Gradient Iterative Method with Optimal Convergent Factor for Solving a Generalized Sylvester Matrix Equation with Applications to Diffusion Equations

• Main Authors: Nunthakarn Boonruangkan, Pattrawut Chansangiam
• Format: Article
• Language: English
• Published: MDPI AG 2020-10-01
• Series: Symmetry
• Subjects: gradient; linear iterative process; matrix norm; generalized Sylvester matrix equation; convection–diffusion equation
• Online Access: https://www.mdpi.com/2073-8994/12/10/1732

We introduce a gradient iterative scheme with an optimal convergent factor for solving a generalized Sylvester matrix equation <inline-formula><math display="inline"><semantics><mrow><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>p</mi></msubsup><msub><mi>A</mi><mi>i</mi></msub><mi>X</mi><msub><mi>B</mi><mi>i</mi></msub><mo>=</mo><mi>F</mi></mrow></semantics></math></inline-formula>, where <inline-formula><math display="inline"><semantics><mrow><msub><mi>A</mi><mi>i</mi></msub><mo>,</mo><msub><mi>B</mi><mi>i</mi></msub></mrow></semantics></math></inline-formula> and <i>F</i> are conformable rectangular matrices. The iterative scheme is derived from the gradients of the squared norm-errors of the associated subsystems for the equation. The convergence analysis reveals that the sequence of approximated solutions converge to the exact solution for any initial value if and only if the convergent factor is chosen properly in terms of the spectral radius of the associated iteration matrix. We also discuss the convergent rate and error estimations. Moreover, we determine the fastest convergent factor so that the associated iteration matrix has the smallest spectral radius. Furthermore, we provide numerical examples to illustrate the capability and efficiency of this method. Finally, we apply the proposed scheme to discretized equations for boundary value problems involving convection and diffusion.