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Li, Z., Huang, H. and Wei, Y. (2011) Ill-Conditioning of the Truncated Singular Value Decomposition, Tikhonov Regularization and Their Applications to Numerical Partial Differential Equations. Numerical Linear Algebra with Applications, 18, 205-221.
https://doi.org/10.1002/nla.766

has been cited by the following article:

• TITLE: A Novel Modified TSVD Method and Truncated Reorthogonalized Golub-Kahan Bidiagonalization Method for Discrete Ill-Posed Problems

  AUTHORS: Yin Wu, Lei Zhao

  KEYWORDS: Discrete Ill-Posed Problems, Truncated Singular Value Decomposition, Reorthogonalized Golub-Kahan Bidiagonalization, Gram-Schmidt

  JOURNAL NAME: Open Journal of Applied Sciences, Vol.16 No.2, February 14, 2026

  ABSTRACT: Truncated singular value decomposition (TSVD) and Golub-Kahan diagonalization are two elementary techniques for solving a least squares problem from a linear discrete ill-posed problems. For small to medium sized ill-posed problems, we propose a novel Modified Truncated Singular Value Decomposition (NMTSVD) based on the MTSVD method. This method presents three approaches to select truncation indices, leading to three approximate matrices of the coefficient matrix A . And the relationships of these approximate matrices are established in terms of the spectral condition number. For large-scale unstable discrete ill-posed problems, we propose a truncated Reorthogonalized Golub-Kahan Bidiagonalization (RGKB) method by integrating the Golub-Kahan bidiagonalization process, truncation index, and the Gram-Schmidt reorthogonalization procedure. Numerical experiments are presented to illustrate the effectiveness of the methods NMTSVD and RGKB.