[Paper Review] Numerical computation of formal solutions to interval linear systems of equations

Sergey P. Shary|arXiv (Cornell University)|Mar 15, 2019

Numerical Methods and Algorithms4 citations

TL;DR

This paper presents numerical methods for computing formal solutions to interval linear systems in Kaucher interval arithmetic using stationary iterative solvers based on matrix splitting. The proposed ARMSplit method achieves high accuracy (up to 13 significant digits) for large systems, with convergence behavior analyzed and compared to subdifferential Newton methods, demonstrating robustness even for singular or ill-conditioned matrices.

ABSTRACT

The work is devoted to the development of numerical methods for computing "formal solutions" of interval systems of linear algebraic equations. These solutions are found in Kaucher interval arithmetic, which extends and completes the classical interval arithmetic algebraically. The need to solve these problems naturally arises, for example, in inner and outer estimation of various solution sets to interval linear systems of equations. The work develops two approaches to the construction of stationary iterative methods for computing the formal solutions that are based on splitting the matrix of the system. We consider their convergence and implementation issues, compare with the other approaches to computing formal solutions.

Motivation & Objective

• To develop efficient numerical methods for computing formal solutions to interval linear systems in Kaucher interval arithmetic.
• To analyze the convergence and implementation of stationary iterative methods based on matrix splitting for formal solutions.
• To compare the performance and applicability of ARMSplit with other approaches, including subdifferential Newton methods.
• To evaluate the robustness of these methods on large-scale and ill-conditioned interval systems, including singular matrices.

Proposed method

• The paper employs matrix splitting techniques to construct stationary iterative methods for solving interval linear systems in Kaucher arithmetic.
• The ARMSplit algorithm is developed based on point splitting of the interval matrix, enabling iterative approximation of formal solutions.
• Convergence analysis is performed for the iterative schemes, with theoretical conditions derived for convergence in Kaucher arithmetic.
• The subdifferential Newton method is applied as a comparison, using a damping factor τ = 1 for improved convergence in difficult cases.
• Numerical experiments use interval matrices from standard test sets, including tridiagonal and Neumaier matrices, with right-hand sides derived from perturbed exact solutions.
• Implementation focuses on computing interval vectors that satisfy the system exactly under Kaucher arithmetic operations.

Experimental results

Research questions

• RQ1 How can formal solutions to interval linear systems be computed efficiently in Kaucher interval arithmetic?
• RQ2 What are the convergence properties of stationary iterative methods based on matrix splitting in Kaucher arithmetic?
• RQ3 How does the ARMSplit method compare in accuracy and convergence speed to the subdifferential Newton method for formal solutions?
• RQ4 Can iterative methods like ARMSplit reliably compute formal solutions for large-scale or singular interval systems?
• RQ5 What is the impact of matrix structure and condition on the convergence behavior of iterative solvers in formal solution computation?

Key findings

• The ARMSplit method achieves 3 valid significant digits after 10 iterations and 6 correct significant digits after 20 iterations for a 2×2 system, matching the efficiency of prior iterative methods.
• For a 40×40 tridiagonal interval system, ARMSplit computes formal solutions with 12–13 correct significant digits after 16 iterations for both original and dualized matrices.
• In the Neumaier 40×40 system, ARMSplit reaches an accuracy of about 10⁻⁸ after 40 iterations, successfully computing the formal solution despite matrix singularity.
• For a 7×7 system with a non-convergent ARMSplit, the subdifferential Newton method converges in 9 iterations with τ = 1, demonstrating superior convergence robustness.
• When the (7,7) element of the 7×7 matrix is narrowed to [8,82], ARMSplit converges slowly, requiring about 100 iterations for 5 correct significant digits.
• The subdifferential Newton method outperforms ARMSplit in both convergence speed and applicability scope, especially for systems where iterative methods diverge or converge slowly.