Conjugate gradient-like algorithms for constrained operator equation related to quadratic inverse eigenvalue problems

As is well known, linear operator equations have wide applications in many areas of engineering and applied mathematics. In the present paper, we are interested in solving the linear operator equation $$\begin{aligned} {\mathscr {F}}(J)+{\mathscr {G}}(K)+{\mathscr {H}}(L)=N, \end{aligned}$$ where J, K and L should be partially bisymmetric under a prescribed submatrix constraint. Three conjugate gradient-like algorithms are derived for solving this constrained operator equation including the Lyapunov, Stein and Sylvester matrix equations and the quadratic inverse eigenvalue problem as special cases. The algorithms converge to the solutions of the linear operator equation within a finite number of iterations in the absence of round-off errors. At the end, the accuracy and efficiency of the introduced algorithms are demonstrated numerically with three examples.