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

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.

众所周知，线性算子方程在工程和应用数学的许多领域都有广泛的应用。在本文中，我们对求解线性算子方程感兴趣

其中J，K和L在规定的子矩阵约束下应该是部分双对称的。推导了三种类似共轭梯度的算法来求解该约束算子方程，其中包括Lyapunov，Stein和Sylvester矩阵方程以及特殊情况下的二次逆特征值问题。在没有舍入误差的情况下，算法在有限的迭代次数内收敛到线性算子方程的解。最后，通过三个例子对所引入算法的准确性和效率进行了数值论证。