Solving Tri-Diagonal Linear Systems using Crout Factorization Algorithm, Assignments of Mathematics

Download Solving Tri-Diagonal Linear Systems using Crout Factorization Algorithm and more Assignments Mathematics in PDF only on Docsity! M348 Homework #13 Burden and Faires. Section 6.6 (#2abc, 4b, 6b). Ortega and Grimshaw. Chapter 19: #19.1, 19.4. Programming mini-project. In contrast to initial-value problems, where the approximate solution is obtained by marching forward, boundary-value problems lead to a system of equations. For example, when discretized using central differences on a uniform grid, the boundary-value problem u′′(x) − q(x)u(x) = f(x), 0 ≤ x ≤ 1, u(0) = α, u(1) = β, x x 10 0 n+1 leads to the tri-diagonal linear system of equations         d1 1 0 0 0 0 1 d2 1 0 0 0 0 1 d3 1 0 0 . . . 0 0 0 1 dn−1 1 0 0 0 0 1 dn                 w1 w2 w3 ... wn−1 wn         =         h2f1 − α h2f2 h2f3 ... h2fn−1 h2fn − β         where di = −(2+h 2qi) and h = 1/(n+1) is the grid spacing. Here we use a linear system solver to study the approximate solution wi as the grid is refined. We consider the case when q(x) = 4, f(x) = 2x, α = 0, β = 1 and for reference we note that the exact solution is u(x) = (3e2−2x − 3e2+2x + xe4 − x)/(2− 2e4). (a) Write a C++ program to implement Algorithm 6.7 from the Burden and Faires text (Crout Fac- torization Algorithm) for a tri-diagonal linear system Aw = b of arbitrary size n. (b) Use your program to compute w for n = 10, 20 and 40 and compare the approximate and actual solutions at x = 1/2. Does the error decrease as n is increased? For what value of n is the error below 10−5?