Computing

, 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
Article

Abstract

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.

Keywords

Computational Mathematic General Analysis Practical Consideration Error Perturbation 

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

Zusammenfassung

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.

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Copyright information

© Springer-Verlag 1974

Authors and Affiliations

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

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