Methods of Numerical and Nonlinear Analysis with Applications

We consider approximation of matrix functions f(A)v by Vf(H)e 1, where A is a large and sparse matrix, H is an orthogonal projection of A into some subspace S and V is a matrix whose columns form an orthonormal basis of S. We develop the algorithm which computes matrices V and in the general case when f has any finite number of poles. Further, we develop generalized anti-Gauss quadrature rules which will help us to compute reliable upper and lower estimates for matrix functionals u^Tf(A)v for a large variety of functions f and matrices A. We will also derive new error bounds for quadratures of Gauss type following two approaches: complex-variable method for analytic integrands and a difference between rule itself and its extension such as the averaged Gauss or Kronrod formula. Abstract (cone) metric spaces and abstract normed spaces, partially ordered metric spaces, cones and nonconvex analysis occupy important place in Nonlinear Analysis and have many applications. We expect results on contractions in abstract metric spaces and weak contractions on (ordered) metric spaces, namely theorems on fixed points of Boyd-Wong, Meir-Keeler, Sehgal-Guseemen and Hardy-Rogers types, as well as theorems on common fixed points of omega-compatible mappings in abstract spaces. Results about the sets of fixed points of multi-functions on ordered metric spaces are also expected, resulting in proving existence of solutions of partial differential equations of parabolic type.