By connecting the LU factorization and the Gram-Schmidt orthogonalization without any normalization, closed-forms for the coefficients of the ordinary least squares estimates are presented. Instead of using matrix inversion explicitly, each of the coefficients is expressed and computed directly as a linear combination of non-normalized Gram-Schmidt vectors and the original data matrix and also in terms of the upper triangular factor from LU factorization. The coefficients may computed iteratively using backward or forward algorithms given.

翻译：通过建立LU分解与无需标准化的Gram-Schmidt正交化之间的关联，提出了普通最小二乘估计系数的闭式表达式。该方法无需显式使用矩阵求逆，每个系数均可直接表示为非标准化Gram-Schmidt向量与原数据矩阵的线性组合，亦可基于LU分解的上三角因子形式表达。所给出的向后或向前迭代算法可递归计算这些系数。