MatLab homework - root solving

Simulation on the Nanoscale ENG-M03: Assignment 1

General information

In the following you will be asked to complete a set of tasks using MATLAB. To help you

accomplish these tasks many questions will include MATLAB code (blue text), this can be

directly copied and pasted into MATLAB. Each question will carry a mark which is indicated

at the questions end. For each question document your answers, be that a numerical or

graphical output in an associated word document, which will be submitted as part of your

continual assessment at the of the semester.

Lab. 1

The aim of this laboratory is to cover the basic set-up used throughout this and following

MATLAB (ML) exercises. First we will review the script-subroutine format of ML by

considering a simple example. Following this we will construct a suite of functions that able

to differentiate and integrate a user defined function numerically using MATLAB.

Using a script file (I)

In this first task we are simply going to plot a function using a script file.

(a) Open a MATLAB script file, save it to your personnel directory as lab1_sin.m. This

script will be used to call intrinsic and user-defined functions in ML.

(b) Type:

clear - clears all variables from the workspace

format short g - formats the outputted numerical data

(c) MATLAB has many ‘in-built’ functions that can be easily used; this task we will plot

the sine function over the interval [-π π].

(d) Define the vector/array/table x as:

N = 100;

x = linspace(-pi,pi,N)’; or as

x = (-pi:2*pi/(N-1):pi)’;

NB: the semi-colon suppresses output to the command window and the apostrophe

transposes the default column vector to a row vector.

(e) Use the following code to plot sin(x) as a function of x:

y = sin(x);

figure(1), plot(x,y)

[1 mark]

Defining a function (II)

A function or subroutine in any programming language is essentially a black-box operator –

i.e. you give the function some value and then it operates on that value and returns the

result of that operation. The sine function in the previous task is an example of this already

built-in to MATLAB. In this task we will define our own MATLAB function, which will scale the

argument given to the sine function, i.e. ⁄ , where a will be user defined.

(a) Open a MATLAB function file

(b) The function file is markedly different to the script file you previously opened and

will look something like:

function [ output_args ] = Untitled( input_args ) %UNTITLED Summary of this function goes here % Detailed explanation goes here:

End

 The function starts with the word function and ends with the word end

 Untitled – is the default name of the function – the function name MUST be the same as

the name you save it has!!!

 Input arguments into the function are written in the curved brackets (separated by

commas) next the function name.

 Output arguments must be defined in both the body of the function and in the square

brackets next to the function declaration.

A simple example of a function: function [ y1,y2 ] = multiply_10_and_100( x1,x2 ) % This function multiplies the first input argument by 10 and the second by 100 then return their values in y1 and y2 respectively.

y1 = x1×10; y2 = x2×100;

end

(c) Save the above function as sin_scale.m – make sure you alter the function name to

reflect this change. If you do not do this MATLAB will underline the function name in

yellow.

(d) Modify the function so that you have 1 output and 2 input arguments, your new

function will look like:

function y = sin_scale (x,sf)

% Input: scalar or vector x, sf scale factor

% Output: scalar or vector containing corresponding sine values.

% y = sin(x); ML intrinsic sine function

y = sin(x./sf);

% The ‘.’ before an operator implies a vector/array operation

end

(e) Modify the script previously used in task 1 to plot sin(x) as a function of x, where we

scale x by 0.1.

sf = 0.1

y = sin_scale(x,sf);

figure(2), plot(x,y)

[1 mark]

(f) Using a ‘for loop’ plot the values of sine for the following values sf = [0.1 1 0.5 2]; of

the scale factor. Use the intrinsic ‘zeros’ (type: ‘help zeros’ in the command

window) function to set-aside space for the values of sin(x) for the three values of

scale factor. Use the following code in your script file to help you.

sf=[0.1 0.5 2]; % Scale factor values in an array named sf

N_sf = numel(sf); % Number of elements within sf (ML intrinsic

% function)

sin_array = zeros(N,N_sf);

%Initialise an array of zeros to store data. % Notice the size of the array is predicted by

%the number of values, N and the number of scale factors to be analysed, N_sf.

for ndx = 1: N_sf

sin_array(:,ndx) = sin_function_sf(x,sf(ndx));

end

figure(3),plot(x,sin_array)

title(['A Sine fn; limits = [-pi pi]; Scale factors = ',num2str(sf)])

N.B. you can name your variables as you like, but be careful not to use a ML intrinsic

function. To check this type ‘help variable_name’ if ‘function is not found’ is displayed

you are okay to use the variable name, else change the name.

[2 marks]

Numerical Differentiation

Using the forward difference technique (FDT) we may differentiate a continuous function

over a specified domain. Here, we will consider differentiating the cosine function over the

domain [-pi pi]. Using the FDT the derivative of a function may be expressed as

i

ii

i h

xfxf xf

)()( )('

1 

  [1]

where i is the index along the x-direction and hi is the step-size between xi+1 and xi; here, we

assume this is a constant.

(a) Define the following input vector x = linspace(-pi,pi,N)’; where N=100; in a script file.

(b) Calculate y = cos(x) – cosine is another built-in MATLAB function.

(c) Define function that inputs both the x and y vectors.

(d) Using equation 1 estimate the cosine derivative.

(e) Plot your answer against the intrinsic ML function ‘-sin’ over the same interval.

[2 marks]

Numerical Integration

In this section we develop some numerical techniques to solve definite integrals of the form

dxxfI b

a

 )( [2]

The evaluation of such integrals is usually called quadrature and we will consider two

techniques, namely, the Trapezoidal and Simpson methods.

1. Trapezoidal Method

The area under the curve f(x) from x =a to x = b is approximated by the area beneath a

straight line drawn between the points f(xa,fa) and f(xb,fb). the area below this line is then an

approximation to the integral I, i.e.

))(( 2

1 abffI

ba  [3]

This can be evaluated using the following code:

function int=trap1(f,a,b)

%Input - f is the integrand input as a string 'f'

% - a and b are upper and lower limits of integration

%Output - s is the trapezoidal rule sum, equation 3

if a>b

error(‘b must be greater than a’)

end

h=(b-a);

int=h*(feval(f,a)+feval(f,b))/2;

a) Write a function dxxxI  2/

0

)sin(



and evaluate using trap1.

[2 marks]

b) Calculate the analytic value and compare with the numerical answer, i.e. calculate

the percentage error.

[2 marks]

2. Composite Trapezium Rule

Improvement of the accuracy of this technique can be obtained by simply subdividing the

interval of interest into a large number of smaller intervals. For say n regularly spaced

intervals the integral may be approximated by the following expression:

)2( 2

1 1

1

b

n

i

ian fffxI  





[4]

Where nabx n

/)(  and i

f is the function evaluated at each of the interior points, i.e.

)( ni

xiaxff  .

a) Write a ML function to evaluate equation 4.

b) Calculate the integral in the previous section for 100 intervals and compare with the

analytic value, i.e. determine the percentage error.

c) Create a function ⁄ and evaluate it over the interval , with the following number of intervals 5, 50, 100.

[3 marks]

d) Compare each result with the analytic answer i.e. ln(2), i.e. put in a table.

[1 marks]

e) Determine the number of sub-intervals needed to achieve an accuracy to 6 decimal

places.

[2 mark]

3. Simpson’s Rule

An improvement to the composite trapezoidal rule can be achieved by simply replacing the

straight line segments across each interval by parabolic segments i.e. use Simpson’s rule.

From the lecture notes the approximation to the integral across the interval [a b] can be

expressed as follows:

)24( 3

1 1

1

1

1

b

n

onlyeven i

i

n

onlyodd i

ian ffffxI  









[5]

a) Write a ML function to evaluate equation 5.

b) Calculate the value of the integral for ⁄ and evaluate it over the interval

, with the following number of intervals 5, 50, 100.

[3 marks]

c) Compare each result with the analytic answer i.e. ln(2), i.e. put in a table.

[1 marks]

d) Determine the number of sub-intervals needed to achieve accuracy to 6 decimal

places.

[2 mark]

MATLAB/Assignment 2/Simulation on the Nanoscale Lab 2.pdf