Difference between revisions of "LU decomposition using Gaussian elimination with pivoting"
Jump to navigation
Jump to search
| [unchecked revision] | [quality revision] |
| Line 3: | Line 3: | ||
'''Gaussian elimination with pivoting''': for a given square nonsingular matrix A, this method obtains the LU decomposition (<math>P_{1} A P_{2} = L U</math>) of a permuted matrix <math>P_{1} A P_{2}</math>. All the variants of the method are based on the idea of eliminating nonzero entries using elementary matrices. | '''Gaussian elimination with pivoting''': for a given square nonsingular matrix A, this method obtains the LU decomposition (<math>P_{1} A P_{2} = L U</math>) of a permuted matrix <math>P_{1} A P_{2}</math>. All the variants of the method are based on the idea of eliminating nonzero entries using elementary matrices. | ||
| − | There exist variants of the method with the choice of the pivot within the pivot column ([[LU-разложение методом Гаусса с выбором ведущего элемента по столбцу|column pivoting]]) (<math>P_{2}=E</math>), within the pivot row [[LU-разложение методом Гаусса с выбором ведущего элемента по строке|row pivoting]] (<math>P_{1}=E</math>), within the active submatrix [[LU-разложение методом Гаусса с выбором ведущего элемента по всей матрице|complete pivoting]] (<math>P_{1} \ne P_{2}</math>), and on the principal diagonal [[LU-разложение методом Гаусса с выбором ведущего элемента по главной диагонали|diagonal pivoting]] (<math>P_{1} = P_{2}</math>). The first two variants are most popular and most common in program libraries. | + | There exist variants of the method with the choice of the pivot within the pivot column ([[LU-разложение методом Гаусса с выбором ведущего элемента по столбцу|column pivoting]]) (<math>P_{2}=E</math>), within the pivot row ([[LU-разложение методом Гаусса с выбором ведущего элемента по строке|row pivoting]]) (<math>P_{1}=E</math>), within the active submatrix ([[LU-разложение методом Гаусса с выбором ведущего элемента по всей матрице|complete pivoting]]) (<math>P_{1} \ne P_{2}</math>), and on the principal diagonal ([[LU-разложение методом Гаусса с выбором ведущего элемента по главной диагонали|diagonal pivoting]]) (<math>P_{1} = P_{2}</math>). The first two variants are most popular and most common in program libraries. |
[[Category:Finished articles]] | [[Category:Finished articles]] | ||
[[ru:LU-разложение методом Гаусса с перестановками]] | [[ru:LU-разложение методом Гаусса с перестановками]] | ||
Latest revision as of 11:40, 5 March 2018
Gaussian elimination with pivoting: for a given square nonsingular matrix A, this method obtains the LU decomposition ([math]P_{1} A P_{2} = L U[/math]) of a permuted matrix [math]P_{1} A P_{2}[/math]. All the variants of the method are based on the idea of eliminating nonzero entries using elementary matrices.
There exist variants of the method with the choice of the pivot within the pivot column (column pivoting) ([math]P_{2}=E[/math]), within the pivot row (row pivoting) ([math]P_{1}=E[/math]), within the active submatrix (complete pivoting) ([math]P_{1} \ne P_{2}[/math]), and on the principal diagonal (diagonal pivoting) ([math]P_{1} = P_{2}[/math]). The first two variants are most popular and most common in program libraries.
