Strang-type Preconditioners for Solving Systems of Odes by Boundary Value Methods

Raymond H. CHAN*, Xiao-Qing JIN, Yue-Hung TAM

*Corresponding author for this work

Research output: Journal PublicationsJournal Article (refereed)peer-review


In this paper, we survey some of the latest developments in using boundary value methods for solving systems of ordinary differential equations with initial values. These methods require the solutions of one or more nonsymmetric, large and sparse linear systems. The GMRES method with the Strang-type preconditioner is proposed for solving these linear systems. One of the main results is that if an Aν1,ν2 -stable boundary value method is used for an m-by-m system of ODEs, then the preconditioner is invertible and the preconditioned matrix can be decomposed as I + L where I is the identity matrix and the rank of L is at most 2m(ν1 + ν2). It follows that when the GMRES method is applied to solving the preconditioned systems, the method will converge in at most 2m(ν12)+1 iterations. Applications to differential algebraic equations and delay differential equations are also given.
Original languageEnglish
Pages (from-to)14-46
Number of pages33
JournalElectronic Journal of Mathematical and Physical Sciences
Issue number1
Publication statusPublished - 22 Aug 2002
Externally publishedYes


  • Boundary value method
  • ordinary differential equation
  • Strang-type preconditioner


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