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Effective condition number for numerical partial differential equations. (English) Zbl 1212.65410

Summary: New computational formulas are derived for the effective condition number Cond\(\_\)eff, and the new error bounds involved in both Cond and Cond\(\_\)eff are developed. A theoretical analysis is provided to support some conclusions in the paper by J. M.Banoczi, N. Ch. Chiu, G. E. Cho and C. F. Ipsen [SIAM J. Sci. Comput. 20, No. 1, 203–227 (1998; Zbl 0914.65047)]. For linear algebraic equations solved by Gaussian elimination or QR factorization, the direction of the right-hand vector is insignificant for the solution errors, but such a conclusion is invalid for the finite difference method for Poisson’s equation. The effective condition number is important to numerical partial differential equations, because the discretization errors are dominant.

MSC:

65N06 Finite difference methods for boundary value problems involving PDEs
65F05 Direct numerical methods for linear systems and matrix inversion
65F35 Numerical computation of matrix norms, conditioning, scaling
65N15 Error bounds for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation

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References:

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