modified versions are superior to the original are tentatively suggested.
Journal: Mathematics of Computation - Math. Comput. , vol. 19, no. 92, pp. 577-577, 1965
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    • ...The most successful quasi-Newton update for the nonlinear equations problem, often referred to as the secant update [1, 10], is the matrix solution to the convex optimization problem,...
    • ...The above two theorems establish that our proposed method for rectangular problems has the same convergence properties as the Broyden method in [1, 3]...
    • ...The initial guess, x0, is a random vector equally distributed on interval [0, 1]. Table 1 displays a part of the performance results of these three solvers...
    • ...The initial guess, x0 is a random vector distributed on interval [0, 1]. As we know, the minimizer of min � g(x)� 2 is x ∗ =[ 1, 1 ,..., 1, 1] T and � g(x ∗ )� 2 = 0. Similar to...
    • ...The initial guess, x0 is also chosen randomly, whose entries are distributed uniformly on interval [0, 1]. In order to increase the evaluation cost of the objective function, we ‘artificially’ compute the objective function and its corresponding derivatives ten...

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    • ...Note that (19) is the scalar formulation of the Broyden update [32]...

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    • ...We employ a version of Broyden’s method (Broyden 1965) to solve for an equilibrium...

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    • ...Aggressive SM (ASM) [2] exploits a quasi-Newton iteration with the classical Broyden formula [3] to estimate the mapping...

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