A Gradient Based Iterative Solutions for Sylvester Tensor Equations

Algorithm 2

The modified gradient based iterative algorithm (MGI).

Input:𝐴,𝐵,𝐶,𝒟, and 𝜅.

Output:𝒳.

Initialize 𝒳⁰.

For 𝑘=1,2,…,𝑘_max

 𝒳^𝑘₁=𝒳^𝑘−1+𝜅(𝒟−𝒳^𝑘−1×₁𝐴−𝒳^𝑘−1×₂𝐵−𝒳^𝑘−1×₃𝐶)×₁𝐴^𝑇,

 𝒳^𝑘−1=(𝒳^𝑘₁+𝒳₂^𝑘−1+𝒳₃^𝑘−1)/3,

 𝒳^𝑘₂=𝒳^𝑘−1+𝜅(𝒟−𝒳^𝑘−1×₁𝐴−𝒳^𝑘−1×₂𝐵−𝒳^𝑘−1×₃𝐶)×₂𝐵^𝑇,

 𝒳^𝑘−1=(𝒳^𝑘₁+𝒳^𝑘₂+𝒳₃^𝑘−1)/3,

 𝒳^𝑘₃=𝒳^𝑘−1+𝜅(𝒟−𝒳^𝑘−1×₁𝐴−𝒳^𝑘−1×₂𝐵−𝒳^𝑘−1×₃𝐶)×₃𝐶^𝑇,

 𝒳^𝑘=(𝒳^𝑘₁+𝒳^𝑘₂+𝒳^𝑘₃)/3.

End