Conjugate Gradient Algorithm for Least-Squares Solutions of a Generalized Sylvester-Transpose Matrix Equation

Abstract

We derive a conjugate-gradient type algorithm to produce approximate least-squares (LS) solutions for an inconsistent generalized Sylvester-transpose matrix equation. The algorithm is always applicable for any given initial matrix and will arrive at an LS solution within finite steps. When the matrix equation has many LS solutions, the algorithm can search for the one with minimal Frobenius-norm. Moreover, given a matrix Y, the algorithm can find a unique LS solution closest to Y. Numerical experiments show the relevance of the algorithm for square/non-square dense/sparse matrices of medium/large sizes. The algorithm works well in both the number of iterations and the computation time, compared to the direct Kronecker linearization and well-known iterative methods.

Keywords

• generalized Sylvester-transpose matrix equation
• conjugate gradient algorithm
• least-squares solution
• orthogonality
• Kronecker product