Functions of Matrices and Nearest Correlation Matrices - NAG

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MATLAB: Arbitrary Powers>> A = [1 1e-8; 0 1]A =1.0000e+000 1.0000e-0080 1.0000e+000>> A^0.1ans =1 00 1>> expm(0.1*logm(A))ans =1.0000e+000 1.0000e-0090 1.0000e+000MIMS Nick Higham Matrix <strong>Functions</strong> 19 / 33
MATLAB Arbitrary PowerNew Schur algorithm (H & Lin, 2011) reliablycomputes A p for any real p.Brings improvements over MATLAB A p even fornegative integer p.Alternative Newton-based algorithms available for A 1/qwith q an integer, e.g., forX k+1 = 1 q[(q + 1)Xk − X q+1kA ] X 0 = A,X k → A −1/q .MIMS Nick Higham Matrix <strong>Functions</strong> 20 / 33
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