# homework代写 | math – math

### math

homework代写 | math – 这个题目属于一个math的代写任务, 包括了math等方面, 该题目是值得借鉴的homework代写的题目

`````` math 5640/6640 - Introduction to Numerical Analysis II
Thi-Thao-Phuong Hoang, Spring 2022
homework 6, DueFriday, Mar 25
``````

There are totally 4 problems (2 pages).Submit the code for Problems 2 and 3 in the appendix.

Problem 1 [25 points](Pen and paper) Consider solving the linear systemAxxx=bbbiteratively, where

``````A=
``````

#### 1 1 3

`````` and bbb=
``````

#### .

``````(a) Apply the Jacobi iterative method with initial guessxxx 0 = [1 0 0]T. Compute the first two
iteratesxxx 1 andxxx 2.
(b) Apply the Gauss-Seidel iterative method with initial guessxxx 0 = [1 0 0]T. Compute the first
iteratexxx 1.
``````

Problem 2 25 points Write the code to perform Jacobi and Gauss-Seidel methods for solving the linear system in Problem 1. Usingrrrk 2 < 10 ^6 as the stopping criteria for both methods. Print out the residual normrrrk 2 for all iterations and the final solution.

Problem 3 25 points Consider solving the linear systemAxxx=bbbiteratively, where

#### 0 1 4

`````` and bbb=
``````

#### .

``````(a) Letbe a vector given by= 1 : 0.05 : 1.95 = [1 1.05 1. 10... 1 .95]. Denote byjthejth
entry in the vector, for 1j20. For eachj, apply the successive over-relaxation (SOR)
method to solve the linear system with the relaxation parameteri. Usingrrrk 2 < 10 ^8 as
the stopping criteria and record the total iteration numbers neededNjto achieve the desired
tolerance for eachj.
``````
• Print out total iteration numbersNjfor allj.
• What is the corresponding relaxation parameterjwhenNjobtains the minimum?
``````(b) What is the optimal relaxation parameteroptfor the SOR method for solving this linear
system? Compare the valueoptwith the optimal relaxation parameterjobtained in (a).
``````

Problem 4 [25 points](Paper and pen) Consider solving the linear system Axxx = bbbby the stationary iterative method

``````xxxk+1=xxxk+M^1 (bbbAxxxk) =Txxxk+M^1 bbb,
``````

#### 1

whereT=IM^1 Ais the iteration matrix. Let

#### 1 2 5

`````` and bbb=
``````

#### .

``````(a) Calculate the normsTandT 1 when the Jacobi method is applied. Will the Jacobi
``````(b) Calculate the normsTandT 1 when the Gauss-Seidel method is applied. Will the