This is numerical analysis

1. Construct an efcient routine for solving linear systems. Let 10 7 0 2 4 2 4 U = O 2 6 , A = 4 9 3 , b = 8 2 1 7 6 0 0 5 (a) Create two routines for an n x n matrix, one that implements back substitution, and the other forward substituion. Each should take a matrix U and a vector b as input, and should output a. vector :1: such that U a: = I). Apply your functions to solve U a: = b and U Tm = b and show your results. (b) Create a routine that, when given a matrix A, computes the LU factorization (PA = LU) with partial pivoting. The function should output both L and U, and should also output P. Apply your function to A above and show the result. (c) Create a routine that takes a matrix A, a vector b and outputs a vector :5 such that An: = b. Your code should use the programmes from (a) and (b). Apply your routine to the pair A, b above and show the results