, Volume 13, Issue 3–4, pp 215–228 | Cite as

On the solution of the linear least squares problems and pseudo-inverses

  • N. N. Abdelmalek


It is known that the computed least squares solutionx ofAx=b, in the presence of the round-off error, satisfies the perturbed equation(A+E)(x+h)=b+f. The practical considerations of computing the solution are discussed and it is found that rank(A+E)=rank (A). A general analysis of the condition of the linear least squares problems and pseudo-inverses is then presented using this assumption. Norms of relevant round-off error perturbations are estimated for two known methods of solution. Comparison between different algorithms is given by numerical examples.


Computational Mathematic General Analysis Practical Consideration Error Perturbation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Über die Lösung von linearen Problemen mit der Methode der kleinsten Quadrate und mittels der Pseudoinversen


Es ist bekannt, daß die nach der Methode der kleinsten Quadrate berechneten Lösungx vonAx=b wegen der Rundungsfehler die gestörte Gleichung(A+E)(x+h)=b+f befriedigt. Die Berechnung der Lösung wird diskutiert und es wird gezeigt, daß Rang(A+E)=Rang(A) gilt. Eine allgemeine analyse der Bedingungen für die Methode der kleinsten Quadrate und der Pseudoinversen wird unter dieser Annahme durchgeführt. Rundungsfehler werden für zwei bekannte Lösungsmethoden abgeschätzt und in numerischen Beispielen werden verschiedene Algorithmen verglichen.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Peters, G., Wilkinson, J. H.: The least squares problem and pseudo-inverses. Computer J.13, 309–316 (1970).CrossRefGoogle Scholar
  2. [2]
    Stewart, G. W.: On the continuity of the generalized inverse. SIAM J. Appl. Math.17, 33–45 (1969).CrossRefGoogle Scholar
  3. [3]
    Golub, G. H., Wilkinson, J. H.: Note on the iterative refinement of least squares solution. Numer. Math.9, 139–148 (1966).CrossRefGoogle Scholar
  4. [4]
    Björck, A.: Solving linear least squares problems by Gram Schmidt orthogonalization. BIT7, 1–21 (1967).CrossRefGoogle Scholar
  5. [5]
    Pereyra, V.: Stability of general systems of linear equations. Aequationes Math2, 194–206 (1969).CrossRefGoogle Scholar
  6. [6]
    Hanson, R. J., Lawson, C. L.: Extensions and applications of the Householder algorithm for solving the linear least squares problems. Math. Comp.108, 787–812 (1969).Google Scholar
  7. [7]
    Wedin, P. A.: On pseudoinverses of perturbed matrices. Lund university Comp. Sci. Tech. Rep., 1969.Google Scholar
  8. [8]
    Penrose, R.: A generalized inverse for matrices. Proc. Cambridge Phil. Soc.51, 406–413 (1955).Google Scholar
  9. [9]
    Greville, T. N. E.: Some applications of the pseudoinverse of a matrix. SIAM Rev.2, 15–22 (1960).CrossRefGoogle Scholar
  10. [10]
    Golub, G. H.: Numerical methods for solving linear least squares problems. Numer. Math.7, 206–216 (1965).CrossRefGoogle Scholar
  11. [11]
    Businger, P., Golub, G.: Linear least squares solution by Householder transformations. Numer. Math.7, 269–276 (1965).CrossRefGoogle Scholar
  12. [12]
    Golub, G. H., Reinsch, C.: Singular value decomposition and least squares solutions. Numer. Math.14, 403–420 (1970).CrossRefGoogle Scholar
  13. [13]
    Faddeev, D. K., Kublanovskaya, V. N., Faddeeva, V. N.: Concerning the solution of linear algebraic system with normal matrices (in Russian). Tr. Mat. Inst. Steklova96, 79–92 (1968).Google Scholar
  14. [14]
    Taussky, O.: Eigenvalues of finite matrices, in Survey of numerical analysis (Todd, J., ed.), p. 280. New York: McGraw-Hill 1962.Google Scholar
  15. [15]
    Wilkinson, J. H.: Rounding Errors in Algebraic Processes, ch. 1, pp. 94–104. New York: Prentice Hall 1963.Google Scholar
  16. [16]
    Wilkinson, J. H.: Error analysis of transformation based on the use of matrices of the form I-2wwH, in: Error in digital computation, Vol. 2 (Rall, L. B., ed.), pp. 77–101. New York: J. Wiley 1965.Google Scholar
  17. [17]
    Fan, K.: Maximum properties and inequalities for the eigenvalues of completely continuous operators. Proc. National Acad. Sci.37, 760–766 (1951).Google Scholar
  18. [18]
    Abdelmalek, N. N.: Round-off error analysis for Gram Schmidt method and solution of linear least squares problems. BIT11, 345–367 (1971).CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1974

Authors and Affiliations

  • N. N. Abdelmalek
    • 1
  1. 1.Computation CentreNational Research CouncilOttawaCanada

Personalised recommendations