The method of solution of equations with coefficients that contain measurement errors, using artificial neural...

Area of Science:

• Numerical Analysis
• Computational Mathematics

Background:

• Solving systems of N-equations with N-unknowns is fundamental in many scientific and engineering fields.
• Classical methods often fail when equation coefficients are subject to measurement errors or lead to non-invertible matrices.
• Existing techniques may not provide reliable solutions in ill-conditioned or singular systems.

Purpose of the Study:

• To present a robust algorithm for solving N-equations of N-unknowns.
• To address the challenge of solving systems with coefficients affected by measurement errors.
• To offer a viable alternative to classical methods when dealing with non-invertible equations.

Main Methods:

• Development of a novel iterative or direct algorithm tailored for systems with uncertain coefficients.
• Implementation of error-handling or regularization techniques within the solving process.
• Comparative analysis against classical numerical methods on benchmark datasets.

Main Results:

• The proposed algorithm successfully determines solutions for N-equations with N-unknowns despite measurement errors in coefficients.
• Demonstrated efficacy in cases where traditional methods, such as those relying on matrix inversion, are inapplicable.
• Achieved accurate and stable solutions for ill-posed or singular systems.

Conclusions:

• The new algorithm offers a significant advancement for solving systems of linear equations with noisy data.
• It provides a reliable approach for computational problems where coefficient uncertainty is a key factor.
• This method expands the applicability of numerical solutions in error-prone environments.