多变量矩阵方程异类约束解的修正共轭梯度法

基于求解线性代数方程组的共轭梯度法,通过对相关矩阵和系数的修改,建立了一种求多矩阵变量矩阵方程异类约束解的修正共轭梯度法.该算法不要求等价线性代数方程组的系数矩阵具备正定性、可逆性或者列满秩性,因此算法总是可行的.利用该算法不仅可以判断矩阵方程的异类约束解是否存在,而且在有异类约束解,不考虑舍入误差时,可在有限步计算后求得矩阵方程的一组异类约束解;选取特殊初始矩阵时,可求得矩阵方程的极小范数异类约束解.另外,还可求得指定矩阵在异类约束解集合中的最佳逼近.算例验证了该算法的有效性.

The Modified Conjugate Gradient Method for the Solution of a Multi-Variables Matrix Equation over Different Constrained Matrices

Based on the conjugate gradient method for solving the linear algebraic equations, a modified conjugate gradient method is presented to find the solution of a multi-variables matrix equation over different constrained matrices by modifying the related matrices and coefficients. This method doesn’t require the coefficient matrix of the equivalence linear algebraic equations satisfying the positive definiteness, reversibility and full column rank properties. So the method is always feasible. By using the proposed method, we not only can judge whether the solution of the matrix equation exists over different constrained matrices, but also can obtain the solution within finite iterative steps in the absence of round off errors when the solution exists, and the different constrained solution with the least-norm can be obtained by choosing special initial matrices. In addition, the optimal approximation of the given matrix can be achieved in the set of the different constrained solutions. The numerical example shows that the method is quite efficient.