NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS Numer. Linear Algebra Appl. 2011; 18:205–221 Published online 3 February 2011 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nla.766 Ill-conditioning of the truncated singular value decomposition, Tikhonov regularization and their applications to numerical partial differential equations Zi-Cai Li 1 , Hung-Tsai Huang 2 and Yimin Wei 3,4, ∗, † 1 Department of Applied Mathematics and Department of Computer Science and Engineering, National Sun Yat-sen University, Kaohsiung, Taiwan 80424, Taiwan 2 Department of Applied Mathematics, I-Shou University, Kaohsiung City 84001, Taiwan 3 School of Mathematical Sciences, Fudan University, Shanghai 200433, People’s Republic of China 4 Shanghai Key Laboratory of Contemporary Applied Mathematics, Shanghai 200433, People’s Republic of China SUMMARY This paper explores some intrinsic characteristics of accuracy and stability for the truncated singular value decomposition (TSVD) and the Tikhonov regularization (TR), which can be applied to numerical solutions of partial differential equations (numerical PDE). The ill-conditioning is a severe issue for numerical methods, in particular when the minimal singular value  min of the stiffness matrix is close to zero, and when the singular vector u min of  min is highly oscillating. TSVD and TR can be used as numerical techniques for seeking stable solutions of linear algebraic equations. In this paper, new bounds are derived for the condition number and the effective condition number which can be used to improve ill-conditioning by TSVD and TR. A brief error analysis of TSVD and TR is also made, since both errors and condition number are essential for the numerical solution of PDE. Numerical experiments are reported for the discrete Laplace operator by the method of fundamental solutions. Copyright  2011 John Wiley & Sons, Ltd. Received 11 October 2009; Revised 12 November 2010; Accepted 17 November 2010 KEY WORDS: ill-conditioning; fundamental solutions; truncated singular value decomposition; Tikhonov regularization; effective condition number; collocation Trefftz method 1. INTRODUCTION This paper explores some intrinsic characteristics of the accuracy and stability of the truncated singular value decomposition (TSVD) and the Tikhonov regularization (TR), which can be applied to numerical methods for solving partial differential equations (numerical PDE). The condition number is an important issue for those methods, especially when the minimal singular value  min of the stiffness matrix is close to zero and the left singular vector u min corresponding to  min highly oscillates. TSVD and TR can be used as stable numerical techniques for solving the linear algebraic equations arising from numerical PDE. In this paper, we first derive new bounds for the condition number and the effective condition number, which can be used to improve conditioning by TSVD and TR. Since in numerical PDE, arising from image processing, a high accuracy is more important than the stability, the link and balance between stability and accuracy must be ∗ Correspondence to: Yimin Wei, School of Mathematical Sciences, Fudan University, Shanghai 200433, People’s Republic of China. † E-mail: ymwei@fudan.edu.cn Copyright  2011 John Wiley & Sons, Ltd.