A condition analysis of the weighted linear least squares problem using dual norms

References

• Lawson CL, Hanson RJ. Solving least squares problems. Englewood Cliffs (NJ): Prentice-Hall Inc.; 1974.
   Google Scholar
• Rice J. A theory of condition. SIAM J Numer Anal. 1966;3:287–310.
   Google Scholar
• Björck Å. Numerical methods for least squares problems. Philadelphia: SIAM; 1996.
   Google Scholar
• Chandrasekaran S, Ipsen ICF. On the sensitivity of solution components in linear systems of equations. SIAM J Matrix Anal Appl. 1995;16(1):93–112.
   Web of Science ®Google Scholar
• Rump SM. Structured perturbations part I: normwise distances. SIAM J Matrix Anal Appl. 2003;25(1):31–56.
   Web of Science ®Google Scholar
• Björck Å. Component-wise perturbation analysis and error bounds for linear least squares solutions. BIT Numer Math. 1991;31(2):238–244.
   Google Scholar
• Kenney CS, Laub AJ, Reese MS. Statistical condition estimation for linear least squares. SIAM J Matrix Anal Appl. 1998;19(4):906–923.
   Web of Science ®Google Scholar
• Gratton S. On the condition number of linear least squares problems in a weighted Frobenius norm. BIT Numer Math. 1996;36(3):523–530.
   Web of Science ®Google Scholar
• Arioli M, Baboulin M, Gratton S. A partial condition number for linear least squares problems. SIAM J Matrix Anal Appl. 2007;29(2):413–433.
   Web of Science ®Google Scholar
• Baboulin M, Gratton S. Using dual techniques to derive componentwise and mixed condition numbers for a linear function of a linear least squares solution. BIT Numer Math. 2009;49(1):3–19.
   Web of Science ®Google Scholar
• Stewart GW. On the perturbation of pseudo-inverses, projections and linear least squares problems. SIAM Rev. 1977;19(4):634–662.
   Web of Science ®Google Scholar
• Cucker F, Diao H, Wei Y. On mixed and componentwise condition numbers for Moore-Penrose inverse and linear least squares problems. Math Comput. 2007;76(258):947–963.
   Web of Science ®Google Scholar
• Wei Y, Diao H, Qiao S. Condition number for weighted linear least squares problem. J Comput Math. 2007;25(5):561–572.
   Web of Science ®Google Scholar
• Cucker F, Diao H. Mixed and componentwise condition numbers for rectangular structured matrices. Calcolo. 2007;44(2):89–115.
   Web of Science ®Google Scholar
• Diao H, Wei Y. On Frobenius normwise condition numbers for Moore-Penrose inverse and linear least-squares problems. Numer Linear Algebra Appl. 2007;14(8):603–610.
   Web of Science ®Google Scholar
• Diao H, Wang W, Wei Y, et al. On condition numbers for Moore-Penrose inverse and linear least squares problem involving Kronecker products. Numer Linear Algebra Appl. 2013;20(1):44–59.
   Web of Science ®Google Scholar
• Gulliksson M, Wedin P-Å. Modifying the QR-decomposition to constrained and weighted linear least squares. SIAM J Matrix Anal Appl. 1992;13(4):1298–1313.
   Web of Science ®Google Scholar
• Gulliksson M. Backward error analysis for the constrained and weighted linear least squares problem when using the weighted QR factorization. SIAM J Matrix Anal Appl. 1995;16(2):675–687.
   Web of Science ®Google Scholar
• Gulliksson M, Jin X-Q, Wei Y-M. Perturbation bounds for constrained and weighted least squares problems. Linear Algebra Appl. 2002;349:221–232.
   Web of Science ®Google Scholar
• Wei Y, Wang D. Condition numbers and perturbation of the weighted Moore-Penrose inverse and weighted linear least squares problem. Appl Math Comput. 2003;145(1):45–58.
   Web of Science ®Google Scholar
• Wang Sf, Zheng B, Xiong Zp, et al. The condition numbers for weighted Moore-Penrose inverse and weighted linear least squares problem. Appl Math Comput. 2009;215(1):197–205.
   Web of Science ®Google Scholar
• Li Z, Sun J. Mixed and componentwise condition numbers for weighted Moore-Penrose inverse and weighted least squares problems. Filomat. 2009;23(1):43–59.
   Web of Science ®Google Scholar
• Yang H, Wang S. A flexible condition number for weighted linear least squares problem and its statistical estimation. J Comput Appl Math. 2016;292:320–328.
   Web of Science ®Google Scholar
• Wang G, Wei Y, Qiao S. Generalized inverses: theory and computations. Beijing, New York (NY): Science Press; 2004.
   Google Scholar
• Stewart GW, Sun Jg. Matrix perturbation theory. Boston (MA): Academic Press; 1990.
   Google Scholar
• Lancaster P, Tismenetsky M. The theory of matrices: with applications. 2nd ed. Orlando (FL): Academic Press; 1985.
   Google Scholar
• Graham A. Kronecker products and matrix calculus: with applications. New York (NY): Halsted Press; 1981.
   Google Scholar
• Golub GH, Van Loan CF. Matrix computations. 3rd ed. Baltimore (MD): The Johns Hopkins University Press; 1996.
   Google Scholar
• Paige CC. Computer solution and perturbation analysis of generalized linear least squares problems. Math Comput. 1979;33(145):171–183.
   Web of Science ®Google Scholar
• Paige CC. Fast numerically stable computations for generalized linear least squares problems. SIAM J Numer Anal. 1979;16(1):165–171. Available from: http://epubs.siam.org/doi/abs/10.1137/0716012.
   Web of Science ®Google Scholar
• Higham NJ. Accuracy and stability of numerical algorithms. 2nd ed. Philadelphia (PA): SIAM; 2002.
   Google Scholar
• Arioli M, Duff IS, de Rijk PPM. On the augmented system approach to sparse least squares problems. Numer Math. 1989;55(6):667–684.
   Web of Science ®Google Scholar
• Skeel RD. Scaling for numerical stability in Gaussian elimination. J ACM. 1979;26(3):494–526.
   Web of Science ®Google Scholar