LU DESCOMPOSITION

In linear algebra, the LU decomposition is a matrix decomposition which writes a matrix as the product of a lower triangular matrix and an upper triangular matrix. The product sometimes includes a permutation matrix as well. This decomposition is used in numerical analysis to solve systems of linear equations or calculate the determinant

Solving linear equations

Given a matrix equation Ax=LUx=b we want to solve the equation for a given A and b. In this case the solution is done in two logical steps:

1.First, we solve the equation Ly = b for y
2.Second, we solve the equation Ux = y for x.
Note that in both cases we have triangular matrices (lower and upper) which can be solved directly using forward and backward substitution without using the Gaussian elimination process (however we need this process or equivalent to compute the LU decomposition itself). Thus the LU decomposition is computationally efficient only when we have to solve a matrix equation multiple times for different b; it is faster in this case to do an LU decomposition of the matrix A once and then solve the triangular matrices for the different b, than to use Gaussian elimination each time.




LU DESCOMPOSITION



LU DESCOMPOSITION.