Using the ternary balanced number system to improve the accuracy of calculations

The paper describes the use of a ternary balanced number system for calculating the elements of the inverse matrix for ill-conditioned matrices. The conditionality of a matrix characterizes how strongly the solution of a linear equations system can change depending on small perturbations in the data. The higher the conditionality value, the more sensitive the matrix is to small changes in the data. As an example of an ill-conditioned matrix in this paper the three-by-three Hilbert matrix is considered. Based on the known expression, the true values of the elements of the inverse Hilbert matrix are calculated. An assessment of the errors in calculating the elements of the inverse Hilbert matrix, obtained with varying degrees of calculation accuracy in the binary number system (using a computer, software implementation in C language) and in the ternary balanced number system (calculations were performed manually), is given. Comparison of calculation results is performed in the decimal number system. It is shown that the use of a ternary balanced number system allows to reduce the calculation error of ill-conditioned matrices elements by several times (by 3 or more times for low-precision data and by 1,5 or more times for more precise data).

1. Kalitkin N.N., Yukhno L.F., Kuzmina L.V. The Quantitative Conditionality Criterium for the Systems of Linear Algebraic Equations. Mathematical Models and Computer Simulations. 2011;23(2):3–26. (In Russ.).

2. Kashapova L.A., Stepanova M.D., Tolstykh O.D. On the Question of Matrix Conditionality and Stability Linear Algebraic Systems. Young Science of Siberia. 2021;(2). (In Russ.). URL: https://mnv.irgups.ru/sites/default/files/articles_pdf_files/kashapova_stepanova_tolstyh.pdf

3. Maystrenko A.V., Svetlakov A.A., Cherepanov R.O. The Analysis of Properties of Hilbert Matrix and the Reasons for its Weak Conditionality. Omsk Scientific Bulletin. 2011;(3):265–269. (In Russ.).

4. Molodtsov V.S. A Method for Finding the Inverse Matrix Electricity Distribution. Networks Power Systems. Russian Electromechanics. 2018;61(3):60–67. (In Russ.). https://doi.org/10.17213/0136-3360-2018-3-60-67

5. Partko S.A., Groshev L.M., Sirotenko A.N. Mobile Machine Design Through Dynamic Load Simulation on Their Drive Units. Vestnik of Don State Technical University. 2020;20(2):155–161. https://doi.org/10.23947/1992-5980-2020-20-2-155-161

6. Cheney E.W., Kincaid D.R. Numerical Mathematics and Computing. Thomson Brooks/Cole; 2008. 784 p.

7. Fuks D.B., Fuks M.B. Arifmetika binomial'nykh koeffitsientov. Kvant. 1970;(6):17–25. (In Russ.).

8. Forsythe G.E., Moler C.B. Computer Solution of Linear Algebraic Systems. Moscow: Mir; 1969. 168 p. (In Russ.).

9. Zubkov S.V. Assembler dlya DOS, Windows i UNIX. Moscow: DMK Press; 2000. 608 p. (In Russ.).

10. Brusentsov N.P., Maslov S.P., Rozin V.P., Tishulina A.M. Malaya tsifrovaya vychislitel'naya mashina "Setun'". Moscow: Izd-vo MGU; 1965. 145 p. (In Russ.).

11. Giniyatullin V.M., Salikhova M.A. Effect of Rounding Error Compensation in the Ternary Balanced Number System. Proceedings in Cybernetics. 2020;(4):14–20. (In Russ.). https://doi.org/10.34822/1999-7604-2020-4-14-20