2] and the

Euclidean norm for a matrix by [[parallel] * [parallel].

The semiring N of nonnegative integers is Euclidean if we define the Euclidean norm [delta] by [delta] : n [?

Then there exists a left Euclidean norm [delta] : R\{0} [right arrow] N defined by setting [delta]([SIGMA][a.

In the case of the Euclidean norm, the upper bounding function is the envelope of n-dimensional cones (Fig.

In (Paulavicius and Zilinskas 2006), we have shown that, for dimension n = 2, a combination of bounds based on 2 extreme (infinite and first) norms gives by 22% smaller number of function evaluations than the bound based on the Euclidean norm, and, for dimension n = 3, combination (7) gives 39% smaller number of function evaluations than in the case the Euclidean norm is used alone.

At]]]], induced by the Euclidean norm (Expression 6) that we use to measure the length of vectors (see, for example, Horn and Johnson 1985).

Throughout this paper we have measured the magnitude of a vector x by its Euclidean norm (Expression 6).

3], and [parallel]x[parallel] is the

Euclidean norm, is a Banach space.

8), taking the

Euclidean norm and using again the equality [A.

For both GB and BB, they introduced an error matrix and showed the reduction of its

Euclidean norm.

We denote its solution of minimal

Euclidean norm by [?

They also proved identities for the A-norm and the

Euclidean norm of the error which could justify the stopping criteria [24, Theorems 6:1 and 6:3, p.