A Fixed Point Theorem for Monotone Maps and Its Applications to Nonlinear Matrix Equations

Algorithm 1

Gaofixedpoint.m

function  [X, counter, err]=Gaofixedpoint(A, Q, C, k, q, errtol)

% Solving X = kQ + A⁢^∧⁢⁢(Xhat − C)⁢^∧⁢q⁢A, with Xhat = kronecker(I,X)

% Input: matrices  , , , k > 1, 0 < q < 1, errtol

% Output: solution  , iteration counter,  final equation relative error

m, n = size (A); [p, w] = size (Q); [r, s] = size (C); % Input size etc checks

if floor (m/n) ^~= m/n n ^~= p n ^~= w r ^~= m s ^~= m k

  <= 1 … q <= 0 q>= 1,

  error (’incompatible inputs’),

  return,

end

I = eye (m/n); X = kQ; counter = 0; err = 10000;  % Initialize

while err >= errtol Iterate

 X = kQ + A’(kron(I, X) − C)^∧qA; update X

 S = X − kQ − A’(kron(I, X) − C)^∧qA; form error matrix S

 err = norm(S, 1)/norm(X, 1); relative iteration error

 counter = counter + 1; Iteration counter

end

X = (X + X’)/2; make sure X is symmetric

S = X − kQ − A’(kron(I, X) − C)^∧qA; form final error matrix S

err = norm(S, 1)/norm(X, 1);