Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Discrete the solving model of time-variant standard Sylvester-conjugate matrix equations using Euler-forward formula

Jiakuang He, Dongqing Wu

Abstract

Time-variant standard Sylvester-conjugate matrix equations are presented as early time-variant versions of the complex conjugate matrix equations. Current solving methods include Con-CZND1 and Con-CZND2 models, both of which use ode45 for continuous model. Given practical computational considerations, discrete these models is also important. Based on Euler-forward formula discretion, Con-DZND1-2i model and Con-DZND2-2i model are proposed. Numerical experiments using step sizes of 0.1 and 0.001. The above experiments show that Con-DZND1-2i model and Con-DZND2-2i model exhibit different neural dynamics compared to their continuous counterparts, such as trajectory correction in Con-DZND2-2i model and the swallowing phenomenon in Con-DZND1-2i model, with convergence affected by step size. These experiments highlight the differences between optimizing sampling discretion errors and space compressive approximation errors in neural dynamics.

Discrete the solving model of time-variant standard Sylvester-conjugate matrix equations using Euler-forward formula

Abstract

Time-variant standard Sylvester-conjugate matrix equations are presented as early time-variant versions of the complex conjugate matrix equations. Current solving methods include Con-CZND1 and Con-CZND2 models, both of which use ode45 for continuous model. Given practical computational considerations, discrete these models is also important. Based on Euler-forward formula discretion, Con-DZND1-2i model and Con-DZND2-2i model are proposed. Numerical experiments using step sizes of 0.1 and 0.001. The above experiments show that Con-DZND1-2i model and Con-DZND2-2i model exhibit different neural dynamics compared to their continuous counterparts, such as trajectory correction in Con-DZND2-2i model and the swallowing phenomenon in Con-DZND1-2i model, with convergence affected by step size. These experiments highlight the differences between optimizing sampling discretion errors and space compressive approximation errors in neural dynamics.

Paper Structure

This paper contains 13 sections, 7 theorems, 41 equations, 21 figures.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Con-CZND1 model
  4. Con-CZND2 model
  5. Euler-forward formula, Con-DZND1-2i and Con-DZND2-2i
  6. Numerical experiments and validation
  7. For Con-DZND1-2i model and Con-DZND2-2i model using step sizes $\varepsilon$ of 0.1 and 0.001 with $\gamma$ equals 10
  8. The difference in Con-DZND1-2i model using step sizes $\varepsilon$ of 0.1 and 0.001 for $\gamma$ equals 10$+$20$\mathrm{i}$ and 10$-$20$\mathrm{i}$ respectively
  9. Discussion
  10. Conclusion
  11. Gradient descent
  12. Linear N-step method
  13. Principle of space attack with "No free lunch"

Key Result

Lemma 1

He2024ZeroingNDHe2024RevisitingTC Where $\tau \ge 0$ represents the real-time, $A(\tau)\in \mathbb{C}^{m\times n}$, $B(\tau)\in \mathbb{C}^{s\times t}$, $X(\tau)\in \mathbb{C}^{n\times s}$, are time-variant matrices, the following equation can be obtained:

Figures (21)

  • Figure 1: Differences between SSCME(a) and TVSSCME(b).
  • Figure 2: Different between Con-CZND1 He2024ZeroingND model and Con-CZND2 He2024ZeroingND model. \ref{['fig.1.structure']} Con-CZND1 model. \ref{['fig.2.structure']} Con-CZND2 model.
  • Figure 3: Solution $X(\tau)$ computed by Con-DZND1-2i \ref{['eq.euler.forward.solve.linearerrconcznd1']} model in Example \ref{['example1']} where $\gamma$ equals 10 and $\varepsilon$ equals 0.1.
  • Figure 4: Solution $X(\tau)$ computed by Con-DZND2-2i \ref{['eq.euler.forward.solve.linearerrconcznd2']} model in Example \ref{['example1']} where $\gamma$ equals 10 and $\varepsilon$ equals 0.1.
  • Figure 5: Solution $X(\tau)$ computed by Con-DZND1-2i \ref{['eq.euler.forward.solve.linearerrconcznd1']} model in Example \ref{['example1']} where $\gamma$ equals 10 and $\varepsilon$ equals 0.001.
  • ...and 16 more figures

Theorems & Definitions (14)

  • Definition 1
  • Lemma 1
  • Corollary 1
  • Theorem 1: Convergence theorem
  • proof
  • Theorem 2: Stability theorem
  • proof
  • Theorem 3: Residual theorem
  • proof
  • Lemma 2
  • ...and 4 more