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Linear system solvers for boundary value ODEs. (English) Zbl 0780.65047

Summary: We investigate the stability properties of several linear system solvers for solving boundary value ordinary differential equations. We consider the compactification algorithm, Gaussian elimination with row partial pivoting, and a QR algorithm applied to linear systems arising from solving boundary value problems for which the matrix is block-bidiagonal except for bordering along the last \(n\) rows and columns.
We particularly compare AUTO’s original linear solver (an LU decomposition with partial pivoting) and our implementation of the analogous QR algorithm to AUTO. Two other factors (the underlying continuation strategy and mesh selection strategy) may affect the stability of the linear system solver for ODE continuation codes as well and are also discussed in our numerical investigations.

MSC:

65L10 Numerical solution of boundary value problems involving ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
65F05 Direct numerical methods for linear systems and matrix inversion
34B05 Linear boundary value problems for ordinary differential equations
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