Models of Computer Calculations

Abstract

Introduction. The complexity of computational algorithms for solving typical problems of computational, applied, and discrete mathematics is analyzed from the perspective of the theory of computation, depending on the computer architecture and the used computing model: single-processor, multiprocessor, and quantum. The following classes of problems are considered: systems of linear algebraic equations, the Cauchy problem for systems of ordinary differential equations, numerical integration, boundary value problems for ordinary differential equations, factorization of numbers, finding the discrete logarithm of a number in multiplicative integer groups, searching for the necessary record in an unordered database, etc. The purposes of the paper are: 1. To investigate how the computational complexity depends on the computer architecture and the computational model. 2. To show that the construction of the computational process under the given conditions of calculations is related to the solution of the following problems: – the existence ε-solution to the problem; – the existence of T-effective computing algorithms; – the possibility of building a real computing process under the given computing conditions. 3. To investigate the effect of rounding numbers on computational complexity (especially when solving problems of transcomputational complexity). 4. To give the complexity estimates and total error of the computational algorithm for a number of typical problems of computational, applied, and discrete mathematics. The results. The complexity estimates of computational algorithms of the listed classes of problems for single-processor, multiprocessor and quantum computing models are given. The main focus is on high-performance computing: using the principles of parallel data processing and quantum mechanics. Conclusions. The connection of complexity estimates of computational algorithms with the architecture of computers and models of calculations is demonstrated. The characteristics of the first quantum computers (2016 – 2022), which have gone beyond laboratory research, are given. Keywords: computer technologies, rounding error, sequential, parallel and quantum computing models, complexity estimate.