Preconditioning reduced matrices. (English) Zbl 0851.65018

The authors are interested in solving linear systems of the form \(Z^T GZp= d\). Systems of this form are obtained from the constraint minimization problem \[ \text{minimize}_x\;f(x)= \textstyle{{1\over 2}} x^T Gx- c^T x\quad\text{subject to } Ax= b. \] Here \(Z\) is a matrix whose columns form a basis for the kernel of \(A\). A preconditioner of the form \(W^T M^{- 1} W\approx (Z^T G^Z)^{- 1}\) is proposed, where \(W^T\) is a left inverse for \(Z\) and \(M\) approximates \(G\). [Obiously, the computation of \(Z\) may be expensive and the assumptions are restrictive. Therefore we refer to the alternative method of R. E. Bank, B. D. Welfert and H. Yserentant, Numer. Math. 56, No. 7, 645-666 (1990; Zbl 0684.65031).] The paper contains some remarks on applications by Newton’s method and on Neumann series.
Reviewer: D.Braess (Bochum)


65F10 Iterative numerical methods for linear systems
90C20 Quadratic programming
65K05 Numerical mathematical programming methods


Zbl 0684.65031
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