TY - JOUR

T1 - Two classes of multisecant methods for nonlinear acceleration

AU - Fang, Haw Ren

AU - Saad, Yousef

PY - 2009/3

Y1 - 2009/3

N2 - Many applications in science and engineering lead to models that 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 allow 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'.

AB - Many applications in science and engineering lead to models that 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 allow 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'.

KW - Anderson mixing

KW - Broyden's methods

KW - Fixed point problems

KW - Quasi-Newton methods

KW - Self-consistent field (SCF) iteration

UR - http://www.scopus.com/inward/record.url?scp=65749083859&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=65749083859&partnerID=8YFLogxK

U2 - 10.1002/nla.617

DO - 10.1002/nla.617

M3 - Article

AN - SCOPUS:65749083859

SN - 1070-5325

VL - 16

SP - 197

EP - 221

JO - Numerical Linear Algebra with Applications

JF - Numerical Linear Algebra with Applications

IS - 3

ER -