"数值分析第二次习题课闵旭：Jacobi 和 G-S 迭代法的收敛性问题"

The second exercise of the second tutorial; Chapter 4 Exercise 3 examines the convergence of the Jacobi and G-S iterative methods; the initial value [0, 0, 0]⊤ is used to solve the Jacobi and G-S iterative methods. When ∥x(k 1) − x(k)∥ < 10−2, the iteration terminates. The problem in the fourth chapter: 5x1 + 2x2 + x3 = -12, -x1 + 4x2 + 2x3 = 20, 2x1 - 3x2 + 10x3 = 3; the convergence of the Jacobi and G-S iterative methods is considered; the initial value [0, 0, 0]⊤ is used to solve the Jacobi and G-S iterative methods. When ∥x(k 1) − x(k)∥ < 10−2, the iteration terminates. According to Theorem 4.11, the matrix A is strictly diagonally dominant, so it converges. Jacobi iteration: x(k 1)1= -25x(k)2- 15x(k)3- 125x(k 1)2= 14x(k)1- 12x(k)3- 125x(k 1)3= -3 + 3x(k)1+ 25x(k)2 G-S iterative method: x(k 1)1= -25x(k)2- 05x(k)3- 125x(k 1)2= 15x(k 1)1- 10x(k)3- 125x(k 1)3= -3 + x(k)1+ 25x(k 1)2 The iterations of the Jacobi and G-S methods are calculated, and the equation is iterated until the value of x(k 1) and x(k) satisfies the given condition. The convergence of the iterations is determined by comparing the result of each iteration with the set threshold of 10−2. In summary, the exercise involves examining the convergence of the Jacobi and G-S iterative methods for a set of linear equations. The initial value [0, 0, 0]⊤ is used to solve the Jacobi and G-S iterative methods. The convergence of the iterations is determined by comparing the results of each iteration with a threshold, and the iterations continue until the threshold is met. The exercise serves as a practical application of the theoretical concepts of iterative methods and convergence in numerical analysis. Through this exercise, students can gain a better understanding of the practical implementation and convergence properties of Jacobi and G-S iterative methods.