Resolution of Linear Systems Using Interval Arithmetic and Cholesky Decomposition

Abstract

This article presents an innovative approach to solving linear systems with interval coefficients efficiently. The use of intervals allows the uncertainty and measurement errors inherent in many practical applications to be considered. We focus on the solution algorithm based on the Cholesky decomposition applied to positive symmetric matrices and illustrate its efficiency by applying it to the Leontief economic model. First, we use Sylvester's criterion to check whether a symmetric matrix is positive, which is an essential condition for the Cholesky decomposition to be applicable. It guarantees the validity of our solution algorithm and avoids undesirable errors. Using theoretical analyses and numerical simulations, we show that our algorithm based on the Cholesky decomposition performs remarkably well in terms of accuracy. To evaluate our method in concrete terms, we apply it to the Leontief economic model. This model is widely used to analyze the economic interdependencies between different sectors of an economy. By considering the uncertainty in the coefficients, our approach offers a more realistic and reliable solution to the Leontief model. The results obtained demonstrate the relevance and effectiveness of our algorithm for solving linear systems with interval coefficients, as well as its successful application to the Leontief model. These advances are crucial for fields such as economics, engineering, and the social sciences, where data uncertainty can greatly affect the results of analyses. In summary, this article highlights the importance of interval arithmetic and Cholesky's method in solving linear systems with interval coefficients. Applying these tools to the Leontief model can help you better understand the impact of uncertainty and make informed decisions in a variety of fields, including economics and engineering.