Inversion Free Algorithms for Computing the Principal Square Root of a Matrix

Table 1

Principal square root iterative algorithms.

	Inversion use		Inversion free
1a	𝑌𝑛+1=12𝑌𝑛+𝑌𝑛^−1𝐴,𝑌0=𝐴,𝑌𝑛⟶𝐴^1/2	1b	𝑍𝑛+1=12𝑍𝑛+𝐴𝑋𝑛,𝑍0=𝐴,𝑍𝑛⟶𝐴^1/2𝑋𝑘+1=𝑋𝑘2𝐼−𝑍𝑛𝑋𝑘,𝑋0=𝐼𝑍𝑛∞,𝑋𝑘⟶𝑋𝑛
2a	𝑍𝑛+1=12𝑍𝑛+𝐴𝑍𝑛^−1,𝑍0=𝐴,𝑍𝑛⟶𝐴^1/2	2b	𝑍𝑛+1=12𝑍𝑛+𝐴𝑋𝑛,𝑍0=𝐴,𝑍𝑛⟶𝐴^1/2𝑋𝑘+1=𝑋𝑘2𝐼−𝑍𝑛𝑋𝑘,𝑋0=𝐼𝑍𝑛∞,𝑋𝑘⟶𝑋𝑛
3a	𝑀𝑛+1=12𝑀𝑛+𝑁𝑛^−1,𝑀0=𝐴,𝑀𝑛⟶𝐴^1/2𝑁𝑛+1=12𝑁𝑛+𝑀𝑛^−1,𝑁0=𝐼,𝑁𝑛⟶𝐴^−1/2	3b	𝑀𝑛+1=12𝑀𝑛+𝑋𝑛,𝑀0=𝐴,𝑀𝑛⟶𝐴^1/2𝑋𝑘+1=𝑋𝑘2𝐼−𝑁𝑛𝑋𝑘,𝑋0=𝐼𝑁𝑛∞,𝑋𝑘⟶𝑋𝑛𝑁𝑛+1=12𝑁𝑛+𝑌𝑛,𝑁0=𝐼,𝑁𝑛⟶𝐴^−1/2𝑌𝑘+1=𝑋𝑘2𝐼−𝑀𝑛𝑌𝑘,𝑌0=𝐼𝑀𝑛∞,𝑌𝑘⟶𝑌𝑛
4a	𝐺𝑛+1=4𝐺𝑛𝐼+𝐺𝑛^−2,𝐺0=𝐴,𝐺𝑛𝑅⟶𝐼𝑛+1=12𝑅𝑛𝐼+𝐺𝑛,𝑅0=𝐼,𝑅𝑛⟶𝐴^1/2	4b	𝐺𝑛+1=4𝐺𝑛𝑋^2𝑛,𝐺0=𝐴,𝐺𝑛𝑋⟶𝐼𝑛=𝐼+𝐺𝑛𝑌𝑘+1=𝑌𝑘2𝐼−𝑋𝑛𝑌𝑘,𝑌0=𝐼𝑋𝑛∞,𝑌𝑘⟶𝑋𝑛𝑅𝑛+1=12𝑅𝑛𝐼+𝐺𝑛,𝑅0=𝐼,𝑅𝑛⟶𝐴^1/2
5a	𝑋𝑛+1=12𝐼+𝑋𝑛^−1−12𝑋𝑛^−2,𝑋0=12(𝐼+𝐴),𝑋𝑛𝑊⟶𝐼𝑛+1=𝑊𝑛𝑋𝑛,𝑊0=𝐼,𝑊𝑛⟶𝐴^1/2	5b	𝑋𝑛+1=12𝐼+𝑌𝑛−12𝑌^2𝑛,𝑋0=12(𝐼+𝐴),𝑋𝑛𝑌⟶𝐼𝑘+1=𝑌𝑘2𝐼−𝑋𝑛𝑌𝑘,𝑌0=𝐼𝑋𝑛∞,𝑌𝑘⟶𝑌𝑛𝑊𝑛+1=𝑊𝑛𝑋𝑛,𝑊0=𝐼,𝑊𝑛⟶𝐴^1/2
6a	𝑈𝑛+1=𝑈𝑛+𝑉𝑛,𝑈0𝑉=𝐼𝑛+1=𝐴𝑈𝑛+𝑉𝑛,𝑉0𝑃=0𝑛=𝑉𝑛𝑈𝑛^−1𝑃𝑛⟶𝐴^1/2	6b	𝑈𝑛+1=𝑈𝑛+𝑉𝑛,𝑈0𝑉=𝐼𝑛+1=𝐴𝑈𝑛+𝑉𝑛,𝑉0𝛿=0𝑛=𝑉𝑛+1𝑈𝑛−𝑉𝑛𝑈𝑛+1𝑉⟶0𝑛𝑈𝑛^−1⟶𝐴^1/2
7a	𝑎𝑛+1=𝑎^2𝑛+𝐴𝑏^2𝑛,𝑎0𝑏=𝐼𝑛+1=2𝑎𝑛𝑏𝑛,𝑏0𝑃=𝐼𝑛=𝐴𝑏𝑛𝑎𝑛^−1𝑃𝑛⟶𝐴^1/2	7b	𝑎𝑛+1=𝑎^2𝑛+𝐴𝑏^2𝑛,𝑎0𝑏=𝐼𝑛+1=2𝑎𝑛𝑏𝑛,𝑏0𝑒=𝐼𝑛+1=𝑒^2𝑛,𝑒0𝑒=𝐼−𝐴𝑛=𝑎^2𝑛−𝐴𝑏^2𝑛⟶0𝐴𝑏𝑛𝑎𝑛^−1⟶𝐴^1/2
8a	𝑃𝑛+1=2𝑃𝑛^−1+𝑃𝑛𝐴^−1−1,𝑃0=𝐴,𝑃𝑛⟶𝐴^1/2	8b	𝑃𝑛+1=2𝑅𝑛,𝑃0=𝐴,𝑃𝑛⟶𝐴^1/2𝑍𝑛=𝑋𝑛+𝑃𝑛𝐴^−1𝑋𝑘+1=𝑋𝑘2𝐼−𝑍𝑛𝑋𝑘,𝑋0=𝐼𝑍𝑛∞,𝑋𝑘⟶𝑅𝑛𝑌𝑘+1=𝑋𝑘2𝐼−𝑃𝑛𝑌𝑘,𝑌0=𝐼𝑃𝑛∞,𝑌𝑘⟶𝑋𝑛