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MATH4602 Scientific Computing
创建日期：2024-03-29 10:11:38
所属分类：数学mathematics
标签： MATH4602
Scientific Computing


MATH4602 Scientific Computing

Mimi Project

Due Date: 23 April 2024 (5:00 pm)

In this MATLAB coding assignment, we focus on solving the following structured linear system of equations:

Ax = b

where A is an n2  × n2  matrix taking the following form.

Here In  is the n × n identity matrix and

It is a block-Toeplitz matrix with Toeplitz blocks.  We first consider solving the linear system by using an iterative method, the Gauss-Seidel method.

(a) (i) Show that A is a symmetric positive definite matrix for d ≥ 2.  Therefore Gauss- Seidel method converges when applied to solving the linear system Ax = b.

(ii) Let the right-hand side vector b = Ae where

e = [1 1  ...  1]T

and the true solution of Ax = b is e.  Write a MATLAB program to implement the Gauss-Seidel method for solving Ax = b with the initial guess x0  = 0 and the stopping criterion is as follows:

||xn - e|| 2  < 10-6

where xn  is the approximate solution obtained in the nth iteration.

(iii) What is the computational cost in each iteration of the Gauss-Seidel method?

(iv) Report the number of iterations for convergence for the following pairs of (n,d), n = 10, 20, 30, 40 and d = 2, 3, 4, 5. Discuss your observations.

(b) We then consider solving the same linear system of equations by using the Block Jacobi method. The idea is to consider the splitting of the matrix A as the sum of the diagonal block bimatrix

and the bi-diagonal block matrix (A - D).

(i) Write a MATLAB program to implement the Block  Jacobi  method for solving Ax = b with the initial guess x0  = 0 and the same stopping criterion

||xn - e|| 2  < 10-6

where xn  is the approximate solution obtained in the nth iteration.

(ii) What is the computational cost in each iteration of the block Jacobi method?

(iii) Report the number of iterations for convergence for the following pairs of (n,d), n = 10, 20, 30, 40 and d = 2, 3, 4, 5. Discuss your observations.

(c) We then consider solving the same linear system of equations by using the Block Gauss-Seidel method. The idea is to consider the splitting of the matrix A as the sum of the lower triangular blocks

and the upper triangular blocks (A - L).

(i) Write a MATLAB program to implement the block Gauss-Seidel method for solving Ax = b with the initial guess x0  = 0 and the same stopping criterion

||xn - e|| 2  < 10-6

where xn  is the approximate solution obtained in the nth iteration.

(ii) What is the computational cost in each iteration of the block Gauss-Seidel method?  (iii) Report the number of iterations for convergence for the following pairs of (n,d), n = 10, 20, 30, 40 and d = 2, 3, 4, 5. Discuss your observations.

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MATH 239 QF5203 INFR08010 IT114105 FIT2093 ECON3124 ECON1195 MATH 337 STAT 3006 COMP202 COMP3130 MATH3609 COMP 250 CS6140 IEEE ELE00041I FIN40090 CIV 5899 MATH3066 IGNMENT MATH 2009 STAT3405 HIST 101 MATH 3281 CEGE0042 ICT532 LAB 7 LAB 5 MCO455 CSIT214 Q4 IFB104 A1 ECO220Y5Y DA5020 CMPSC473 FIN9007 DSCI553 MSCI534 COMP3308 CSE-112 CMPT 310 EE104 F 2C P20 ALY2010 BUSM060 STA258 ECO00030M R BIOSTAT 274 COS6012-B FIT3155 CMPUT 340 EC421 INFS4205 ECOM135 NATS 1740 COMP809 MBE 6005 STA465 CSSE2310 CIVL2700 DATA4700 CMPSC 473 ISE 529 CS5044 MAST30020 COMP2207 ENGG1001 COMP0046 FIT3165 STAT GR5241 MECH0007 vATHK1001 ANALYTIC ATHK1001 MATH352001 HW3 S1 BMAN 24662 COMP 636 Stat-461 STAT 302 STAT 2132 STAR534 ENG4137 CMP3752M CS3103 BISM3222 STA 135 ME155A IIMT3601 ECON3371 ECON 3371 HW4 FINA 5840 DT211C CTEC3902 MAT005 MATH2148 S261F Econ 332 MATH3734 MIS41270 7CCMFM13 SYST 664 CIS434 CENG10014 MT4111 MT4527 ECMM424 CMT219 Math 136 MATH 3120 COMP10002 MAST20009 MATH0058 BEEM117 ISYE 6420 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