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For a numerical method, its stability is a crucial issue in the
sense that an unstable algorithm can render it useless in practical
computation. For an over-determined linear algebraic equation , its
stability is typically evaluated by using the traditional condition
number, Cond= , where and are the maximal and the minimal singular
values of matrix , respectively. In this book, the concept of an
effective condition number, Cond_eff= , is introduced. Cond_eff is
smaller, and generally much smaller, than Cond, and is a better
stability criterion. The Cond_eff can be used as an estimation on
stability for the numerical solutions of partial differential
equations (PDEs) using algorithms, such as the collocation Trefftz
method, the spectral method, the finite difference method, and the
finite element method. An analysis of stability integrated with
errors is the focus of this book.
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