week 9 matlab: control structures, functions, 3d …en1811/15s1/lectures/week09.pdf• matlab has...

ENGG1811 © UNSW, CRICOS Provider No: 00098G1 W9 slide 1

Week 9

Matlab: control structures, functions,

3D plotting

ENGG1811 Computing for Engineers

ENGG1811 © UNSW, CRICOS Provider No: 00098G W9 slide 2

References

• Chapman (2012), parts of chapters 3 to 6, 8

– section 3.1 to 3.5

– section 4.1 to 4.4.3

– section 5.1, 5.2, 5.6

– section 6.1 to 6.2

– section 7.2

– section 8.3

• online documentation at mathworks.com

http://www.mathworks.com.au/help/matlab/index.html

http://www.mathworks.com.au/help/matlab/index.html


Plot appearance

• Line and marker styles

– values encoded in a short optional string following each x, y pair in a plot-type command

• See Chapman Table 2-10

>> plot(x, y, 'b*-'); % more x, y, stylestrings can follow

Line colour, r=red, b=blue, g=green, w=white, c=cyan, m=magenta, y=yellow, k=black

Marker style, .=point, o=circle, x=x, *=star, s=square, v=down triangle, ^=up triangle, etc

Line style, -=solid, --=dashed, :=dotted, -.=dash dot


Additional control

• Additional characteristics of the lines and markers are

specified by extra pairs of arguments to plot of the

form

'propertyname', value,

Properties include

LineWidth – in pixels (integer)

MarkerSize – in pixels

MarkerEdgeColor – string, same codes as for lines

MarkerFaceColor – string, same codes as for lines


LaTeX formatting

• Labels, titles and legend text can be formatted using a

scheme used for typesetting mathematical documents

• LaTeX (pronounced ‘lay-teck’) embeds codes or escape

sequences using backslash and other special characters

\it{text} \bf{text} – italic, boldface*

_{text} ^{text} – subscript, superscript

\lambda \SIGMA – Greek lowercase (), upper ()

\circ \pm \neq \infty – symbols

\rightarrow \uparrow – arrow symbols (also left, down)

\_ \^ \{ \} \\ – literal _ ^ { } or \

• Can omit { } if a single character, for example \itx

• See Chapman Table 3-2

* Font style changes do not properly terminate in Matlab versions to 2012b


LaTeX equations in published docs

• The full LaTeX equation formatter is available when an

M-file is published, special sequences include

\frac{num}{denom} – fraction

\sqrt{eqn} – square root

\sum_{min}^{max} – summation

\int_{min}^{max} – integral

• Inline equations enclosed in $…$ inside comment

• Block equations enclosed in $$…$$ inside comment

• Inside these, algebraic variables are italicised while

standard functions like sin or tanh are not.


LaTeX equation examples

gives this result (though

equations are images, not

scalable text):

If you are really interested,

try help latex (part of the

syms package, converts a

symbolic expression to a

LaTeX equation)

%% % Einstein's famous equation is $E = m c^2$. % % $$\sum_{i=1}^{\infty} \frac{1}{2^i} = 1$$ % % $$\int_{0}^{\infty} x^2 e^{-x^2} dx = \frac{\sqrt{\pi}}{4}$$


Polar plots

• Polar plots have angles 0 to 2 in place of x, and

magnitude (distance from the origin) in place of y

• Chapman Example 3.3 – cardioid microphone response

– gain (relative sensitivity) varies with angle (0 = directly in front) according to the formula

>> g = 0.5; % gain coefficient, characteristic of mic

>> theta = linspace(0, 2*pi, 121); % 360/(121-1) = 3 degrees

>> gain = 2.*g .* (1 + cos(theta)); % * also OK

>> polar(theta, gain, 'r-'); % red, solid, 1px line

>> title('\bfCardioid microphone gain versus angle \it{\theta}');

)cos1(2 ggain


Annotations

Not a full graphics editor, so results are only approximate

Figure window has a toolbar

to add elements and save

Plot browser

Add/edit legend

Select tool

View – Plot Edit Toolbar to access annotation tools such as arrows, shapes and text boxes

File – Save As (*.fig)

See Chapman section 3.3


Other plot types

• Chart types similar to OpenOffice Calc/Excel

bar(x,y) – vertical bar chart

barh(x,y) – horizontal bar chart

stem(x,y) – marker and vertical line

stairs(x,y) – like bar, but only top of skyline shown

pie(x, explode) – use with caution, or preferably not at all

compass(x,y) – polar with arrow to each Cartesian point

• If y is a matrix instead of a vector, each column is a

separate plot

– applies to all plot types that accept y values as second argument

• See Chapman 3.4 and 3.5


Decisions

• Boolean constants are true, false

internally represented by 1 (or nonzero) and zero

• Relational operators are almost the same, except

== is the equality operator for numbers (not =)

~= is the inequality operator for numbers (not <>)

• Boolean operators also use a different notation

&& is the and operator (not And)

|| is the or operator (not Or)

~ is the negation operator (not Not)

• && and || are shortcut operators, right hand side is not

evaluated if the answer is established by the left operand

non-shortcut operators are just & and |

quad2or3 = (x == 0 || y/x <= 0); % quadrant 2 or 4, incl axes


Boolean operations

• Don’t use == or ~= for strings, use strcmp(s1, s2) or

isempty(s) instead

• Built-in Boolean functions include

isempty(a) – a has no elements (empty array), includes

empty strings

ischar(s) – s is a string (character array)

isinf(v) – v is infinite (the special value Inf)

isnan(v) – v is not a defined number (the special value NaN)

isnumeric(v) – v is a numeric array

logical(x) – interpret x as a Boolean, any non-zero is

true


Selection

• Matlab has if, switch and try/catch

– only if is essential for ENGG1811

• Same principle as for OOB but syntax differs slightly

if boolexpr1 statements1 elseif boolexpr2 statements2 elseif …

else

statementsn end

these are

optional

of course

% example (eps is predefined)

if abs(a-b) < eps

disp('a,b match');

else

disp('a,b differ');

end


Validation

• To avoid misleading or meaningless results, always

ensure that the problem instance parameters are

within the limits of the model

– the error function works like fprintf, but displays the

formatted message and where it occurred, then stops

– may not be practical to check all data values, especially a large externally supplied array

if numDataValues < 0

error('expected non-negative value for data size,

%d was specified\n', numDataValues);

end


Iteration – while

• Same principle as the while statement in OOB, syntax

is almost the same too

while BooleanExpression

statements

end

• Problem solving techniques that iteratively apply a

rule and ideally converge to a solution include

Goal seek (OO Calc)

Solver (OO Calc + Matlab’s ODE solvers)

Newton-Raphson (general)

Bisection (general, see next slide)


Bisection revisited

• Simpler than Newton-Raphson, but similar approach to

finding x such that f (x) = 0 for a known function f.

• Requires two points x1 and x2 such that the sign of f (x1)

is different from the sign of f (x2)

• By maintaining the opposite signs while moving the x

values closer, they converge on the root (at the rate of

log10 2 0.3 significant figures per iteration)

• Convergence criterion: relative difference <= eps

• Algorithm: evaluate f at the

midpoint

• Depending on its sign,

move either x1 or x2

2

21 xxxmid


Bisection implementation

• Full version in demos/bisection.m

% get function (as string funcExpr) and initial limits

% x1 and x2 from user

fx1 = fxeval(funcExpr, x1); % evaluate at x1

fx2 = fxeval(funcExpr, x2);

while ~ converged(x1, x2)

xMid = (x1 + x2) / 2;

fxMid = fxeval(funcExpr, xMid);

if sign(fxMid) == sign(fx1)

x1 = xMid;

else

x2 = xMid;

end

end

% x2 is the best estimate of the root

fxeval and converged

are in the en1811 folder;

sign is the built-in signum

function (returns –1, 0, or

+1 according to whether

its argument is negative,

zero or positive)

Schoenherr Line*

• Old formula estimates the coefficient of friction CF

of a ship, given Reynolds number , where

V – speed of vessel (m/s),

L – length of vessel (m) – kinematic viscosity of

seawater at 15°C, about 1.2210–6 m2/s

• For a 300m cruise ship travelling at 24kt

(12.3m/s), R ≈ 3.0e9

• Formula is

• Use the bisection script to solve for CF, which lies

between 1e–5 and 1e–1

– Note that the function expression can’t use any variables apart from the algebraic symbol x


VLR

0242.0

).(log10 F

FC

CR

* Example suggested by Phil Helmore, School of Mech & Manf Eng. Formula dates from 1947.


Iteration – for

• Because Matlab is heavily array-based, the for loop

variable is assigned each value in an array in turn:

– The ArrayExpression is typically a vector generated with the colon operator

– the index variable is set to each value in turn (or each column if it’s a matrix)

– textbooks tend to use ii for row index variables and jj as

column index variables, reflecting the Mi j mathematical

notation but i and j are used to define complex numbers

– ENGG1811 convention is to use row and col (like OOB), or rowi and colj to maintain the matrix notation link

for index = ArrayExpression

statements

end


Examples

order = 10; % generate identity matrix M = eye(order)

for rowi = 1:order % can omit the increment as it’s 1

for colj = 1:order % nested loop: process columns

if rowi == colj

M(rowi, colj) = 1;

else

M(rowi, colj) = 0;

end

end

end

% factorial

n = 9; fact = 1;

for k = 1:1:n % we put the increment in this time

fact = fact * k;

end


Functions

• Functions come in different flavours

– global functions in an M-file with matching name

– local functions (or subfunctions), inside a function M-file

– nested functions (not used in ENGG1811)

– anonymous functions (not used in ENGG1811)

– function handles (useful in advanced programming, not in ENGG1811)

• Declaration is quirky:

• inputs are parameters, passed by value

• outputs can be any number of variables, including none

[ ] thus makes the function equivalent to an OO Basic Sub

function [outputs…] = functionname(inputs)

function body, using local variables

end

Tip: keep the [ ] even when there’s only one

return variable, avoids mistaking it for the

function name.


Inside the function

• Statements/commands have their usual meaning

– changes global environment through other functions calls, so plot() etc are fine

– local workspace, so new variables are local (as are parameters and output variables)

• Purpose: use input values, assign result(s) to output

variables

– return statement terminates the function early

– ENGG1811 convention for single return value is the name

retval (or return_value)

• To conform to doc standards, use the ENGG1811

newfunc command and fill in the comments

>> newfunc 'mygcd'; % edits new function file mygcd.m

% containing the function mygcd


mygcd

• Things to note about the implementation

– fill in the doc first, not as an afterthought

– acknowledge Euclid, he was the first to publish the elegant

converging GCD algorithm

– built-in nargchk validates number of arguments

• you can also allow optional arguments and test with

nargin, this is done in converged.m

– validation to avoid infinite loop if non-positive

• technically the algorithm would calculate gcd(0,0) = 0,

but gcd(0,b), b > 0 will fail. We reject all zero cases.

– OK to modify parameters, they are independent local

variables unlike OO Basic (and they can be any type)

– remember to return the result through retval (or other

named output variable(s)), Matlab only warns if there’s no

assignment to outputs anywhere


Local functions (subfunctions)

• If a function has a subtask that’s not general enough

for its own file, add a local function to the end

– accessible only within the parent file

– regrettably doesn’t apply to scripts (lazy people don’t want to create extra files, so they tend to bloat the script)

function [retval] = dostuff(args) % consists of three subtasks

subtask1

var1 = subtask2func(arr)

subtask3(var1)

…

function [ ] = subtask1() % no return value ~= OO Basic sub … end

function [retval] = subtask2func(vec) … end

function [ ] = subtask3(yyy) … end

Example – extend mygcd

• Add local validation function isint to check if

args are integers

– how? use floor(x) == x

– first version written to expect a single scalar, but can be extended to arrays:

[1,5] == [1,5] doesn’t return true as expected,

it returns [true, true] instead (actually [1,1])

isequal(x,y) used to compare arrays to yield a

single Boolean result



Workspace sharing

• Passing large arrays between functions can slow down

the program as a copy has to be made on each call

• Often you have several functions successively updating

an array

– variables can be put in a shared global workspace, separate from the workspace local to each script/function

– all functions and scripts need to declare global variables

– global variables use upper case by convention

function [ ] = updateImage() global CUR_IMAGE % alter some elements of this var

…

CUR_IMAGE(rowi, colj) = …

end


Data load/save

• Data can be saved in two simple formats

– .mat, a Matlab-specific format that preserves names, types and values from one or more variables

– ascii (text) format, data values only but readable with other applications

>> save mydata.mat image h vec % .mat format

>> save export.dat image –ascii % data only, space separated

>> load mydata.mat % reloads image, h and vec

>> load export.dat % stores result in variable export

• Loading from a mat file restores the saved variables,

overwriting them if necessary

• Loading from a text file (any extension except .mat)

stores into a variable named after the file


Other data sources

• Most data will come from other than Matlab

– Matlab can read from an Excel worksheet, but only numeric data (without using Matlab cell arrays, NCIE*)

– Array has extent based on first and last number found on any row or column, intervening values are NaN

>> data = xlsread('data.xls', 'sample'); % filename and sheet

>> size(data)

ans =

9 4 data = 123 NaN NaN 456 134 NaN NaN 834 145 NaN NaN 34 200 NaN NaN 125 211 NaN NaN 34 222 NaN NaN 456 233 NaN NaN 17 NaN NaN NaN 8 NaN NaN NaN 6

* NCIE means “not covered in ENGG1811”


Importing data from text files

• Other sources may be numeric (only) data in text files

– use csvread if data is CSV (comma-separated values)

– use dlmread if data has some other delimiter such as tab

– can use textread to obtain each column in separate

vectors, but this command is deprecated (obsolescent)

• Most flexible is textscan, but more complex file

operations and returns cell arrays (NCIE)

>> coords = dlmread('ThreeMileDamCoords.txt');

>> size(coords)

ans =

142 2

>> plot(coords) % oops, that’s not what we wanted!

>> plot(coords(:,1), coords(:,2)) % better

>> axis equal % restore the aspect ratio to 1.0

Coordinate axes would look better if the common multiple of 1000m were subtracted first. Matlab also has a mapping toolbox.

file is in a folder that is on Matlab’s path


3D line plots

• 3 dimensional line plots can be generated by passing

three equal sized vectors to plot3

• Example: decaying oscillations in a mechanical system

in two dimensions (Chapman 8.3.1)

t = linspace(0, 10, 200);

x = exp(-0.2*t) .* cos(2*t);

y = exp(-0.2*t) .* sin(2*t);

plot(x, y, 'Linewidth', 2); % 2D plot

title ('\bf2D Line Plot');

xlabel('\itx'); ylabel('\ity'); grid on;

…

plot3(x, y, t, 'Linewidth', 2); % 3D plot, time is z (up)

title ('\bf3D Line Plot');

xlabel('\itx'); ylabel('\ity'); zlabel('\itt'); grid on;

tetytetx tt 2sin)(2cos)( 2.02.0


3D time-based plot

• 3D plot shows the effect of time arguably better than

as distance along the 2D plot

• Even better, the 3D view

has interactive rotation


3D mesh and surface plots

• Data that has two independent variables (for example,

temperature measured at many coordinates in 2D

space) can be visualised as a 3D plot

– x and y are normally the independent variables

– z is normally the dependent variable

– can be displayed in three ways

1. as a mesh or wireframe of individual line plots

2. as a continuous surface, with colouring to highlight the slope at each point

3. as a series of contours, slices parallel to the x-y plane

– the catch is how the x, y and z arrays have to be specified

mesh(x, y, z);

surf(x, y, z);

contour(x, y, z);


3D plot arrays

• All three arrays have exactly the same shape

– the number of rows is equal to the number of y values

– the number of columns is equal to the number of x values

– this is slightly counter-intuitive but built-in tools help

– the x array specifies the x values in each row, duplicated

– the y array specifies the y values in each column

– the z array specifies the dependent value at each point

• Example, evaluated at x = 1, 2, 3

and y = 1.5, 3, 4.5, 6.

22),( yxyxz

708263246608286

408359244460984

242646056316233

354135000280281

666

5.45.45.4

333

5.15.15.1

321

321

321

321

. . .

. . .

. . .

. . .

zyx


3D plot arrays, continued

• Having the arrays structured this way allows you to

calculate the z value using array operators

• Fortunately you can easily construct the x and y arrays

from their respective vectors using meshgrid:

>> [x,y] = meshgrid([1,2,3], [1.5,3,4.5,6]); >> z = sqrt(x.^2 + y.^2);

>> surf(x,y,z);

>> xlabel('\itx');

>> ylabel('\ity');

>> zlabel('\itz');

Mesh doesn’t have surface

shading:


Problem instance parameters

• Good software development practices keep data and

processing logic separate as far as practicable

– data sets stored in external files, unless trivial

– constants in a separate class file NC, PM etc

– problem parameters should also be separately identifiable, distinguished from intermediate variables

• ENGG1811 convention: define parameters in a

separate section of the solution script

– prefix parameter name with p, then title case (since

they’re usually descriptive rather than algebraic)

– OK to have multiple sets, fully commented, the last set represents the ones in effect

– parameter initialisation and descriptive comment go together, so not needed in data dictionary


Example: boat hull*

A simple model for the hull of a boat is given by

where y is the width of the

hull from the centre line, x is

the distance along the centre

line, and z is the depth of the

hull. B is the beam (max

width), L is the length and D

is the draft (max depth).

This hull is 8m long, with 3m

beam and 1.5m draft.

* Based on Holloway, p.144 Project 15.

22

12

12 D

z

L

xBy


Implementing the model

• x and z are the independent variables, and range from

–L/2 to L/2, and –D to 0 respectively

– use linspace and meshgrid to generate these arrays

• The first half of the y values (the starboard side) are

produced by the formula, the second set of y values

has the opposite sign

– to display both halves (thus avoiding a huge maritime

disaster!), use the command hold on between plot

commands

– axis equal maintains aspect ratio (uniform scales)

• see boathull.m in the lecture package

– use the 3D rotation tool to examine the shape

– does mesh or surf give the better view?

– experiment with shading (faceted, flat or interp)

Coming up

• Assign 1 deadline Friday 5pm

– late penalty 15% off the max available mark per day

• Testing will start on the weekend

– every crash and dialogue box has to be acknowledged manually, could take a few days

• Assign 2 (Matlab, simple harmonic motion)

available by Thursday pm

– can be done singly, or in pairs from the same lab group

– due Friday week 11

– peer-assessed against objective criteria in lab12

• Final exam is scheduled for 8:45am Thu 18 June

– more info later, including sample exam + solutions


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