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Question 3. (30 Pts) Let LU is the Crout LU decomposition of the nx n tridiagonal matrix 01. 1012 02.1 222 223 03.2 23,3 43,4 Gr-1,-2 -1,-1 On-1,60 BIL 332/532 - Numerical methods Homework 2 Note that the Crout decomposition implies that the diagonal of U consists of 1s. Besides, ky 0 if j < (i-1) and j>i and wij = 0 ifj<i and j> (i+1) because of the nonzero structure of A. Write a pseudocode to compute the nonzero elements of L and U by deriving the a series of recursive formulas for the nonzero elements starting with aus eius fori,j=1,2,..., M. Question 4. (10 Pts) Write the element-based formula for the Gauss-Seidel method that solves the linear system 01. 101.2 ... 01. (2,1 02.2 ... 02. L., 01.2... One where x denotes ith element of the solution vector at iteration k.

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Answer 1

Solution - We write method iteration We write Grun seide Gauss- Seide as - by 3 x (1) _ I [6,- (att (by giren matrix we write) Clo (k 14 they = | [be - (a, et ta thot ... thanx)] + (k o [bn– Cani xaw + Qorxon (k+1) (kty O meony A- ann- this method is called the method displacement. Since, the elements of are successive replaced successively