Stability of iterative algorithms in inner product spaces: An analysis based on the Cauchy-schwarz inequality

Abstract

This paper examines the role of the Cauchy-Schwarz inequality in ensuring the stability of iterative algorithms within inner product spaces. The inequality ∣⟨𝑢, 𝑣⟩∣ ≤ ∥𝑢∥⋅∥𝑣∥ provides a fundamental bound on the inner product of two vectors, influencing the convergence and stability of various iterative methods employed in computational mathematics and numerical analysis.