有效條件數在過去的論文中有被應用於超定系統中，在此我們研究有效條件數在欠定系統中會有什麼不同，並且將其應用於Neumann 問題。在傳統的Neumann 問題裡面，我們必須滿足離散協調性條件，才能保證解的存在性。當然也可以利用固定變數法或者刪除某一方程式的方法，將離散協調性條件移除，簡化演算法。然而我們固定哪一個變數，刪除哪一個方程，可以得到比較好的收斂性或穩定性，是我們研究的重點。

In this thesis, for the under-determined system Fx = b with the matrix
F ∈m×n (m ≤ n), new error bounds involving the traditional condition number
and the effective condition number are established. Such error bounds are
simple than those of over-determined system. The errors results implies that
for stability, the condition number and the effective condition numbers are
important if the perturbation of matrix F and vector b are dominant, respectively.
This thesis is also devoted to the application of Neumann problems,
where the consistent condition holds to guarantee the existence of multiple
solutions. For the traditional Neumann conditions, the discrete consistent
condition has to be satisfied to guarantee the existence of numerical solutions.
Such a discrete consistent condition can be removed, to greatly simplify the
numerical algorithms, and to retain the same convergence rates. For Neumann
Problems, we may solve its ordinal discrete linear equations, or the
underdetermined systems by ignoring some dependent equations, or the fixed
variables methods. Moreover, we may choose different equations to be ignored,
and different variables to be fixed. The comparisons of these different
methods and choices are important in applications. In this thesis, the new
comparisons and relations of stability and accuracy are first explored, and
some interesting results and new discoveries are found. Numerical examples
of Neumann problem in 1D are carried out, to support the analysis made.
However, the algorithms and stability analysis can be applied to the complicated
Nuemann problems in 2D and 3D, such as the traction problems in
linear elastic problems.

1 Introduction 6
2 Effective Condition number 8
3 Error Bounds for Perturbations (2.0.9) 12
4 Neumann Problems 15
4.1 1D Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.3 Comparisons with Other Methods . . . . . . . . . . . . . . . . . . . . . . 22
5 Choice of Ignoring Equations 26
6 The Identities of Singular Values among Matrices A,Fopt and Aopt
1 36
6.1 Uniform Grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
6.2 Non-Uniform Grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
7 Extensions 43
7.1 Other Lower Bounds of σmin . . . . . . . . . . . . . . . . . . . . . . . . . 43
7.2 Relations of σmin for General Underdetermined Systems . . . . . . . . . . 45
8 Error Bounds and Numerical Experiments by Different Approaches for
Neumann Problems 47
8.1 Five Approaches of Numerical Algorithms . . . . . . . . . . . . . . . . . 48
8.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
8.3 Numerical Comparisons and Conclusions . . . . . . . . . . . . . . . . . . 54
8.4 Error Bounds via Stability Analysis . . . . . . . . . . . . . . . . . . . . . 55
8.5 Arguments via Stability Analysis for Removal of the Discrete Consistent
Condition (4.1.7) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
9 Applications to Neumann Problems in 2D 59
9.1 Partition without obtuse triangles . . . . . . . . . . . . . . . . . . . . . . 60
9.2 Partition with Delaunay triangulation . . . . . . . . . . . . . . . . . . . . 62
10 Concluding Remarks 67

[1] Atkinson, K. E, An Introduction to Numerical Analysis(2nd Edition), Johns Wiley
& Sons, New York, 1989.
[2] Banoczi, J. M., Chiu, M., Cho, G. E. and Ipsen, C. F., The lack of influence of
right-hand side on accuracy of linear system solution, SIAM J. Sci. Computing, Vol.
27, 203 - 227, 1998.
[3] Bruckstein, A. M., Donoho, D. L. and Elad, M., From sparse solutions of systems
of equations to sparse modeling of signals and images, SIAM Review, vol 51, pp. 34
– 81, 2009.
[4] Ciarlet, P.G., Basic error estimates for elliptic problems in Finite Element Methods
(Part 1) Eds by P.G. Ciarlet and J. L. Lions, pp. 17-351, North-Holland, Amsterdam,
1991.
[5] Cucker, F., Diao, H. and Wei, Y., On mixed and componentwise condition numbers
for Moore-Penrose inverse and linear least squares problems, Math. Comput., Vol.
76, pp. 947-963, 2007.
[6] Demmel, J. N. and Higham, N. J., Improved error bounds or under-determined
system solvers, SIAM J. Matrix Anal. Appl., vol 14, pp. 1 – 14, 1993.
[7] Donoho, D. L., Elad, M. and Temlyakov, V. N., Stability recovery of sparse over-
complete representations in the presence noise, IEEE Trans. on Information Theory,
vol. 52. pp. 6 - 18, 2006.
[8] Golub, G. H. and van Loan, C. F., Matrix Computations (2nd Edition), The Johns
Hopkins, Baltimore and London, 1989.
[9] Higham, N. J., Accuracy and Stability of Numerical Algorithms, SIAM Philadelphia,
1996.
[10] Huang, H.T., Li, Z. C. and Yan, N., New error estimates of Adini’s elements for
Poisson’s equation, Applied Numerical Mathematics, Vol. 50, pp. 49-74, 2004.
[11] Li, Z. C., Wong, S. W., Wei, Y. and Jin, X. Q., Results on condition numbers,
Proceeding of a book, Nova Publication, to appear in 2009.
[12] Li, Z. C., Combined Methods for Elliptic Equations with Singularities, Interface and
Infinities, Kluwer Academic Publishers, Boston, Amsterdam, 1998.
[13] Li, Z. C., Chien, C. S. and Huang., H. T., Effective Condition number for finite
difference method, J. Comp. and Appl. Math., vol. 198, pp. 208-235, 2007.
[14] Li, Z. C. and Huang, H. T., Effective condition number for partial differential equa-
tions, Numerical Linear Algebra with Applications, vol. 15, pp. 575 – 594, 2008.
[15] Li, Z. C. and Huang, H. T., Effective condition number for finite element method
using local refinements, Applied Numer. Math., vol. 59, pp. 1179-1795, 2009.
[16] Li. Z. C. and Wang, S., The finite volume method and application in combinations,
J. Comp. and Appl. Math., Vol. 106, pp. 21-53, 1999.
[17] Li, Z. C. andWei Y., Stability with nodalwise and mean norms via effective condition
number, Technical report, Department of Applied Mathematics, National Sun Yatsen
University, Kaohsiung, Taiwan, 2010.
[18] Li, Z. C., Yamamoto, T. and Fang, Q., Superconvergence of solution derivatives for
the Shortley-Weller difference approximation of Poisson’s equation, Part I Smooth-
ness problems, J. Comp. and Appl. Math., Vol. 151, no.2, 307-333, 2003.
[19] Varga, R. S., Matrix Iterative Analysis, Prentice-Hall Inc., Englewood Cliffs, New
York, 1962.
[20] Yin, W., Osher, S., Goldfarb, D. and Dorborn, J., Bregman iterative algorithms for
ℓ1− minimization with applications to compressed sensing, SIAM, Imaging Science,
vol. 1, pp. 143 – 168, 2008.