Approximations of matrix-valued functions the form $W^Tf(A)W$, where $A \in {\mathbb R}^{m\times m}$ is symmetric, $W\in{\mathbb k}$, with $m$ large and $ k\ll m$, has orthonormal columns, $f$ a function, can be computed by applying few steps symmetric block Lanczos method to $A$ initial block-vector k}$. Golub Meurant have shown that approximants obtained in this manner may considered Gauss quadrature rules associated measure. This paper generalizes anti-Gauss rules, introduced Laurie for...

Large-scale networks arise in many applications. It is often of interest to be able identify the most important nodes a network or ascertain ease traveling between nodes. These and related quantities can determined by evaluating expressions form $\mathbf{u}^Tf(A)\mathbf{w}$, where $A$ adjacency matrix that represents graph network, $f$ nonlinear function, such as exponential $\mathbf{u}$ $\mathbf{w}$ are vectors, for instance, axis vectors. This paper describes novel technique determining...

It is known that Clarke's tangent cone at any point of subset R n always both unique and convex. By contrast, nearly all other notions convex in the literature are monotone sense if a K x 0 set C ⊆ , then K′ K, C′ automatically implies same type for . This carries rider such cones not and, general, even maximal (convex) unique. In this paper it shown most types, one can identify preferred (for ), called core type,

If X and Y are real Hilbert spaces, $A:X \to Y$ is a bounded linear operator, $\Gamma \subseteq closed convex cone, an immediate sufficient condition for quadratic form Q on to be positive subject the constraint $Ax \in \Gamma $, that decomposable as sum $Q(x) = C(Ax) + S(x)$, where C which S definite X. The necessity of such decomposition not obvious, but established here class forms commonly occur in variational problems—the Legendre forms. proof furnishes formulas explicit apart from...

In a recent article in this TRANSACTIONS Walach and Zeheb presented test for the positivity of real multivariable polynomials on rectangular domains. While basic idea Walach-Zeheb is sound potentially useful, case unbounded domains need reexamination, and, what more important, restriction to can be dispensed with entirely. Using Fritz John Multiplier Rule, present note extends scope include arbitrary closed domains, clarifies entailed particular it shown that actually tests strong...

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ABSTRACT Second order optimality conditions may require the verification of a conditional definiteness condition form where Q(·) is quadratic on Rn and A an m x n matrix. The prototype case, apart from that positive definiteness, in which identity matrix—in this case conditionally definite forms are called strictly copositive forms. It has recently been shown by authors, using theorem Finsler forms, general can be reduced to strict copositivity, (≠) holds if only there exist C(·) S(·) such...