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MATH0033 Numerical Method
创建日期：2024-08-12 17:50:38
所属分类：数学mathematics
标签： MATH0033
Numerical Method

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MATH0033 Numerical Method

Theoretical exercise sheet 2

Linear systems

Exercises 1 and 5 (marked *) to be submitted via *Crowdmark* in pdf

format (either handwritten and scanned, or typeset using LaTeX).

EXERCISE 1(*) Let.

(a) The Jacobi and Gauss-Seidel methods for the approximation of the solution to the linear

system Ax = b can both be written in the form

Pxk+1 = Nxk + b.

Write down the matrices P and N for each of the two methods. What are the associated

iteration matrices BJ and BGS?

(b) Compute the vector x1 obtained after one iteration with the Jacobi method starting from

the initial vector x0 .

(c) Do both methods converge? Which gives the iteration matrix with the smallest spectral

radius?

(d) Prove that both methods converge linearly with respect to the norm ‖ · ‖∞, and compare

the convergence constants.

EXERCISE 2 Let.

(a) Write the Gauss-Seidel method for the solution of the linear system Ax = b. What is the

associated iteration matrix BGS?

(b) What can be said about the convergence of the Gauss-Seidel method for this system?

(c) Now consider the iteration

D(xk+1 ? xk) = αrk k ≥ 0,

where D is the diagonal of A and rk = b?Axk is the residual. This method is sometimes

called the Jacobi over-relaxation (JOR) method; note that for α = 1 the method is

equivalent to the Jacobi method.

Write down the associated iteration matrix BJOR, and find the optimal choice of α min-

imising the spectral radius ρ(BJOR).

(d) Starting from the initial vector x(0) = (1, 1)T , compute the first iteration x1 of the JOR

method using the optimal choice of α you derived in part (c).

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关键词一 关键词二 关键词三 关键词四 关键词五 和法规和的发更好的风格和 ECON 615 FINC 612 CSC148 COMP 639 ACCT 203 ACCT 211 PSYC 101 MTH2009 COMP52315 PSYCO 432 statistics MATH4063 11 xxx 1 MA1MSP MAST30028 MATH3888 CS341 EEE30002 ES2F5 MCD4160 CC0933 ECON 1101 MCHM3001 CHEM3118 COMMGMT 2511 COMP41280 STAT 631 24T1 IRE379 SENG365 ECON433 FIT3139 MATH4602 LING5621M FINS5516 ECON3208 MN2177 L2 SM EEET1415 Finance DS4023 COMP3004 7CCSMRTS K1S5B6 Stat230 CSCI4211 CSI2132 ECON30130 MAS6100 ENVS222 COM2107 INFO1112 STAT 4004 FIT2100 comp2022 COMP4691 Physics 307 Econ 659 probability CSCI316 fir29 CSCI203 COMP3160 COMP6714 INFO1113 Physics 7B Physics Photonics S203031 STAT2005 COMP90015 CSCI 2110 CSCC46 Information CSC477 COMP4650 Statistical COMP 346 COMP1017 ELEC340 MATH1007 Programming function STAT 244 CS 134 Java cse13s Math 108A FW20 ECON3389 CSE 130 Science 331 COMP9020 engineering COSC 445 CSE 5523 GAME2012 visualization MATH 235 Math 370 CSC343 COGS 108 EE524 COMP10200 STAT 416 3FK4 MSCI 311 ECON6113 ACC40700 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