Approximating the Matrix Sign Function Using a Novel Iterative Method

Algorithm 2

The new (inversion-free) method for computing the matrix sign.

(1) Given a suitable 𝑋₀∈ℂ^𝑛×𝑛

(2) use (12) until 𝑋²_𝑘−𝐼_𝐹<𝜀=1

(3) set 𝑋₀=𝑋_𝑘

(4) for 𝑘=0,1,… until convergence do

(5)  𝐶_𝑘=𝐼−40𝑋²_𝑘−450𝑋⁴_𝑘−1004𝑋⁶_𝑘−705𝑋⁸_𝑘−100𝑋_𝑘¹⁰−4𝑋_𝑘¹²

(6)  𝑋_𝑘+1=𝑋_𝑘7𝐼+30𝑋²_𝑘+11𝑋⁴_𝑘𝐼+20𝑋²_𝑘+25𝑋⁴_𝑘+2𝑋⁶_𝑘𝐶_𝑘

(7) end for