We consider the problem of solving least squares problems involving a matrix >M of small displacement rank with respect to two matrices >Z1 and >Z2. We develop formulas for the generators of the matrix >M H>M in terms of the generators of >M and show that the Cholesky factorization of the matrix >M H>M can be computed quickly if >Z1 is close to unitary and >Z2 is triangular and nilpotent. These conditions are satisfied for several classes of matrices, including Toeplitz, block Toeplitz, Hankel, and block Hankel, and for matrices whose blocks have such structure. Fast Cholesky factorization enables fast solution of least squares problems, total least squares problems, and regularized total least squares problems involving these classes of matrices.
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机译：我们考虑解决涉及两个矩阵> Z 1和> Z 2的，位移等级小的矩阵> M 的最小二乘问题。我们根据> M 的生成器为矩阵> M H > M 的生成器开发公式，并证明如果> Z 1接近于and且 sup> H > M ，则可以快速计算矩阵> M H > M strong> Z 2是三角形且幂等的。对于几类矩阵（包括Toeplitz，块Toeplitz，Hankel和Block Hankel）以及块具有这种结构的矩阵，可以满足这些条件。快速的Cholesky因式分解可快速求解涉及这些类别的矩阵的最小二乘问题，总最小二乘问题和正则化总最小二乘问题。
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