Successive Matrix Squaring Algorithm for Computing the Generalized Inverse 𝐴⁽²⁾_𝑇,𝑆

Algorithm 1

SMS algorithm for computing the generalized inverse 𝐴⁽²⁾_𝑇,𝑆.

Input: Input the initial value matrices 𝐴,𝑋₀,𝑃_{𝐴(𝑇),𝑆} and the accurate value 𝜖;

Output: The algorithm export the matrix, that is 𝑋≈𝐴⁽²⁾_𝑇,𝑆;

Begin: Assignment the matrix 𝑄 by the initial value matrix 𝑋₀, that is 𝑄⇐𝑋₀;

Assigned the matrix 𝑃 by 𝑃_{𝐴(𝑇),𝑆}−𝑃_{𝐴(𝑇),𝑆}𝐴𝑄, that is 𝑃⇐𝑃_{𝐴(𝑇),𝑆}−𝑃_{𝐴(𝑇),𝑆}𝐴𝑄;

Computed matrix 𝑋₁, that is 𝑋₁⇐𝐼+𝑃;

Computed the error between 𝑋₁ and 𝑋₀, that is 𝑒⇐‖𝑋₁−𝑋₀‖;

Judged that whether 𝑒 is lower than 𝜖 or not,

that is while 𝑒<𝜖, do 𝑃⇐𝑃⋅𝑃;

Defined the loop function: 𝑋_𝑘+1⇐𝑋_𝑘+𝑃;

Computed the error between 𝑋_𝑘 and 𝑋_𝑘+1, that is 𝑒⇐‖𝑋_𝑘+1−𝑋_𝑘‖;

Finished the loop function

The 𝑘+1 matrix 𝑋_𝑘+1 multiplied by 𝑄 and assigned to 𝑋, that is 𝑋⇐𝑄𝑋_𝑘+1;

End the algorithm.