Matrix 高斯消元Gaussian elimination 中的complete pivoting和partial pivoting

首先科普下Pivoting的含義

通常翻譯爲「主元」,在對矩陣作某種算法時,首先進行的部分元素.在線性規劃的單純形法中常見.
wiki的解釋以下：
Pivot element
（the first element distinct from zero in a matrix in echelon form）
The pivot or pivot element is the element of a matrix,which is selected first by an algorithm (e.g.Gaussian elimination,Quicksort,Simplex algorithm),to do certain calculations with the matrix.
The above mentioned matrix algorithms require an entry distinct from zero in pivot position to work properly or at all respectively.Depending on the algorithm either one (random) element distinct from zero or the element with the greatest absolute value in a row or column is chosen.This is called pivotization.The row containing the pivot element is called pivot row,the pivot element's column is called pivot column.
Pivot element in Quicksort means the element that is selected as boundary for partitioning.Quicksort sorts all elements „left「 and „right「 of the pivot element recursively.

 

區別：

（1）

complete pivoting（即full pivoting），就是在矩陣分解或高斯消元的過程當中，主元是在未進行變換的全部行和列之間進行選擇。也就是須要同時進行行交換和列交換。
（2）

partial pivoting就是隻在當前進行變換的列中選擇主元，只須要進行行交換

 

通常full pivoting相比partial pivoting更加穩定，可是更麻煩，速度會慢。
就是速度和精度之間的問題。