避免奇异解联合对角化的两种高效算法

Two efficient algorithms for joint diagonalization with exception of the singular solution

针对避免奇异解的联合对角化算法计算量大的问题,提出两种改进的高效算法.在第一种改进算法中,将对角化矩阵行列式按当前更新的列展开,从而避免了计算行列式过程中的矩阵求逆.另一种改进算法将列交换后的对角化矩阵进行QR分解,由分解得到的上三角矩阵计算对角化矩阵的行列式.由于两种改进算法减少了一次矩阵求逆,因此降低了原算法的计算量.仿真结果表明,当目标矩阵个数和维数较大时,两种改进算法的计算量分别为原算法的18.9%和13.5%.

Two improved efficient algorithms are proposed to reduce the computational load for nonorthogonal joint diagonalization free of singular solutions. In the first improved algorithm, the determinate of the diagonalization matrix is expanded by its current updating column to avoid matrix inverse operation in this computational phase. The other improved algorithm is developed by QR factorization of the column exchanged diagonalization matrix, whose determinate is computed by the product of the diagonal elements of the resulting upper triangular matrix. As a result, the improved algorithms exhibit a low computational load since both of them reduce one matrix inverse operation in comparison to the original one. Finally, numerical simulation results show that the two improved algorithms have a computational load equal to 18.9% and 13.5% that of the original algorithm when the objective matrices are of large dimension and number.