Encyclopedia of Optimization

2001 Edition
| Editors: Christodoulos A. Floudas, Panos M. Pardalos

ABS Algorithms for Linear Equations and Linear Least Squares

Abaffi–Broyden–Spedicato Algorithms for Linear Equations and Linear Least Squares
  • Emilio Spedicato
Reference work entry
DOI: https://doi.org/10.1007/0-306-48332-7_2
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The Scaled ABS Class: General Properties

ABS methods were introduced by [1], in a paper dealing originally only with solving linear equations via what is now called the basic or unscaled ABS class . The basic ABS class was later generalized to the so-called scaled ABS class and subsequently applied to linear least squares, nonlinear equations and optimization problems, see [2]. Preliminary work has also been initiated concerning Diophantine equations , with possible extensions to combinatorial optimization, and the eigenvalue problem. There are presently (1998) over 350 papers in the ABS field, see [11]. In this contribution we will review the basic properties and results of ABS methods for solving linear determined or underdetermined systems and overdetermined linear systems in the least squares sense.

Let us consider the linear determined or underdetermined system, where rank( A) is arbitrary
65K05 65K10 
linear algebraic equations linear least squares ABS methods Abaffian matrices Huang algorithm implicit LU algorithm implicit LX algorithm 
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References

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Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • Emilio Spedicato
    • 1
  1. 1.Dept. Math.Univ. BergamoBergamoItaly