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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
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A Class of Methods for Solving Nonlinear Simultaneous Equations
(
Citations: 453
)
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C. G. Broyden
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, and 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. 577577, 1965
DOI:
10.2307/2003941
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Citation Context
(169)
...The first secant update for the nonlinear equation problem was proposed by Broyden in 1965
2
as Immediately, researchers considered the general class of rank1 updates given by the formula for any
Joanna M. Papakonstantinou
,
et al.
Generation of classes of symmetric rank2 secant updates and the maxim...
...The most successful quasiNewton 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...
Wei Xu
,
et al.
A secant method for nonlinear leastsquares minimization
...Note that (19) is the scalar formulation of the Broyden update [
32
]...
Tobias Wartzek
,
et al.
Triboelectricity in Capacitive Biopotential Measurements
...We employ a version of Broyden’s method (
Broyden 1965
) to solve for an equilibrium...
John Duggan
,
et al.
A Newton collocation method for solving dynamic bargaining games
...Aggressive SM (ASM) [2] exploits a quasiNewton iteration with the classical Broyden formula [
3
] to estimate the mapping...
Ali Khalatpour
,
et al.
Accelerating Space Mapping Optimization with Adjoint Sensitivities
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Citations
(453)
Generation of classes of symmetric rank2 secant updates and the maximality of the Davidon class
Joanna M. Papakonstantinou
,
Richard A. Tapia
Journal:
Optimization Methods & Software  OPTIM METHOD SOFTW
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A secant method for nonlinear leastsquares minimization
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