COMP 116: Introduction to Scientific Programming

Lecture 36: Final Review I

Final Exam Details

The final will be ◦Comprehensive◦Roughly like the midterms, but will

have more stuff ◦Short programs (5-10 lines of code)◦No linear programming, image

manipulation, audio filtering

Course objectivesTo teach you how to write code

◦We’ve mainly used MATLAB as a vehicle for this

int my_function(){ int data[ ] = {6, 3, 3, 8, 43, 2, 19, 7, 5, 88}; int num_elements = 10;

int res = data[0]; for (int i = 1; i < num_elements; ++i) { if (data[i] < res) { res = data[i]; } } return res;}

What does this function do?

Course objectivesTo teach you how to write code

◦We’ve mainly used MATLAB as a vehicle for this

◦…but the concepts you’ve learnt in this class carry over to almost every other programming language

Course objectivesWhat I hope you’ve learnt: If you have some data

◦Read it into MATLAB◦Process it in some way◦Display/plot it in some way◦Write out the results

Keep going!◦Many other great courses in Comp. Science◦ In today’s world, programming is a

fantastic skill to have

Important points:Operator precedence When in doubt use parenthesesSemicolon suppresses output

◦5+3; % does not create output◦5+3 % creates output

Matlab as a Calculator

Super-important: Distinguish between element by element operators and matrix operators.

Operators / Full Precedencedoc ops

Precedence

Operator

1 (high) Parentheses ( )

* Nesting (inner to outer)

2 Transpose .' or ' , Power .^ or ^

3 Unary minus/plus -, + Negation ~

4 Multiplication .* or *Right / Left division ./ or /, .\ or \

5 Addition +, Subtraction -

6 Colon operator :

7 Comparisons < <= == >= >

8 Logical ‘And’ &

9 Logical ‘Or’ |

10 Logical short-circuit ‘And’ &&

11 (low) Logical short-circuit ‘Or’ ||

* Left to right rule applies

Operators

Arithmetic:

Unary: -, +

Binary: +, -, *, /, \, ^ .*, ./, .\, .^ Assignment: =

Colon: :Logical: ~, &, |, &&, ||, XORRelational: <, <=, ==, ~=, >=, >

Special CharactersChar Name Usage

===

Equal Sign

: Colon

; Semi-colon

( ) Parenthesis

[ ] Brackets

{} Curly braces

. Period

' Apostrophe

% Percentage Sign

Special CharactersChar Name Usage

===

Equal Sign 1. Assignment2. Equality test

: Colon 1. <begin>:<step>:<end>

; Semi-colon 1. Suppress command output2. Start a new row in an array

( ) Parenthesis 1. Force Desired precedence in expressions2. Indexing Operator3. Used with function calls

[ ] Brackets 1. Create Arrays (Vectors & Matrices)2. Multiple outputs from function calls

{} Curly braces 1. Construct cell arrays2. Access cell array content (vs. accessing a

sub-cell array with () ).

. Period 1. Decimal point in floating point numbers2. Access field names in structures

' Apostrophe 1. Matrix transpose2. Construct Character array (string)

% Percentage Sign

1. Comments (%% == grouping comments)

Array & Vector Operations

Working with Vectors / Matrices

Colon Operator ‘:’ v = <start>:<stop>

v = <start>:<step>:<stop>

>> A = [1.1:1.1:4.4; 5.5:1.1:8.8; ...

9.9:1.1:13.2];

Linspace command:>> linspace(0,2*pi,100);

Indexing Operator ()A(7) % a(n) = 1D Indexing

A(3,2) % a(r,c) = 2D Indexing

Important Matrix FunctionsFunction Example Note

zeros zeros( 3, 5 )

ones ones( 4, 3 )

eye eye( 4 )

diag diag( [1 2 3 4] )

rand rand(5, 7)

size size( m )

length length( m )

numel numel(m)

' m'

reshape reshape(m,2,6)

repmat repmat(m,3,2)

<empty>

E = []

deletion E(2:4,:) = []

Important Matrix FunctionsFunction Example Note

zeros zeros( 3, 5 ) Creates m x n matrix of all zeros

ones ones( 4, 3 ) Creates m x n matrix of all ones

eye eye( 4 ) Creates n x n Identity matrix

diag diag( [1 2 3 4] )

Creates n x n matrix with vector on diag

rand rand(5, 7) Creates m x n matrix of random values

size size( m ) Returns Vector of size of each dimension

length length( m ) Returns scalar = maximum dimension size

numel numel(m) Returns total elements in matrix

' m' Transpose matrix (rows ↔ columns)

reshape reshape(m,2,6)

Reshapes matrix to new r x c layout

repmat repmat(m,3,2)

Builds large matrix from r x c copies of specified small matrix, IE ‘m’

<empty>

E = [] create an empty matrix or vector variable

deletion E(2:4,:) = [] Delete 2nd thru 4th rows from array

ConcatenationVectors and matrices can only be

concatenated if the number of rows/columns line up

[1 2 3; 4 5 6] % works

[1 2 3; 4 5] % does not work

Matrix OperatorsOperator

Purpose

Matrix operations

+, - Addition, subtraction

.*, ./, .\ Multiply, divide (r. vs. l)

.^ Exponentiation

More matrix operations

C = A*B Matrix Multiply

C = A/B Matrix R. Divide

x = A\b Matrix L. Divide (Solve)

C = A^3 Matrix Power

solves a linear equation systemof the form Ax = b

element-by-element operations

Compatibility Rules:

C4x5 = A4x5 .* B4x5A, B must be same size and shape

Matrix Multiplication Rules: C4x7 = A4x3 * B3x7A.nCols == B.nRowsInner dimensions agree, outer dimensions become new size & shape

An example

Note: Not necessary to compute actualvalues. Simply determine size of the individual factors.

rand(3,4)*[1 2; 3 4; 5 6; 7 8]*ones(2,3)

(3x4) * (4x2) * (2x3)

Results in: (3x3)

Resulting Matrix Sizes of Matrix-value ExpressionsGets trickier if e.g. .* and * are used.Example:

rand(3,4)*[1 2; 3 4; 5 6; 7 8].*ones(3,2)

(3x4) * (4x2) .* (3x2)

(3x2)

(3x2)Evaluates from the left to the right!

Matrix indexingA is a preassigned matrix>>B=A(u,v)B(i,j)=A(u(i),v(j))

What does A(1:2:end,1:2:end) return?

How do you extract the 3x3 matrix around the (4,5) element of A?

Solving Systems of EquationsAx = b

% System of equations5x + 10y – 4z + w = 2

x - 2y + z = 1

2x + 3y + z – 6w = 3

7x + 2z + 3w = 4

% Set up coefficient matrixA = [5 10 -4 1; 1 -2 1 0; ...

2 3 1 -6; 7 0 2 3];

b = [2 1 3 4]; % Setup solution vector

x = A \ b'; % Solve for unknowns

Plotting & Publishing

2D Plotting BasicsTitle

x label

y label

Legend

y axis

x axis

Marker

Plottingplot( x, y, ‘LineSpec’, ‘PropertyName’, ‘PropertyValue’, … );

Plot Attributes line attributes ( ‘-.ro’ )

Line style, Color, Marker other properties

Line Width, Marker Colors

Formatting: xlabel, ylabel, title legend, text, …

More Commands◦axis <min,max>, square, equal

◦ gcf, clf, grid [on|off], …

Multiple plots Overlaying plots

plot( x, [y1 y2…], … ) plot( plot1, plot2,… ) hold on|off

subplot( 3, 2, 3 )

More Plots hist, semilogx, loglog bar, pie, polar, …

Conditional Logic:

Relational OperatorsTests relationship between two objects or arrays

Result is always a logical data type or array of logical data types

Name Operators Examples

Equivalence (binary) Operators

Equality == 5 == 5, x == y

Inequality ~= 5 ~= 5, z == (x^2 + y^2)

Comparison (binary) Operators

Less Than < 5 < 3, sin(2) < cos(y)

Less Than or Equal

<= 4 <= 4,

Greater Than or Equal

>= 7 >= 10

Greater Than > 10 > 7

Logical OperatorsBoolean operatorsName Operators Examples

Unary Operators

Logical Negation (NOT)

~ ~ (3 == 5) = 1 (true)

Binary Operators

Logical And (AND)Short circuit (AND)

& or &&

5 & 7 = 1 (true)

Logical Or (OR)Short circuit (OR)

| or ||

0 | 1 = 1 (true)

Exclusive Or (XOR) XOR xor(8, 5) = 0 (false)

Performs binary logic on two logical data type operands (or arrays) to return a logical result.

Formulating Logical ExpressionMost expressions can be written

directly: E.g. Point is within or on the unit circle◦Condition: x^2+y^2<=1

Sometimes differs from standard math notation◦(2<x)&(x<3) % is correct (Program Style)◦2<x<3 % is incorrect (Math Style)

When in doubt: use parentheses.

if Statements

Single Testif <test> commands; % Section 1end

Test between Two Choicesif <test> commands; % Choice #1 (test was true)else commands; % Choice #2 (test == false)end

if Statements

Chained Tests

if <Test1> commands1; % T1 trueelseif <Test2> commands2; % T1 false T2 %trueelseif <Test3> commands3; % T1, T2 %false, T3 trueelse commands4; % T1,T2,T3falseend

Nested Tests

if <Test1> if <Test2> commands1; % T1,T2 both true else commands2; % T1=1, T2=0 endelse if <Test3> commands3; % T1=0, T3=1 else commands4; % T1,T3 both false endend

Finding Elements/Indexing

Finding Elements/Indexing

Two different ways of finding elementsUsing the find command ind = find( x>3 );

% returns indices of elements in x satisfying test

If you want to extract the respective elements use◦ x( ind ) % returns elements at specified indices

Finding Elements/Indexing

If you want to find the maximum or minimum element in a vector or matrix you can get both the value and indices through min or max.

[minVal, minIndex] = min( x )[maxVal, maxIndex] = max( x )

Functions

Writing Simple functionfunction [o1, o2]=funcName( i1, i2 )% Function Comments… % Body (implementation)end %optional

• Can have multiple inputs (i1) and multiple outputs (o2)

function [] = funcName()function o1 = funcName()function o1 = funcName( i1 )function o1 = funcName( i1, i2 )function [o1, o2] = funcName( i1, i2, i3)

WorkspaceGlobal vs. Local StorageGlobal Workspace

◦Shared by Command Window and script commands

Local Workspace◦Created locally on entry to each

function◦Disappears on exit from function call.

% This is a scriptradius = 10;area = pi .* radius .^2;

function [area] = circ_area(radius)area = pi .* radius .^ 2;

% Call function in workspacemy_area = circ_area( 10 );

Looping

Loops: for loop statementthe counted loop solution

for <varindex> = <start>:<stop><Body: do some work>

end

for <idx> = <start>:<step>:<stop><Body: do some work>

end

Loops: while loop statementthe conditional loop solution

while <test> <Body: do some work> <Update: make progress towards exiting loop>end• While loops are great if we don’t know how many times

we need to loop, but if we can write a test for when we’re done

• For this to work properly, the test needs to evaluate to a logical value

• The while loop will run as long as test evaluates to true

• The while loop does not have a built-in counter like the for-loop (if you want to count something, you need to implement the counter yourself)

Common Idioms: Looping over a matrix• Use a nested for loop:

function ret = findMaxElement( A )sz = size(A);ret = A(1,1);

for i=1:sz(1) for j=1:sz(2) if ( A(i,j)>ret ) ret = A(i,j); end endend