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Book review:
Effective condition number for numerical partial differential equations
By Zi-Cai Li, Hung-Tsai Huang, Yimin Wei and Alexander H.-D. Cheng
Science Press, Beijing 2015 (second edition)ISBN 978-7-03-036753-2
(reviewed by Life-time Distinguished Professor J T Chen,
Taiwan Ocean University, E-mail: jtchen@mail.ntou.edu.tw)
This book reviewed the traditional condition number first and then
summarized the recent development of effective condition by the authors.
The stability in numerical analysis is a crucial issue in the sense
that the unstable numerical methods can not be accepted in practical
applications. A stable and robust approach is required in engineering applications.
Although the history of condition number is long, mathematicians and engineers did not pay too much attention to the effective condition number.
This book can fill the gap.
For the algorithm of PDE (partial differential
equation), the stability proof is always difficult and
challenging. It provides an easier way to answer
the question of stability. There are many numerical methods considered in this book.
For example, collocation Trefftz method, finite difference method and finite element method. Besides, truncated SVD and Tikhonov regularization are also addressed.
In this book, different numerical methods for
various applications demonstrate the advantages of the effective condition number over the traditional condition number.
From the engineering point view, the effective conditional number considered the
loading vector (the forcing vector) but the traditional one does not.
For the FEM and FDM practice, the traditional condition number Cond
is large. However, the effective condition number may be small to display a better stability of numerical methods. It is particularly useful for the local
refinements for singularity problems.
Therefore, ECN reveals the condition number in a more useful index for engineering applications. This is the reason why some numerical results are acceptable even though the traditional conditional number is very large.
In structural dynamics using the mode superposition approach, the modal participation factor is an important idea for determining the dynamic response. It is interesting to find that this factor also depends on the forcing vector. Based on this point of view, it is possible that students and professors
in the engineering background can enjoy this book with physical sense.
Linear algebraic equations obtained from the FEM can
be solved by using Gaussian elimination, the conjugate gradient methods, or the multigrid methods. Since the significant decimal digits is not unlimited,
the final numerical solutions must have rounding errors.
Even for the computer symbolic software Mathematica, more working digits can be provided but it is also finite in real computation. Stability is the problem that we can not avoid in calculation.
First, the effective condition number is a new index
for numerical stability of numerical PDE, and this book covers the
newest results on this subject. This idea can be connected to modal participation factor by engineers.
The second characteristics is the hybrid disciplines of PDE and numerical analysis. Although there are various books on error analysis,
only a few combine it with stability analysis. Stability focus
is the third characteristics of this book.
This book contains 14 chapters, appendix and epilogue.
Each chapter is based on the authors published papers.
The new edition adds three chapters, MFS, singularly perturbed differential equation and small sample statistical condition estimation for the generalized Sylvester equation.
This book is not only useful for mathematicians but also
for engineers. Besides, either undergraduate or graduate student
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