This paper describes the conception, general architecture, data structure, and main components of an Integrated Computational Environment (ICE) for the high-performance solution of a wide class of numerical algebraic problems on heterogeneous supercomputers with distributed and hierarchical shared memory. The tasks considered include systems of linear algebraic equations (SLAEs), various eigenvalue problems, and transformations of algebraic objects with large sparse matrices. These tasks arise in various approximations of multidimensional initial boundary value problems on unstructured grids. A quite large variety of types of matrices, featuring diverse structural, spectral, and other properties are allowed; there can also be a wide diversity of algorithms for computational algebra. There are relevant issues associated with scalable parallelism through hybrid programming on heterogeneous multiprocessor systems, MPI-processes, multithread computing, and vectorization of operations, including those without formal constraints on the number of degrees of freedom and on the number of computing units. The numerical methods and technologies are implemented in the KRYLOV library, which provides the integrated subsystem of the ICE. There are various technical requirements imposed upon the software: extendibility of the set of problems and algorithms, adaptation to the evolution of supercomputer architecture, ability to reuse external products, and coordinated participation of development groups taking part in the project. The end goal of these requirements is to provide a product featuring a long life cycle, high performance, and general acceptance among end users of diverse professional backgrounds.