Convergence of iterative methods like the Conjugate Gradient Method can be accelerated by the use of a technique called preconditioning. The convergencerates of iterative methods often depend, directly or indirectly, on the condition number of the coefficient matrix A. The idea of preconditioning is to reduce theeffective condition number of the problem.The preconditioned form of the n X n linear system Ax = b iswhere M is an invertible n x n matrix called the preconditioner.When A is a symmetric positive-definite n x n matrix, we will choose a symmetric positive-definite matrix M for use as a preconditioner. A particularly simplechoice is the Jacobi preconditioner M = D, where D is the diagonal of A. The Preconditioned Conjugate Gradient Method is now easy to describe:Replace Ax = b with the preconditioned equation M-¹Ax = M-¹b, and replace the Euclidean inner product with (v, w) M.To convert Algorithm 2 in Section 3.3 to the preconditioned version, let zk = M-¹b - M-¹ Axk = M¹. Then the algorithm is1. Initialize x as any vector. Set ro = b - Axo and u。 = Zo = M-¹ro.2. For k = 0,1,..., n - 1:rizkA. ak =u AukB. Jk+1 = k taklkC. Tk+1=rk - ak AukD. if (k+1 < €):a. breakE. Zk+1 = M-¹7k+1F. bk =THzk+1rT ZkG. Uk+1 = Zk+1 +bkukM¹Ax = M¹b,3. return xk+1. Now, consider the following problem.Let A be the n x n matrix with n = 1000 and entries A(i, i) = i, A(i, i + 1) = A(i + 1, i) = 1/2, A(i, i + 2) = A(i + 2, i) = 1/2 for all i that fit within thematrix.(a) Take a look at the nonzero structure of the matrix using plt.spy(A).(b) Let xe be the vector of n ones (exact solution). Set b = Axe, and apply the Conjugate Gradient Method, without preconditioner, and with the Jacobipreconditioner. Compare errors (using 2-norm) of the two runs in a plot versus step number (using semilogy). (So you need to modify the conjugate gradientcodes to keep track of and return the solutions of all steps.) Use eps = 1e-10.The two methods may converge in different number of steps. Which one do you see is faster?

The Python code to fix the issue is provided here:

1. Initialize xo as any vector. Set ro = b - Axo and uo = zo = M-¹ro. 2. For k = 0, 1,...,n - 1: A. ak = u Auk B. Xк+1 = xk + akuk C. rk+1=rkak Auk D. if (k+1 < €): a. break E. Zk+1 = M-¹rk+1 +1²k+1 F. bk G. Uk+1 = Zk+1+bkuk 3. return *k+1.

1. Initialize xo as any vector. Set ro = b - Axo and uo = zo = M-¹ro. 2. For k = 0, 1,...,n - 1: A. ak = u Auk B. Xк+1 = xk + akuk C. rk+1=rkak Auk D. if (k+1 < €): a. break E. Zk+1 = M-¹rk+1 +1²k+1 F. bk G. Uk+1 = Zk+1+bkuk 3. return *k+1.

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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