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MAST30028 Numerical Methods & Scientific Computing

创建日期：2024-08-12 15:54:57

所属分类：数学mathematics

标签： MAST30028

Numerical Methods & Scientific Computing

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MAST30028 Numerical Methods & Scientific Computing

Assignment 2: Root-finding, linear systems

and least squares fitting.

Note: This assignment is worth 20% of the total assessment in MAST30028. When you submit the assignment,

you should submit two files: one pdf file and one zip file. The pdf file contains the answer you write, the

numerical results you generated including figures or tables(0 mark if numerical results are not included in the

pdf file), the comment or the explanation of the results, and the printout of your code (you may use Matlab

publish or Matlab live scripts). The zip file should only contains all the m-files.

1 Root-finding [10 marks]

a. Use Newton’s method to find roots of the following functions with the given initial guesses x0 and the

absolute and relative tolerance being 10?12. In each case report and explain the results.

(i)

In the case of (iii), investigate the order of convergence.

b. MATLAB has only one built-in function for finding roots : fzero. (The Optimization Toolbox has more).

To find out about it, start up Matlab and type

help fzero

Try out fzero on the examples above. Interrogate the structure output (given by fzero) to find out what

happened. Use the same tolerance as in the Newton’s method.

2 Operation counts [7 marks]

Derive the operation count (multiplications and divisions only) for

a. Choleski factorization (without pivoting)

The detailed algorithm for Choleski factorization is that implemented in CholScalar.

b. Gauss elimination (equivalent to LU factorization) without pivoting of an Upper Hessenberg matrix.

An Upper Hessenberg matrix is one with zeroes below the subdiagonal. Just modify the derivation for

Gauss elimination in lectures.

c. solution of a tridiagonal system by LU factorization without pivoting (also called the Thomas algorithm)

This is the algorithm implemented in the code tridisolve in the Asst2 folder.

3 Sparse matrices. [7 marks]

MATLAB has support for sparse matrices. Banded matrices are an important class of sparse matrices. Here

you explore the sparse matrix facilities by concentrating on the tridiagonal matrix from Moler Exercise 2.19 but

with n arbitrary instead of 100. Let’s call it A. It has a diagonal full of 2s and a sub- and superdiagonal full of

-1s.

Form the matrix A in 2 ways:

a. using 3 calls to diag to form A1

b. using spdiags instead, to form A2

For each of these forms, use tic,toc to benchmark, for n = 200 : 200 : 2000 the time to solve Ax = b for

some b using \. I suggest you time a loop of, say, 100 solves in order to get sensible results. If you have a fast

machine, you may need to increase n.

Also try the tridiagonal solver tridisolve from the Asst2 folder to solve the same problem. It will be

slower because it’s an M-file, not a built-in function.

How does the time taken by each of these 3 solvers scale with n? On the basis of Q2, what do you think

Matlab’s \ is doing? By using whos, describe another benefit of using sparse.

4 Norm equivalence [6 marks]

Prove the following inequalities, for arbitrary m,n. Here x is an m-vector and A is an m× n matrix.

||x||2 ≤ ||x||1

||x||1 ≤

√

m||x||2

||A||2 ≤

√

n||A||1

||A||1 ≤

√

m||A||2

In the first two cases, give an example of a vector x where equality holds.

5 Conditioning [4 marks]

Let X be the n× n matrix defined by

[I,J] = ndgrid(1:n);

X = min(I,J) + 2*eye(n,n) - 2;

Which, if any, of the triangular factorizations chol(X), lu(X), and qr(X) reveal the poor conditioning of X

when n = 20? Explain, using results from the lectures.

6 Are all fits equally good? [9 marks]

Examine the M-file polyfit.m (edit polyfit) and understand what lines 59–67 do.

Do Moler Exercise 5.7 but use the the values tk = (k ? 1)/5, yk = erf(tk), k = 1 : 11.

For part c, plot the errors of the three fitted functions with 5 parameters (i.e. a degree 4 polynomial, a degree

9 odd polynomial and the function in part c) on a loglinear scale. Which form fits the data best?

Hint: You may want to write your own M-file to help generate the least squares fits for parts b and c using

polyfit/polyval as a template (although it’s not necessary to do so).

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1300A ENGG 1300 OLET1139 STAT 3503 STAT 206 FEM11149 ELEN 4720 COMP20003 SOLA1070 CS574 C11CS LINB29H3 MAST90083 ECON7200 STAT 610 LINB29 DSM-5 COMP3161 EEEN40010 MATH 4431 K353 STA 3386 IERG4300 COMS W4167 ENG5298 AMME 2302 CS3910 COMPSCI 320 STATS 369 GR5242 COSC 328 CSCI435 MGTS1301 INFS2200 COMP3900 CIS 375 Module AE03 COMP3121 CS260 HAM503 MGMT5050 INT3075 COSC 285 GEOL0029 GEGE2X01 STAT1201 MATH1061 MMF1941H STAT 5060 CS 520 COMP 4320 MSIN0104 000T FINS3648 ECMT2160 ECON4003 Phys 129L CSE 142 CS 3130 MGEB01 EECS 340 Finance (20443/20444) ECMM149 COMP5310 EEEN60180 CSC 111 AT2 CSCI 4430 STAB57 MA/CSC 427 MATH322 D100 CSCI544 CITS3004 CI6227 CSE 565 APCO 1P01 852L1 ELEC S411F COMP3207 STATS330 ECON7530 FIT5149 CSI4104 CMPT 371 COMP201 SEHS4696 FI4003 MATH6182 CG1 WS MATH401 MGT 205 QTM 110 COMP 0137 MACE12201 FIT5210 ECE 544 BISM7206 HUDM 6055 ECON3203 STATS 210 MSCI 570 MATH6173 COMPSCI4039 Accounting COM3503 AOE2024 ECON213 PSYC10004 Control Data 100/200A PHIL2617 PHYS3116 MIE250 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