内积空间矛盾方程组最小二乘解理论

Least Square Solution Theory of Contradictory Equations in Inner Product Space

基于内积理论对线性矛盾方程组最小二乘解问题进行理论推导,将方程组系数矩阵的各列作为“基函数”在离散点的值,给出法方程组内积表示形式,证明最小二乘解的存在、唯一性.得到线性矛盾方程组系数矩阵列满秩是其最小二乘解存在、唯一的充分条件,两种理论(极值理论与内积理论)所得法方程组是等价的,算例显示用新方法易于求得法方程组.

Defined the base function value at the points as the corresponding column of the coefficient matrix,the normal equations are obtained for the linear contradictory equations in inner product form.The theory is proved on the existence and uniqueness of the solution and the equivalence between the results from the inner product theory and from the extreme value theory.It is the sufficient condition that the coefficient matrix is the full column rank one for the unique least square solution of the linear contradictory equations.Examples show it is easy to calculate the normal equations using the algorithm in this paper.