6. Consider a 5 X 5 matrix A that has only nonzero elements in the main 3 diagonals, that is, adj : O for |i 7 jl > 1. What can you say about the sparsity patterns of the factors L and U of A? Now, consider the same tridiagonal matrix A with a nonzero element in A15. What can you say about the sparsity pattern of the factors L and U of A in this case? 6 Let A be matrix a A - ag3 O a43 aqu O O O 954 955 To calculate LU Decomposition and sparse of matrix it will be difficult to find out them with variable so consider a matrix A' similar to A with constands A = 0 O O A' = LU where O OOO L have non-zeros in the main L= 0 0 diagonal and in the lower diagonal, so sparsity =16/25. 10 O 0 0.5Let A be matrix a O OO a O ag3 0 O O O O 954 955 To calculate LU Decomposition and sparse of matrix it will be difficult to find out them with variable so consider a matrix A' similar to A with constands A 0 L 1 O O A ' = LU where O 0 0 0 7 L= O A - 1 10 O 0.5 1 U have non-zeros in the main diagonal and in the upper diagonal, so sparsity =?/25.O 0 O 0 2 0 O 0 0.5 - and v are calculated as above We know that sparse of matrix is the ratio of zero valued elements to the total #s of elements Hence u = 15 5 0.6 - 60% 25 5 L = 14 O. 56 56% 25 AforA = 1Z = 0.48- 48%% 25 In the second part the matrix A of A' transposed to 10 01 1 100 A or n - 0 1 1 10 O0 1 1 1 O0 011 A = LU0 0 0 0 7 O 1 0 0 0 I O L = 10 10 0 U = OD 0 -1 L will have non-zero in the - 1 0 0 0 2 main diagonal and lower diagonal, but U will have O O , 0.5 non-zero in upper, main diagonal, and all element in the 5th column.| ce Lip = 14 = 0.56 = 56% 25 0 U = 14 = 0.56 - 56% 25 A or A = 11 = 0.44 = 44 20 25 In the first part, spargity of a greater than sparsity of I and A In the second part , sparsity of I and u are equal and greater than A