Talk:Gaussian elimination: Difference between revisions
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What is the relationship of this task to [[LU decomposition]]? –[[User:Dkf|Donal Fellows]] 09:01, 12 February 2012 (UTC) |
What is the relationship of this task to [[LU decomposition]]? –[[User:Dkf|Donal Fellows]] 09:01, 12 February 2012 (UTC) |
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:Basically, Gaussian elimination is the same as LU decomposition followed by backsubstitution. To solve the system AX=B with LU decomposition, you multiply on the left by P: PAX=PB, hence LUX=PB. Then you solve first the lower triangular system LZ=PB, which gives a vector Z (or matrix if B is a matrix, which happens if you want to solve several systems at the same time with the same matrix A). Then you solve the upper triangular system UX=Z, and you are done. |
:Basically, Gaussian elimination is the same as LU decomposition followed by backsubstitution. To solve the system AX=B with LU decomposition, you multiply on the left by P: PAX=PB, hence LUX=PB. Then you solve first the lower triangular system LZ=PB, which gives a vector Z (or matrix if B is a matrix, which happens if you want to solve several systems at the same time with the same matrix A). Then you solve the upper triangular system UX=Z, and you are done. |
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:The difference lies in the order the operations are done: in Gaussian elimination, the lower triangular matrix L is not stored, since the computation of Z (that is, the solution of the lower triangular system LZ=PB) is done once and for all in the first step. Likewise, the permutation matrix P is not needed because the rows of B are swapped at the same time the rows of A are swapped. The backsubstitution step is identical in Gaussian elimination and LU decomposition. |
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:The Stata section shows both methods. |
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:[[User:Eoraptor|Eoraptor]] ([[User talk:Eoraptor|talk]]) 16:22, 26 October 2017 (UTC) |
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Latest revision as of 16:22, 26 October 2017
Task relationship?
What is the relationship of this task to LU decomposition? –Donal Fellows 09:01, 12 February 2012 (UTC)
- Basically, Gaussian elimination is the same as LU decomposition followed by backsubstitution. To solve the system AX=B with LU decomposition, you multiply on the left by P: PAX=PB, hence LUX=PB. Then you solve first the lower triangular system LZ=PB, which gives a vector Z (or matrix if B is a matrix, which happens if you want to solve several systems at the same time with the same matrix A). Then you solve the upper triangular system UX=Z, and you are done.
- The difference lies in the order the operations are done: in Gaussian elimination, the lower triangular matrix L is not stored, since the computation of Z (that is, the solution of the lower triangular system LZ=PB) is done once and for all in the first step. Likewise, the permutation matrix P is not needed because the rows of B are swapped at the same time the rows of A are swapped. The backsubstitution step is identical in Gaussian elimination and LU decomposition.
- The Stata section shows both methods.
- Eoraptor (talk) 16:22, 26 October 2017 (UTC)