A Gradient Based Iterative Solutions for Sylvester Tensor Equations

Algorithm 1

The gradient based iterative algorithm (GI).

Input:𝐴,𝐵,𝐶,𝒟, and 𝜇.

Output:𝒳.

Initialize 𝒳⁰.

For 𝑘=1,2,…,𝑘_max

 Compute ℛ^𝑘−1=𝒟−𝒳^𝑘−1×₁𝐴−𝒳^𝑘−1×₂𝐵−𝒳^𝑘−1×₃𝐶,

 𝒳^𝑘₁=𝒳^𝑘−1+𝜇ℛ^𝑘−1×₁𝐴^𝑇,

 𝒳^𝑘₂=𝒳^𝑘−1+𝜇ℛ^𝑘−1×₂𝐵^𝑇,

 𝒳^𝑘₃=𝒳^𝑘−1+𝜇ℛ^𝑘−1×₃𝐶^𝑇,

 𝒳^𝑘=(𝒳^𝑘₁+𝒳^𝑘₂+𝒳^𝑘₃)/3.

End