348 CHAPTER 12 - 2-norm of the right-hand side vector: ||b- Ax)||/|bl|. Comment on how the number of iterations required to reduce this quantity below a given level, say, 10-6, appears to be related to the matrix size. For a right-hand side vector, take b = (b₁,..., b), where b; = (i/(n + 1))², i = 1,...,n, and take the initial guess x(0) to be the zero vector. (We will see in chapter 13 that this corresponds to a finite difference method for the two-point boundary value problem: u"(x) = x², 0 < x < 1, u(0) = u(1) = 0, whose solution is u(x) = (x - x). The entries in the solution vector (x-x). 12 x of the linear system are approximations to u at the points i/(n + 1), i = 1,..., n.) 10. Implement the conjugate gradient algorithm for the matrix A in Exercise 5; that is, A = 2-1 -1 2. -1 -12 Try some different matrix sizes, say, n = 10, 20, 40, 80, and base your convergence criterion on the 2-norm of the residual divided by the

348 CHAPTER 12 - 2-norm of the right-hand side vector: ||b- Ax)||/|bl|. Comment on how the number of iterations required to reduce this quantity below a given level, say, 10-6, appears to be related to the matrix size. For a right-hand side vector, take b = (b₁,..., b), where b; = (i/(n + 1))², i = 1,...,n, and take the initial guess x(0) to be the zero vector. (We will see in chapter 13 that this corresponds to a finite difference method for the two-point boundary value problem: u"(x) = x², 0 < x < 1, u(0) = u(1) = 0, whose solution is u(x) = (x - x). The entries in the solution vector (x-x). 12 x of the linear system are approximations to u at the points i/(n + 1), i = 1,..., n.) 10. Implement the conjugate gradient algorithm for the matrix A in Exercise 5; that is, A = 2-1 -1 2. -1 -12 Try some different matrix sizes, say, n = 10, 20, 40, 80, and base your convergence criterion on the 2-norm of the residual divided by the

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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