Direct Secant Updates of Matrix Factorizations

Dennis, J. E.; Marwil, Earl S.

doi:10.2307/2007282

Cited by 8 publications

(7 citation statements)

References 5 publications

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“…Direct update of factorizations. An algorithm of this type (e.g., [17,20]) implicitly assumes the Jacobian being factorized as a product of two matrices (typically LU factorization), and updates the two matrices separately to satisfy the secant condition at each iteration. The sparsity of the approximate Jacobian or its factorization is sometimes assumed and taken into account for efficiency.…”

Section: Broyden's Familymentioning

confidence: 99%

Two classes of multisecant methods for nonlinear acceleration

Fang

Saad

2008

Numerical Linear Algebra App

261

264

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Many applications in science and engineering lead to models which require solving large-scale fixed point problems, or equivalently, systems of nonlinear equations. Several successful techniques for handling such problems are based on quasi-Newton methods that implicitly update the approximate Jacobian or inverse Jacobian to satisfy a certain secant condition.We present two classes of multisecant methods which allows to take into account a variable number of secant equations at each iteration. The first is the Broyden-like class, of which Broyden's family is a subclass, and Anderson mixing is a particular member. The second class is that of the nonlinear Eirola-Nevanlinna-type methods.This work was motivated by a problem in electronic structure calculations, whereby a fixed point iteration, known as the self-consistent field (SCF) iteration, is accelerated by various strategies termed 'mixing'.

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Section: Broyden's Familymentioning

confidence: 99%

Two classes of multisecant methods for nonlinear acceleration

Fang

Saad

2008

Numerical Linear Algebra App

261

264

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“…This extension was suggested first for the case of sparse nonlinear equations by Schubert (1970), and was analyzed by Marwil (1978). Discussions of sparse quasi-Newton methods for optimization and nonlinear equations are given in Toint (1977), Dennis and Schnabel (1979), Toint (1979), Shanno (1980), Steihaug (1980), Thapa (1980), Powell (1981), Dennis and Marwil (1982) and Sorensen (1982). In the remainder of this section we give a brief description of sparse quasi-Newton methods applied to unconstrained optimization.…”

Section: Minimize F(x)mentioning

confidence: 99%

Sparse Matrix Methods in Optimization

Gill¹,

Murray²,

Saunders³

et al. 1984

SIAM J. Sci. and Stat. Comput.

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Abstract. Optimization algorithms typically require the solution of many systems of linear equations Bkyk b,. When large numbers of variables or constraints are present, these linear systems could account for much of the total computation time.Both direct and iterative equation solvers are needed in practice. Unfortunately, most of the off-the-shelf solvers are designed for single systems, whereas optimization problems give rise to hundreds or thousands of systems. To avoid refactorization, or to speed the convergence of an iterative method, it is essential to note that B is related to Bk_ 1.We review various sparse matrices that arise in optimization, and discuss compromises that are currently being made in dealing with them. Since significant advances continue to be made with single-system solvers, we give special attention to methods that allow such solvers to be used repeatedly on a sequence of modified systems (e.g., the product-form update; use of the Schur complement). The speed of factorizing a matrix then becomes relatively less important than the efficiency of subsequent solves with very many right-hand sides.At the same time, we hope that future improvements to linear-equation software will be oriented more specifically to the case of related matrices B k.

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“…Since the algorithm begins with an LU factorization of A =F'(x), we are in fact requiring that there exists an e > 0 and a permutation matrix P such that if IIA F'(x*)ll < e and PA is LU factorable without pivoting, then PF'(x*) is also factorable without pivoting. Dennis and Marwil [2] establish that this is indeed the case for threshold pivoting strategies.…”

Section: Computational Considerationsmentioning

confidence: 86%

A Quasi-Newton Method Employing Direct Secant Updates of Matrix Factorizations

Johnson¹,

Austria²

1983

SIAM J. Numer. Anal.

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A quasi-Newton algorithm is presented which employs an initial factorization of the approximate Jacobian matrix. The method then updates the upper and lower triangular factors directly at each step. Iterates are generated using forward and backward substitution employing the updated factorizations. The method is shown to be locally and q-superlinearly convergent, and a numerical algorithm is presented.

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Direct Secant Updates of Matrix Factorizations

Cited by 8 publications

References 5 publications

Two classes of multisecant methods for nonlinear acceleration

Two classes of multisecant methods for nonlinear acceleration

Sparse Matrix Methods in Optimization

A Quasi-Newton Method Employing Direct Secant Updates of Matrix Factorizations

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