A Report on the Accuracy of Some Widely Used Least Squares Computer Programs
若干常用最小二乘计算程序的数值精度报告AI 解读 ↓
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若干常用最小二乘计算程序的数值精度报告
- 研究问题
- 不同算法与计算精度设置，会怎样影响最小二乘系数的数值准确性，广泛使用的计算程序之间是否存在明显差异？
- 方法与思路
- 作者以五次多项式拟合问题测试二十多个程序，在多种计算机及单、双精度下比较不同算法给出的回归系数。
- 材料陈述的贡献
- 摘要称，所测正交化方法通常比消元方法更准确；表现最好的程序采用双精度内积累加与迭代改进。
- 阅读关注点
- 应核对测试问题的病态程度、参考系数和误差指标，以及具体算法实现、精度设置与迭代改进对结果的影响。
- 阅读边界
- 仅据版本未注明的 OpenAlex 索引摘要；结果针对当时程序及测试问题，不能据此评价现今软件或推断刊本变化。
原文摘要
Linear least squares test problems based on fifth degree polynomials have been run on more than twenty different computer programs in order to assess their numerical accuracy. The programs tested, all in present-day use, included representatives from several statistical packages as well as some from the SHARE library. Essentially five different algorithms were used in the various programs to obtain the coefficients of the least squares fits. The tests were run on several different computers, in double precision as well as single precision. By comparing the coefficients reported, it was found that those programs using orthogonal Householder transformations, classical Gram-Schmidt orthonormalization or modified Gram-Schmidt orthogonalization were generally much more accurate than those using elimination algorithms. Programs using orthogonal polynomials (suitable only for polynomial fits) also proved to be superior to those using elimination algorithms. The most successful programs accumulated inner products in double precisionand made use of iterative refinement procedures.