zbMATH — the first resource for mathematics

A geometric property of the least squares solution of linear equations. (English) Zbl 0704.15005
The authors consider the overdetermined system of linear equations \(Ax=b\), where A is an \(m\times n\) real matrix for some \(m>n\), and presents a new representation of the solution of the least squares problem: \[ \min \{(Ax-b)^ T(Ax-b)=\| Ax-b\|^ 2_ 2;\quad x\in R^ n\}. \]
Reviewer: K.Burian

15A09 Theory of matrix inversion and generalized inverses
65F20 Numerical solutions to overdetermined systems, pseudoinverses
Full Text: DOI
[1] Ben-Israel, A.; Greville, T.N.E., Generalized inverses: theory and applications, (1974), Wiley-Interscience New York · Zbl 0305.15001
[2] Gantmacher, F.R., The theory of matrices, (1959), Chelsea, New York · Zbl 0085.01001
[3] Gill, P.E.; Murray, W.; Saunders, M.A.; Tomlin, J.A.; Wright, M.H., On projected Newton barrier methods for linear programming and an equivalence to Karmarkar’s projective method, Math. programming, 36, 183-209, (1986) · Zbl 0624.90062
[4] Goldman, A.J., Resolution and separation theorems for polyhedral convex sets, (), 41-51, Ann. of Math. Stud. 38 · Zbl 0072.37505
[5] Karmarkar, N., A new polynomial-time algorithm for linear programming, Combinatorica, 4, 373-395, (1984) · Zbl 0557.90065
[6] Linnik, Yu.V., Method of least squares and principles of the theory of observations, (1961), Pergamon Oxford · Zbl 0112.11105
[7] Ortega, J.M.; Rheinboldt, W.C., Iterative solution of nonlinear equations in several variables, (1970), Academic San Diego · Zbl 0241.65046
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.