## Estimates in Quadratic Formulas (1994)

Citations: | 26 - 7 self |

### BibTeX

@MISC{Golub94estimatesin,

author = {Gene H. Golub and Zdenek Strakos},

title = {Estimates in Quadratic Formulas},

year = {1994}

}

### OpenURL

### Abstract

Let A be a real symmetric positive definite matrix. We consider three particular questions, namely estimates for the error in linear systems Ax = b, minimizing quadratic functional min x (x T Ax \Gamma 2b T x) subject to the constraint k x k= ff, ff !k A \Gamma1 b k, and estimates for the entries of the matrix inverse A \Gamma1 . All of these questions can be formulated as a problem of finding an estimate or an upper and lower bound on u T F (A)u, where F (A) = A \Gamma1 resp. F (A) = A \Gamma2 , u is a real vector. This problem can be considered in terms of estimates in the Gauß-type quadrature formulas which can be effectively computed exploiting the underlying Lanczos process. Using this approach, we first recall the exact arithmetic solution of the questions formulated above and then analyze the effect of rounding errors in the quadrature calculations. It is proved that the basic relation between the accuracy of Gauß quadrature for f() = \Gamma1 and the rate of ...