An approach to improving the stability of triangular decomposition of a dense positive definite matrix with a large condition number by using the Gauss and Cholesky methods is considered. It is proposed to introduce additions to standard computational schemes with an incomplete inner product of two vectors which is formed by truncating the lower digits of the sum of the products of two numbers. The truncation in the process of decomposition increases the diagonal elements of the triangular matrices by a random number and prevents the appearance of very small numbers during the Gauss decomposition and a negative radical expression in the Cholesky method. The number of additional operations required for obtaining an exact solution is estimated. The results of computational experiments are presented.

中文翻译：

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改善病态矩阵三角分解的稳定性

考虑使用高斯和乔尔斯基方法提高条件数大的稠密正定矩阵三角分解的稳定性的方法。提议将具有两个向量的不完整内积的标准计算方案引入加法运算，这是通过将两个数的积之和的低位数字截断而形成的。分解过程中的截断使三角矩阵的对角元素增加了一个随机数，并防止了在高斯分解过程中出现非常小的数字以及在Cholesky方法中出现负根表示。估计获得精确解所需的其他操作数。给出了计算实验的结果。