Improved methods and starting values to solve the matrix equations πΒ±π΄*πβ»ΒΉπ΄=πΌ iteratively
Ivan Ganchev Ivanov, Vejdi I. Hasanov, Frank Uhlig Β· Mathematics of Computation Β· 2004
The two matrix iterations X k + 1 = I β A β X k β 1 A X_{k+1}=I\mp A^*X_k^{-1}A are known to converge linearly to a positive definite solution of the matrix equations X Β± A β X β 1 A = I X\pm A^*X^{-1}A=I , respectively, for known choices of X 0 X_0 and under certain restrictions on A A . The convergence for previously suggested starting matrices X 0 X_0 is generally very slow. This paper explores different initial choices of X 0 X_0 in both iterations that depend on the extreme singular values of A A and lead to much more rapid convergence. Further, the paper offers a new algorithm for solving the minus sign equation and explores mixed algorithms that use Newtonβs method in part.