Question

Give example with matrix of how LU decomposition works in image processing with explanation.

Answer 1

In numerical analysis and linear algebra, LU decomposition (where ‘LU’ stands for ‘lower upper’, and also called LU factorization) factors a matrix as the product of a lower triangular matrix and an upper triangular matrix. Computers usually solve square systems of linear equations using the LU decomposition, and it is also a key step when inverting a matrix, or computing the determinant of a matrix. The LU decomposition was introduced by mathematician Tadeusz Banachiewicz in 1938.

Let A be a square matrix. An LU factorization refers to the factorization of A, with proper row and/or column orderings or permutations, into two factors, a lower triangular matrix L and an upper triangular matrix U, A=LU.

Doolittle Algorithm :
It is always possible to factor a square matrix into a lower triangular matrix and an upper triangular matrix. That is, [A] = [L][U]

Lower Triangular O UOO 001 U02 A00 A10 A20 A01 A02 All A12 A21 A22 = L10 1 0 L20 121 1 U11 U12 0 122 0 Upper Triangular

Doolittle’s method provides an alternative way to factor A into an LU decomposition without going through the hassle of Gaussian Elimination.

For a general n×n matrix A, we assume that an LU decomposition exists, and write the form of L and U explicitly. We then systematically solve for the entries in L and U from the equations that result from the multiplications necessary for A=LU.

For each i = 0, 1, 2 ....n-1: Ui,k = Ai,k - (Li,j Uj,k) for k=i,i+1...n-1 produces the kth row of u. Ls,« = ( As,k - Ś (Ls; U,x) ) / UK,K for i=k+1, +2, ....n-1 and Li, i = 1 produces the kth column of L


Input : 2 1 27 Output : Lower Triangular Upper Triangular 1 0 0 2 -2 1 0 0 -2 - 110 -1 4 -2 -1 3



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