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In this paper, the sensitivity of the nonlinear complex matrix equation \( X - A^*\root 2^m \of {X^{-1} A} = I,\) \(m \in {\mathbb {N}}\) is studied. Non-local nonlinear perturbation analysis is made. Local bound, based on the condition numbers, as well as non-local perturbation bound are formulated. Estimates for the Frobenius norm of the second order Fréchet derivative of the matrix power function \(X \rightarrow X^{-1/2^m}\) are proposed. The bounds are obtained applying the techniques of the Fréchet derivatives, Lyapunov majorants and fixed point principles.

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Perturbation Analysis of the Matrix Equation \(X - A^*\root 2^m \of {X^{-1}} A = I\)

  • 会议论文

  • 发表日期：

    2025年

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  • Vera Angelova

摘要

In this paper, the sensitivity of the nonlinear complex matrix equation \( X - A^*\root 2^m \of {X^{-1} A} = I,\) \(m \in {\mathbb {N}}\) is studied. Non-local nonlinear perturbation analysis is made. Local bound, based on the condition numbers, as well as non-local perturbation bound are formulated. Estimates for the Frobenius norm of the second order Fréchet derivative of the matrix power function \(X \rightarrow X^{-1/2^m}\) are proposed. The bounds are obtained applying the techniques of the Fréchet derivatives, Lyapunov majorants and fixed point principles.