Solvability Theory and Iteration Method for One Self-Adjoint Polynomial Matrix Equation

Algorithm 1

Given matrices 𝐴,𝑄∈ℂ^𝑛×𝑛 and positive integers 𝑠,𝑡.

Step  1. Compute  𝜆_max(𝐴^∗𝐴), 𝜆_min(𝐴^∗𝐴), 𝜆_max(𝑄), 𝜆_min(𝑄).

Step  2. Input (3).

Step  3. If  𝑠<𝑡, run Steps  4-5; if  𝑡<𝑠, run Steps  6-7; otherwise, run Steps  8-9.

Step  4. Compute the roots  𝛼₁, 𝛼₂  of  𝑔₁(𝑥), and  𝛽₁, 𝛽₂  of  𝑔₂(𝑥), respectively.

Step  5. Let  𝑋₀=𝛽₂𝐼, run (17).

Step  6. Compute the root  𝛾₁  of  𝑔₁(𝑥)  and the root  𝛾₂  of  𝑔₂(𝑥), respectively.

Step  7. Let  𝑍₀=𝛾₁𝐼, run (22).

Step  8. Compute the root  𝛿₁  of  𝑔₁(𝑥)  and the root  𝛿₂  of  𝑔₂(𝑥), respectively.

Step  9. If  𝜆_max(𝐴^∗𝐴)<1  and  𝛿₁≥𝛿₂, then let  𝑋₀=𝛿₁𝐼  and run (22).