Broyden's quasi-Newton methods for a nonlinear system of equations and unconstrained optimization: a review and open problems

Al-Baali, Mehiddin; Spedicato, Emilio; Maggioni, Francesca

doi:10.1080/10556788.2013.856909

Cited by 53 publications

(32 citation statements)

References 59 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We briefly summarize the specific version of Broyden's method which will be the basis of our algorithm on. We use a damped version of Broyden's method, as described e.g., in [1,Section 7]. The derivation follows from the Newton-like update equation By combining (2.1), (2.3) and (2.4), it is clear that z k+1 can be directly computed from…”

Section: Background and Basic Algorithmmentioning

confidence: 99%

Broyden's Method for Nonlinear Eigenproblems

Jarlebring¹

2019

SIAM J. Sci. Comput.

View full text Add to dashboard Cite

Broyden's method is a general method commonly used for nonlinear systems of equations, when very little information is available about the problem. We develop an approach based on Broyden's method for nonlinear eigenvalue problems. Our approach is designed for problems where the evaluation of a matrix vector product is computationally expensive, essentially as expensive as solving the corresponding linear system of equations. We show how the structure of the Jacobian matrix can be incorporated into the algorithm to improve convergence. The algorithm exhibits local superlinear convergence for simple eigenvalues, and we characterize the convergence. We show how deflation can be integrated and combined such that the method can be used to compute several eigenvalues. A specific problem in machine tool milling, coupled with a PDE is used to illustrate the approach. The simulations are done in the julia programming language, and are provided as publicly available module for reproducability.(2.1)where the next approximation is computed with a damped update equationThe choice of the damping parameter γ k will be tuned to our setting, essentially to avoid taking too large steps (as we shall further describe in Remark 3.3). The next matrix J k+1 will satisfy (what is commonly called) the secant conditionwhere J k+1 is a rank-one modification of J k . We will focus on updates of the form,(2.4) J k+1 = J k + 1 ∆x k 2 z k+1 ∆x H .

show abstract

Section: Background and Basic Algorithmmentioning

confidence: 99%

Broyden's Method for Nonlinear Eigenproblems

Jarlebring¹

2019

SIAM J. Sci. Comput.

View full text Add to dashboard Cite

show abstract

“…The method based on the inverse Jacobian, Equation (22), is usually called the "bad" Broyden method because in its original applications, it did not perform as well as the "good" Broyden update, Equation (21) [66,[68][69][70][71][72]. It is more convenient to have an update to the inverse Jacobian than the Jacobian itself since (cf.…”

Section: Methodsmentioning

confidence: 99%

A Diagonally Updated Limited-Memory Quasi-Newton Method for the Weighted Density Approximation

et al. 2017

View full text Add to dashboard Cite

Abstract:We propose a limited-memory quasi-Newton method using the bad Broyden update and apply it to the nonlinear equations that must be solved to determine the effective Fermi momentum in the weighted density approximation for the exchange energy density functional. This algorithm has advantages for nonlinear systems of equations with diagonally dominant Jacobians, because it is easy to generalize the method to allow for periodic updates of the diagonal of the Jacobian. Systematic tests of the method for atoms show that one can determine the effective Fermi momentum at thousands of points in less than fifteen iterations.

show abstract

“…One of the famous method for solving is the Newton method, which generates a sequence { x k } using the formula

x_{k + 1} = x_{k} - J {(x_{k})}^{- 1} F (x_{k}), \forall k = 0, 1, 2, … .

The Newton method has a very good convergence property but have some shortcomings such as Jacobian computation per iteration, which is costly. Other methods for solving are quasi Newton methods, spectral gradient methods, conjugate gradient methods, etc . For simplicity, we denote F k = F ( x k ) and J k = J ( x k ).…”

Section: Introductionmentioning

confidence: 99%

“…Other methods for solving (1) are quasi Newton methods, spectral gradient methods, conjugate gradient methods, etc. [2][3][4][5][6][7][8][9] For simplicity, we denote F k = F(x k ) and J k = J(x k ). Methods for solving symmetric nonlinear problem (1) has been proposed by many authors.…”

mentioning

confidence: 99%

See 1 more Smart Citation

An inexact conjugate gradient method for symmetric nonlinear equations

Abubakar

Awwal

2019

Comp and Math Methods

View full text Add to dashboard Cite

In this article, we present a conjugate gradient method for large‐scale nonlinear equations with symmetric Jacobian. The method is a modification of the descent Dai‐Liao conjugate gradient method for unconstrained optimization problem proposed by Kafaki and Gambari. Under some assumptions, which include symmetric property of the Jacobian, we establish the global convergence of the proposed method and, finally, some numerical results to show its efficiency.

show abstract

Broyden's quasi-Newton methods for a nonlinear system of equations and unconstrained optimization: a review and open problems

Cited by 53 publications

References 59 publications

Broyden's Method for Nonlinear Eigenproblems

Broyden's Method for Nonlinear Eigenproblems

A Diagonally Updated Limited-Memory Quasi-Newton Method for the Weighted Density Approximation

An inexact conjugate gradient method for symmetric nonlinear equations

Contact Info

Product

Resources

About