(44-1) 17 * << * >> * Russian * English * Content * All Issues
On a method for calculating generalized normal solutions
of underdetermined linear systems
A.I. Zhdanov 1, Y.V. Sidorov 1
1 Samara State Technical University, Samara, Russia
PDF, 350 kB
Full text of article: Russian language.
The article presents a novel algorithm for calculating generalized normal solutions of underdetermined systems of linear algebraic equations based on special extended systems. The advantage of this method is the ability to solve very poorly conditioned (possibly sparse) underdetermined linear systems of large dimension using modern versions of the iterative refinement method based on the generalized minimum residual method (GMRES - IT). Results of applying the considered algorithm to solve the problem of balancing chemical equations (mass balance) are presented.
underdetermined linear systems, generalized normal solution, augmented systems.
Zhdanov AI, Sidorov YV. On a method for calculating generalized normal solutions of underdetermined linear systems. Computer Optics 2020; 44(1): 133-136. DOI: 10.18287/2412-6179-CO-607.
- Bakushinsky AB, Goncharsk AV. Illposed tasks. Numerical methods and applications [In Russian]. Moscow: MGU Publisher; 1989.
- Pospelov VV, Chichagov AV. Method for recovering lost signal fragments [In Russian]. Avtometriya 1988; 1: 60-64.
- Rozhkov OV, Piskunov DE, Nosov PA, Pavlov VYu, Khorokhorov AM, Shirankov AF. Bauman MSTU scientific school “Zoom lens design”: features of theory and practice. Computer Optics 2018; 42(1): 72-83. DOI: 10.18287/2412-6179-2018-42-1-72-83.
- Sukru Torun F, Manguoglu M, Aykanat C. Parallel minimum norm solution of sparse block diagonal column overlapped underdetermined systems. ACM Trans Math Softw 2017; 43(4): 31. DOI: 10.1145/3004280.
- Björck Å. Numerical methods in matrix computation. New York: Springer; 2015.
- Golovashkin DL, Pavelyev VS, Soifer VA. The numerical analysis of the light propagation through antireflecting structure within the limits of the electromagnetic theory [In Russian]. Computer Optics 1999; 19: 44-46.
- Björck Å, Elfving T. Accelerated projection methods for computing pseudoinverse solutions of systems of linear equations. BIT Numerical Mathematics 1979, 19(2), 145-163.
- Herman GT, Lent A, Rowland SW. ART: Mathematics and applications. J Theor Biol 1973; 42: 1-32.
- Björck Å. Iterative refinement of linear least squares solutions. BIT Numerical Mathematics 1967; 7(4): 257-278.
- Björck Å. Numerical stability of methods for solving augmented systems. Proceedings of Recent Developments in Optimization Theory and Nonlinear Analysis 1997: 51-60.
- Herman WC. On balancing chemical equations: Past and present. J Chem Edu 1997; 74(11): 1359.
- Sen SK, Agarwal H, Sen K. Chemical equation balancing: An integer programming approach. Mathematical and Computer Modelling 2006; 44(7): 678-691. DOI: 10.1016/j.mcm.2006.02.004.
- Soleimani F, Stanimirović PS, Soleymani F. Some matrix iterations for computing generalized inverses and balancing chemical equations. Algorithms 2015; 8: 982-998.
- Carson E, Higham NJ. A new analysis of iterative refinement and its application to accurate solution of ill-conditioned sparse linear systems. SIAM Journal on Scientific Computing 2017; 39(6): A2834-A2856. DOI: 10.1137/17M1122918.
© 2009, IPSI RAS
151, Molodogvardeiskaya str., Samara, 443001, Russia; E-mail: email@example.com ; Tel: +7 (846) 242-41-24 (Executive secretary), +7 (846) 332-56-22 (Issuing editor), Fax: +7 (846) 332-56-20