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Matrix Algebra pp 265-306 | Cite as

Solution of Linear Systems

  • James E. Gentle
Chapter
  • 5.8k Downloads
Part of the Springer Texts in Statistics book series (STS)

Abstract

One of the most common problems in numerical computing is to solve the linear system
$$\displaystyle{Ax = b;}$$
that is, for given A and b, to find x such that the equation holds. The system is said to be consistent if there exists such an x, and in that case a solution x may be written as Ab, where A is some inverse of A. If A is square and of full rank, we can write the solution as A−1b.

References

  1. Ammann, Larry, and John Van Ness. 1988. A routine for converting regression algorithms into corresponding orthogonal regression algorithms. ACM Transactions on Mathematical Software 14:76–87.MathSciNetCrossRefzbMATHGoogle Scholar
  2. Benzi, Michele. 2002. Preconditioning techniques for large linear systems: A survey. Journal of Computational Physics 182:418–477.MathSciNetCrossRefzbMATHGoogle Scholar
  3. Brown, Peter N., and Homer F. Walker. 1997. GMRES on (nearly) singular systems. SIAM Journal of Matrix Analysis and Applications 18: 37–51.MathSciNetCrossRefzbMATHGoogle Scholar
  4. Bunch, James R., and Linda Kaufman. 1977. Some stable methods for calculating inertia and solving symmetric linear systems. Mathematics of Computation 31:163–179.MathSciNetCrossRefzbMATHGoogle Scholar
  5. Chaitin-Chatelin, Françoise, and Valérie Frayssé. 1996. Lectures on Finite Precision Computations. Philadelphia: Society for Industrial and Applied Mathematics.CrossRefzbMATHGoogle Scholar
  6. Gentleman, W. M. 1974. Algorithm AS 75: Basic procedures for large, sparse or weighted linear least squares problems. Applied Statistics 23:448–454.CrossRefGoogle Scholar
  7. Golub, G. H., and C. F. Van Loan. 1980. An analysis of the total least squares problem. SIAM Journal of Numerical Analysis 17:883–893.MathSciNetCrossRefzbMATHGoogle Scholar
  8. Golub, Gene H., and Charles F. Van Loan. 1996. Matrix Computations, 3rd ed. Baltimore: The Johns Hopkins Press.zbMATHGoogle Scholar
  9. Greenbaum, Anne, and Zdeněk Strakoš. 1992. Predicting the behavior of finite precision Lanczos and conjugate gradient computations. SIAM Journal for Matrix Analysis and Applications 13:121–137.MathSciNetCrossRefzbMATHGoogle Scholar
  10. Heath, M. T., E. Ng, and B. W. Peyton. 1991. Parallel algorithms for sparse linear systems. SIAM Review 33:420–460.MathSciNetCrossRefzbMATHGoogle Scholar
  11. Higham, Nicholas J. 1997. Stability of the diagonal pivoting method with partial pivoting. SIAM Journal of Matrix Analysis and Applications 18:52–65.MathSciNetCrossRefzbMATHGoogle Scholar
  12. Keller-McNulty, Sallie, and W. J. Kennedy. 1986. An error-free generalized matrix inversion and linear least squares method based on bordering. Communications in Statistics — Simulation and Computation 15:769–785.MathSciNetCrossRefzbMATHGoogle Scholar
  13. Kennedy, William J., and James E. Gentle. 1980. Statistical Computing. New York: Marcel Dekker, Inc.zbMATHGoogle Scholar
  14. Kenney, C. S., and A. J. Laub. 1994. Small-sample statistical condition estimates for general matrix functions. SIAM Journal on Scientific Computing 15:191–209.MathSciNetCrossRefzbMATHGoogle Scholar
  15. Kenney, C. S., A. J. Laub, and M. S. Reese. 1998. Statistical condition estimation for linear systems. SIAM Journal on Scientific Computing 19:566–583.MathSciNetCrossRefzbMATHGoogle Scholar
  16. Miller, Alan J. 1992. Algorithm AS 274: Least squares routines to supplement those of Gentleman. Applied Statistics 41:458–478 (Corrections, 1994, ibid. 43:678).Google Scholar
  17. Rust, Bert W. 1994. Perturbation bounds for linear regression problems. Computing Science and Statistics 26:528–532.Google Scholar
  18. Saad, Y., and M. H. Schultz. 1986. GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM Journal on Scientific and Statistical Computing 7:856–869.MathSciNetCrossRefzbMATHGoogle Scholar
  19. Sherman, J., and W. J. Morrison. 1950. Adjustment of an inverse matrix corresponding to a change in one element of a given matrix. Annals of Mathematical Statistics 21:124–127.MathSciNetCrossRefzbMATHGoogle Scholar
  20. Skeel, R. D. 1980. Iterative refinement implies numerical stability for Gaussian elimination. Mathematics of Computation 35:817–832.MathSciNetCrossRefzbMATHGoogle Scholar
  21. Stallings, W. T., and T. L. Boullion. 1972. Computation of pseudo-inverse using residue arithmetic. SIAM Review 14:152–163.MathSciNetCrossRefzbMATHGoogle Scholar
  22. Stewart, G. W. 1990. Stochastic perturbation theory. SIAM Review 32:579–610.MathSciNetCrossRefzbMATHGoogle Scholar
  23. Walker, Homer F. 1988. Implementation of the GMRES method using Householder transformations. SIAM Journal on Scientific and Statistical Computing 9:152–163.MathSciNetCrossRefzbMATHGoogle Scholar
  24. Walker, Homer F., and Lu Zhou. 1994. A simpler GMRES. Numerical Linear Algebra with Applications 1:571–581.MathSciNetCrossRefzbMATHGoogle Scholar
  25. Woodbury, M. A. 1950. “Inverting Modified Matrices”, Memorandum Report 42, Statistical Research Group, Princeton University.Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • James E. Gentle
    • 1
  1. 1.FairfaxUSA

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