Computer oriented methods for fitting tabular data in the linear and nonlinear least squares sense
Computer oriented methods for fitting tabular data in the linear and nonlinear least squares sense
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在线性和非线性最小二乘意义上拟合表格数据的面向计算机的方法
DOI：
10.1145/1480083.1480176
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发表时间：
1972
期刊：
影响因子：
--
通讯作者：
K. Brown
中科院分区：
文献类型：
--
作者：
K. Brown
的例子。参考前面的多项式拟合的例子，(3)中系数矩阵MM的*th、第j个元素由2 ^£i tk~和第i个元素给出。右边My(3)的元素是^2JZI fc*~V*这种看似自然（且数学上精确！）的方法的困难在于方程(3)的系统可能是高度病态的。这意味着系数矩阵MM的微小变化会产生解a的巨大变化。1957年，George Forsythe表明，通过对多项式拟合问题使用正交多项式，可以大大减少与(3)相关的病态量。Gram-Schmidt过程通常用于生成正交多项式。这个过程的优点是，当从n - 1次多项式拟合到n - a次多项式拟合时，不必从头开始
Example. With reference to the previous example of polynomial fitting, the *th, jth element of the coefficient matrix MM in (3) is given by 2 ^ £ i tk~ and the ith. element of the right hand side My of (3) is simply ^2JZI fc*~V*The difficulty with this seemingly natural (and mathematically exact!) approach is that the system of equations (3) can be highly ill-conditioned. This means that slight changes in the coefficient matrix MM produce enormous changes in the solution a. In 1957 George Forsythe showed that by using orthogonal polynomials for the polynomial fitting problem, the amount of ill-conditioning associated with (3) could be reduced considerably. The Gram-Schmidt process is usually used to generate the orthogonal polynomials. This process has the advantage of not having to start from scratch when going from a polynomial fit of degree n— 1 to one of degree n. A number