Why lu decomposition

What was necessity for LU Decomposition, i. A typical example is when you are solving a partial differential equation for different forcing functions. For these different forcing functions, the meshing is usually kept the same. Another example is when you are solving a time dependent problem, where the unknowns evolve with time. Once you have this factorization, the cost of solving i. Why use LU-decomposition? Sign up to join this community.

The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. Asked 8 years, 10 months ago. Active 9 months ago. Moreover, Gaussian elimination in its pure form may be unstable. While LU-decomposition is a useful computational tool, but this does not work for every matrix. Consider even the simple example with matrix. In this case, we say that A has a PLU factorization.

For a permutation matrix P , the product PA is a new matrix whose rows consists of the rows of A rearranged in the new order. Note that a product of permutation matrices is a permutation matrix.

Pivoting for LU factorization is the process of systematically selecting pivots for Gaussian elimination during the LU-decomposition of a matrix. The LU factorization is closely related to Gaussian elimination, which is unstable in its pure form.

To guarantee the elimination process goes to completion, we must ensure that there is a nonzero pivot at every step of the elimination process. This is the reason we need pivoting when computing LU decompositions. But we can do more with pivoting than just making sure Gaussian elimination completes. We can reduce roundoff errors during computation and make our algorithm backward stable by implementing the right pivoting strategy.

Depending on the matrix A , some LU decompositions can become numerically unstable if relatively small pivots are used. Relatively small pivots cause instability because they operate very similar to zeroes during Gaussian elimination. Through the process of pivoting, we can greatly reduce this instability by ensuring that we use relatively large entries as our pivot elements. This prevents large factors from appearing in the computed L and U , which reduces roundoff errors during computation.

The goal of partial pivoting is to use a permutation matrix to place the largest entry of the first column of the matrix at the top of that first column. For this matrix, this means we want 4 to be the first entry of the first column. Here value of l 21 , u 11 etc can be compared and found. Gauss Elimination Method According to the Gauss Elimination method: Any zero row should be at the bottom of the matrix. The first non zero entry of each row should be on the right-hand side of the first non zero entry of the preceding row.

This method reduces the matrix to row echelon form. Now, reduce the coefficient matrix A, i. The matrix so obtained is U. To find L, we have two methods. The other method is that the remaining elements are the multiplier coefficients because of which the respective positions became zero in the U matrix. This method is a little tricky to understand by words but would get clear in the example below Now, we have A the nXn coefficient matrix , L the nXn lower triangular matrix , U the nXn upper triangular matrix , X the nX1 matrix of variables and C the nX1 matrix of numbers on the right-hand side of the equations.

Now, we first consider and convert it to row echelon form using Gauss Elimination Method. So, by doing 1 2 we get Now, by doing 3 we get. So, we have Solving, we get , and. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.

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