Conjugate gradient type methods for unsymmetric and inconsistent systems of linear equations. (English) Zbl 0439.65020


65F10 Iterative numerical methods for linear systems
65F20 Numerical solutions to overdetermined systems, pseudoinverses
65F35 Numerical computation of matrix norms, conditioning, scaling
Full Text: DOI


[1] Hestenes, M. R.; Stiefel, E., Method of conjugate gradients for solving linear systems, J. Res. Nat. Bur. Standards, No. 49, 409-436 (1952) · Zbl 0048.09901
[2] Lanczos, C., Solution of the systems of linear equations by minimized operations, J. Res. Nat. Bur. Standards, No. 49, 33-53 (1952)
[3] Hestenes, M. R., The conjugate gradient method for solving linear systems, Proceedings of the Symposium on Applied Mathematics, Vol. 6, 83-102 (1956), New York
[4] Reid, J. K., On the method of conjugate gradients for the solution of large sparse systems of linear equations, (Reid, J. K., Proceedings of the conference on Large Sparse Sets of Linear Equations (1971), Academic), 231-254
[5] Axelsson, O., A generalized SSOR method, Nordisk Tidskr. Informationsbehandling (BIT), 13, 443-467 (1972) · Zbl 0256.65046
[6] Axelsson, O., On preconditioning and convergence acceleration in sparse matrix problems, CERN 74-10 (1974), Geneva · Zbl 0354.65020
[7] Axelsson, O., A class of iterative methods for finite element equations, Computer Methods in Applied Mechanics and Engineering, 9, 123-137 (1976) · Zbl 0334.65028
[8] Axelsson, O.; Barker, V. A., Solution of linear systems of equations: iterative methods, Sparse Matrix Techniques. (Lecture Notes in Mathematics #572 (1976), Springer). (Lecture Notes in Mathematics #572 (1977), Springer) (1976)
[9] Björck, Å.; Elfving, T., Accelerated projection methods for computing pseudoinverse solutions of systems of linear equations, (Report LITH-MAT-R-1977 (1977), Linköping Univ: Linköping Univ Sweden) · Zbl 0409.65022
[10] Nashed, M. Z., Generalized inverses, normal solvability and iteration for singular operator equations, (Rall, L. B., Nonlinear Functional Analysis and Applications (1971), Univ. of Wisconsin: Univ. of Wisconsin Madison) · Zbl 0236.41015
[11] Fridman, V. M., New methods for solving linear operator equations, Dokl. Akad. Nauk. SSSR, 128, 3, 482-484 (1959) · Zbl 0086.32401
[12] Faddeev, D. K.; Faddeeva, V. N., Computational Methods of Linear Algebra (1963), Freeman: Freeman San Francisco · Zbl 0112.07503
[13] Björck, Å., Methods for sparse linear least squares problems, (Bunch, J.; Rose, D., Sparse Matrix Computations (1976), Academic) · Zbl 0734.65031
[14] Elfving, T., On computing generalized solutions of sparse linear systems with application to some reconstruction problems, (Linköping Studies in Science and Technology (1978), Linköping Univ: Linköping Univ Sweden), Dissertations. No. 27
[15] Craig, E. J., The \(N\)-step iteration procedure, J. Mathematical Phys., 34, 65-73 (1955)
[17] Concus, P.; Golub, G. H., A generalized conjugate gradient method for nonsymmetric systems of linear equations, (Proceedings of the Second International Symposium on Computing Methods in Applied Sciences and Engineering (Dec. 1975), IRIA: IRIA Paris). (Glowinski, R.; Lions, J-L., Lecture Notes in Economics and Mathematical Systems, Vol. 134 (1976), IRIA: IRIA Berlin) · Zbl 0385.65048
[18] Widlund, O., A Lanczos method for a class of nonsymmetric systems of linear equations, SIAM J. Numer. Anal., 15, 801-812 (1978) · Zbl 0398.65030
[19] Axelsson, O.; Gustafsson, I., A modified upwind scheme for convective transport equations and the use of a conjugate gradient method for the solution of non-symmetric systems of eq uations, (Computer Sciences Department 77.12 R (1977), Chalmers University of Technology: Chalmers University of Technology Gothenburg, Sweden). (Computer Sciences Department 77.12 R (1977), Chalmers University of Technology: Chalmers University of Technology Gothenburg, Sweden), J. Inst. Math. Appl., 23, 321-337 (1979) · Zbl 0427.65079
[20] Gustafsson, I., A class of first-order factorization methods, Nordisk Tidskr. Informationsbehandling (BIT), 18, 142-156 (1978) · Zbl 0386.65006
[21] Vinsome, P. K.W., Orthomin, an iterative method for solving sparse sets of simultaneous linear equations, Society of Petroleum Engineers of AIME (1976), paper number SPE 5729
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.