High-performance computing in the residual class system
Y. Nikolaychuk, Ihor Pitukh · Kibernetyka ta Systemnyi Analiz · 2025
The mathematical foundations of data coding and performing modular operations in the residual class system are outlined. The mathematical transformations of the system of orthogonal harmonic functions of the Fourier number-theoretic basis are investigated. Models of forming codes of the residual class system are constructed based on phase portraits of harmonic functions, the frequencies of which meet the conditions of mutual simplicity. Models of forming discrete-quantized sawtooth functions of phase portraits of harmonic frequencies in the residual class module system are presented. Algorithms for forming and processing digital data represented by codes of the smallest non-negative residues are investigated. Algorithms for performing computational operations in codes of integer, normalized, and perfect forms of the residual class system are analyzed. Methods and algorithms for performing high-performance arithmetic and logical operations in codes of the residual class system are proposed. The characteristics of the speed of performing arithmetic and logical operations in the system of residual classes of digital data represented in the Rademacher, Rademacher–Krestenson, and Haar–Krestenson number-theoretic bases are studied. The algorithms for comparing numbers in the codes of the residual class system are analyzed. The methods for forming residue codes based on analog and digital data are studied. A method for converting binary codes of the Rademacher number-theoretic basis into residue codes modulo the residual class system is developed. An algorithm for determining the sample mathematical expectation in the residual class system based on determining the sum of ranks of streaming data processing is presented. A method for converting multi-digit numbers represented in the residual class system into binary codes of the Rademacher number-theoretic basis is studied. The structures of special processors that implement computational tasks in the residual class system are presented. Keywords: Fourier harmonic functions, residual class system, algorithms, high-performance computing, functional diagrams of special processors.