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

Algorithm 2

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

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

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

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

Assigned the matrix 𝑃₁ by 𝑃_{𝐴(𝑇),𝑆}−𝑃_{𝐴(𝑇),𝑆}𝐴𝑄, that is 𝑃₁⇐𝑃_{𝐴(𝑇),𝑆}−𝑃_{𝐴(𝑇),𝑆}𝐴𝑄;

Computed the product of 𝑃₁ and 𝑃₁, and assigned its value to 𝑃₂. that is 𝑃₂⇐𝑃₁⋅𝑃₁;

Similarly, we repeatedly do the computation for the product 𝑃_𝑖 and 𝑃₁ as well as above

the computation, where 𝑖=2,…,𝑚−2.

Computed the product of the matrix 𝑃_𝑚−1 and 𝑃₁, and assigned its value to 𝑃_𝑚 as well

as above computations, that is 𝑃_𝑚⇐𝑃_𝑚−1⋅𝑃₁;

Assigned the matrix 𝑋₁ by the sum of the matrices 𝑃_𝑖, where 𝑖=0,1,2,…,𝑚 and

𝑃₀=𝐼. that is 𝑋₁⇐𝐼+𝑃₁+⋅⋅⋅+𝑃_𝑚−1+𝑃_𝑚;

Take the norm of ‖𝑋₁−𝑋₀‖ and assigned its value to 𝑒. that is 𝑒⇐‖𝑋₁−𝑋₀‖;

while 𝑒<𝜖 do;

We need the iteration not to exceed 500 times. that is 𝑛=500; (In fact (𝑝^𝑘−1)𝑚−1=500)

Do 500 step repeatedly computations in the following.

that is For 𝑖= 1 : 𝑛

Computed the product of the given matrix 𝑃_𝑚=𝑃^𝑚 and the iteration matrix 𝑃_𝑖, and

assigned its value to the new matrix 𝑃_𝑖. that is 𝑃_𝑖⇐𝑃_𝑚⋅𝑃_𝑖;

From the iteration 𝑃_𝑖⇐𝑃_𝑚⋅𝑃_𝑖, we obtain the new matrix 𝑃_𝑖 and add its value to 𝑋_𝑖,

and assigned the sum of 𝑃_𝑖 and 𝑋_𝑖 to the matrix 𝑋_𝑖+1..that is 𝑋_𝑖+1⇐𝑋_𝑖+𝑃_𝑖; After these,

return the the step 𝑃_𝑖=𝑃_𝑚⋅𝑃_𝑖.

Finished the For loop function that is end

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

Finished the While loop function. that is end

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

End the algorithm.