Matlab Tutorial - math.umn.edu · PDF file• We will need some files later in this tutorial. Please download them here: • Save those files into your Matlab working folder if you

Matlab Tutorial

Fanda Yang

University of Minnesota

8/31/2017

File download

• We will need some files later in this tutorial. Please download them here:

http://z.umn.edu/mfm17-matlab

• Save those files into your Matlab working folder if you know what it is. Otherwise, save them to a new folder.

Click

Matlab first impression

• Open Matlab

• Type in “hello”

What we do today?

• Let’s play Sudoku– Matlab basics

– On-screen presentation

• OMG! It’s Linear Algebra– Special matrices

– Read data from Excel spreadsheet

• Hey Matlab, take the derivative for me– M-files

– Anonymous function

– 2D Graphing

• Advanced Topic– Logical values

– fminsearch

LET’S PLAY SUDOKU

Sudoku

8 6

3

2

• Use integers from 1 to 9 to fill out the matrix on the left

• Each row and column sum up to 15

• major diagonal and minor diagonal sum up to 15 too

Sudoku

8 1 6

3 5 7

4 9 2

• magic(3)

Variable declaration

• Claim a variable like you claim shotgun seat– Don’t worry about data type

– Case sensitive

• a = 2

• b = 2^2

Matrix and vector definition

• A matrix is an array of vectors– A 4 by 3 matrix can be viewed as 4 row vectors or 3 column vectors

•

1 4 98 3 54 7 39 4 0

•

− 𝒓𝟏 −− 𝒓𝟐 −− 𝒓𝟑 −− 𝒓𝟒 −

•

| | |𝒄𝟏 𝒄𝟐 𝒄𝟑| | |

• Comma separates Columns; semicolon separates rows

Syntax (row vector)• r1 = [1,4,9]• r2 = [8 3 5]

• r3 = [4 7 3]• r4 = [9 4 0]

Syntax (column vector)• c1 = [1;8;4;9]• c2 = [4;3;7;4]• c3 = [9;5;3;0]

Syntax (matrix)• X = [1 4 9; 8,3,5; 4,7,3; 9 4 0]• X = [r1;r2;r3;r4]• X = [c1,c2,c3]

Exercise

• Define this matrix on your computer1 2 34 0 00 0 6

Heuristic Discussion: how do WE know it’s right?

8 1 6

3 5 7

4 9 2

• Let’s write a small program to verify this answer

• A = magic(3)

WE

A

Heuristic Discussion: how do WE know it’s right?WE

8 1 61. Grab a row out of the matrix

2. Sum up the 3 numbers in the row

3. Check if the sum is 15

Search for how to take sum of an array in matlab

http://www.bing.com

Bing it on!You can google it too if you want.

Heuristic Discussion: how do WE know it’s right?WE

8 1 61. Grab a row out of the matrix

8 1 6sum( ) 2. Sum up the 3 numbers in the row

3. Check if the sum is 158 1 6sum( ) == 15

Heuristic Discussion: how do WE know it’s right?WE

A(1,:)1. Grab a row out of the matrix

A(1,:)sum( ) 2. Sum up the 3 numbers in the row

3. Check if the sum is 15A(1,:)sum( ) == 15

Matrix indexing

• A(i,j) entry at ith row & jth column

• A(i,:) ith row

• A(:,j) jth column

• A(i1:i2,j) rows from i1 to i2 intersect with jth column

• The trick of Boolean / logical indices– Example

• A(A<5)=0

– The above code replaces entries that are smaller than 5 in A with 0

• For more information about matrix indexing– http://www.mathworks.com/help/matlab/math/matrix-indexing.html

Matrix indexing - Caveat

• Matlab does not support A(i,:)(j). This won’t give you the (i,j) entry.

• What you can do is to split up the expression into two lines:– r1 = A(1,:);

– r1(2)

Operators & Equivalent

• Arithmetic– +, -, ^ (power), ’ (transpose), *, /• Inverse: A^-1 or inv(A)

• Transpose alternative: transpose(A)

– dot operators: .*, ./, .^. More on this later

• Relational – <, <=, >, >=, ==, ~= (not equal to)

• Logical– & (logic AND), | (logic OR), ~ (logic NOT), xor (logic XOR)

– any(), all()

• For more information– http://www.mathworks.com/help/matlab/operators-and-elementary-

operations.html

Exercise: how do we check columns all at once?

8 1 6

3 5 7

4 9 2

• all(sum(A,1)==15)

• Also, verify – all rows

– the (major) diagonal

Exercise: how do we check minor diagonal?

8 1 6

3 5 7

4 9 2

• Hint: Use the following functions

• fliplr: flips left to right

• diag: gets diagonal entries of a matrix or forms a diagonal matrix from a vector

Data management (Workspace variables)

• save (filename [, variables])– Saves workspace variables to a file

– filename: name of the file to be saved. File extension “.mat”. Use single quotation marks to enclose the file name.

– variables: a list of variable to be saved. Use quotation marks to enclose each variable name.

• load filename– Loads workspace variables from a .mat file

– filename: name of the file to be saved. File extension “.mat”. Use single quotation marks to enclose the file name.

Present results NICELY

• disp(X)– Display matrix X on screen without printing matrix name

• fprintf(format [, var_list])– Write formatted variable values on screen. C++ style

– format is a string type parameter. You can print literal text together with conversion specifications. You always want to end this parameter with “\n”

– var_list is a list of variables whose values will be printed on screen.

– Examples:

• fprintf('The value of Pi is %.8f.\n',pi)

• fprintf('%d\t%d\t%d\n',A’)

– For more information:• http://www.mathworks.com/help/matlab/ref/fprintf.html#inputarg_formatSpec

Exercise: Display a matrix

• Display matrix B=

11.9 12.345 13.321 22 2331 32 3341 42 43

as shown below:

– matrix B can be loaded from “data.mat”

Display in fixed-point:11.900000 12.345000 13.30000021.000000 22.000000 23.00000031.000000 32.000000 33.00000041.000000 42.000000 43.000000

Display in 2 decimal places:11.90 12.35 13.3021.00 22.00 23.0031.00 32.00 33.0041.00 42.00 43.00

Exercise: Display a matrix

• Display matrix B=

11.9 12.345 13.321 22 2331 32 3341 42 43

as shown below:

– matrix B can be loaded from “data.mat”

Display in exponential notation:1.190000e+01 1.234500e+01 1.330000e+012.100000e+01 2.200000e+01 2.300000e+013.100000e+01 3.200000e+01 3.300000e+014.100000e+01 4.200000e+01 4.300000e+01

Display in exponential notation with 2 decimal places:1.19e+01 1.23e+01 1.33e+012.10e+01 2.20e+01 2.30e+013.10e+01 3.20e+01 3.30e+014.10e+01 4.20e+01 4.30e+01

OMG! IT’S LINEAR ALGEBRA

Nah~ You will be fine

Special Matrices

• Try on your computer and see what they do

• zeros(m,n)– If m=n, the syntax can be simplified as zeros(m)

• ones(m,n)

• eye(n)

• eye(m,n)

creates an m by n zero matrix

forms an m by n matrix of 1’s

creates an n by n identity matrix

generates an m by n zero matrix then fills the major diagonal with 1’s

Handy linear algebra tricks

• Switch rows– I3 = eye(3);

– C = [I3(2,:);I3(1,:);I3(3,:)]

• Switch column 1 and 2– A = magic(3)

– A * C

• Switch row 1 and 2– C * A

Handy linear algebra tricks

• Column sums – ones(1,3) * A

– 1 1 1 ×8 1 63 5 74 9 2

= [15 15 15]

– same as sum(magic(3),1)

• Row sums– Anyone?

Handy linear algebra tricks

• Column average– Anyone?

• Row average

Load data from Excel

• Before you guess, let’s load some data from Excel.

• Syntax:– X = xlsread(file_name [, sheet_name, range])

– file_name: the file name of the Excel spreadsheet

– sheet_name: optional, name of the sheet, a.k.a. tab name of the spreadsheet

– range: optional, data range in the sheet. ie: A3:D8

• In Matlab type in:

– The first line empties Matlab memory (Workspace). The second line reads data from available range of Sheet2 in book1.xlsx and returns a variable called “data”.

clear;

data = xlsread('book1.xlsx','Sheet2');

Guess what we are doing here

• Keep typing:

– You may copy-paste the chuck if you have the pdf version.

• Now guess what this line is doing here:

X = data(:,1:2); %extract first 2 columns into X

Y = data(:,end); %grab the last column

clear data; %destroy variable data

X = [X, ones(size(X,1),1)]; %add a column of 1’s to the

end of X

b = (X'*X)^-1*X'*Y

Guess what we are doing here

• Plot the curve in 2DZ = (1:.1:10)';

Z = [Z, Z.^2, ones(size(Z,1),1)];

Y_hat = Z*b;

plot(Z(:,1), Y_hat,'LineWidth',3);

set(gca,'Fontsize',14);

HEY MATLAB, TAKE THE DERIVATIVE FOR ME

Here’s the problem

• Take the derivative of the following function and evaluate at

𝑥 = 3,2,1

3𝑝𝑖.

𝑓 𝑥 = 𝑥5 + log𝑠𝑖𝑛 𝑥

𝑡𝑎𝑛 𝑥+ 𝑒

𝑥2

2

• Plot the function

Two types of M-files

• Script– Has no input or output arguments

– Stores a sequence of commands to be used repeatedly in future

• Function– May accept input or output arguments

– File name has to be the same as the function name defined inside

• To create an M-file, just type in

• edit file_name

Exercise: Create a script m-file

• Create an m-file named fstscript

• edit fstscript

Anonymous function

• Anonymous function is not stored in a program file and usually is treated like a variable. It accepts inputs and returns outputs.

• Syntax:– function_name = @(input_list) expression

– input_list: list of input arguments separated by colons

– function_name: the name of the function. It is actually a function handle.

• To save some debug time, avoid putting already defined variables into input_list.

x = 5;

myfx = @(x) x^2;

myfx(7)

Exercise: Define the function as anonymous function in m-file

• Take the derivative of the following function and evaluate at

𝑥 = 3,2,1

3𝑝𝑖.

𝑓 𝑥 = 𝑥5 + log𝑠𝑖𝑛 𝑥

𝑡𝑎𝑛 𝑥+ 𝑒

𝑥2

2

• Plot the function

Vectorization

• Our anonymous function probably can only process one number at a time. How about a vector of numbers?

• Prove? – vecX = [3 2 pi/3]

– myfx(vecX)

• Use dot operator to vectorize our function:– Dot operator is a cell-by-cell operator that works best for vectors

– Replace

• ^ to .^

• / to ./

– Most Matlab built-in functions are already vectorized: log, exp, sin, tan…

The derivative

• Simple notion (Differentiation):

– 1𝑠𝑡 𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒 =𝑐ℎ𝑎𝑛𝑔 𝑖𝑛 𝑓

𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑥=

Δ𝑓

Δ𝑥=

f x+Δx −f(x)

Δ𝑥

• Definition:

– 𝑓′ 𝑥 = limℎ→0

𝑓 𝑥+ℎ −𝑓(𝑥)

ℎ

• Numerical differentiation

– 𝑓′ 𝑥 ≈𝑓 𝑥+𝑚 −𝑓(𝑥−𝑚)

2𝑚

– 𝑚 is a very small number, like 0.001

– 𝑥 is the value at which 𝑓′ is evaluated

Function in an m-file

• Create an m-file named “mydev”

• Define function mydev as the following:function [rtn] = mydev(func, eval)

end

• The above code defined a function named “mydev”

• The function takes two input arguments: “func” and “eval”

• It also returns to an output argument “rtn”

• “func” denotes any anonymous function with one input argument

• “eval” denotes the value at which the derivative is evaluated

Exercise: Define a function

• Implement the numerical differentiation equation

𝑓′ 𝑥 ≈𝑓 𝑥 +𝑚 − 𝑓(𝑥 − 𝑚)

2𝑚

• In mydev.m

Exercise: Implement the first bullet point

• Take the derivative of the following function and evaluate at

𝑥 = 3,2,1

3𝑝𝑖.

𝑓 𝑥 = 𝑥5 + log𝑠𝑖𝑛 𝑥

𝑡𝑎𝑛 𝑥+ 𝑒

𝑥2

2

• Plot the function

• In fstscript.m

2D plot

• In order to plot something, you need to create a series of points in Cartesian coordinate system (XY plane)

• The command to make any two-dimensional graph is plot.

• Syntax:– plot(X, Y, LineSpec)

– X: a vector of values on the X-axis

– Y: a vector of values on the Y-axis

– LineSpec: a string that specifies the appearance of the line. For detailed definition, see http://www.mathworks.com/help/matlab/ref/linespec.html.

Exercise: Plot 𝑓(𝑥) from 𝑥 = 0 to 1

• Take the derivative of the following function and evaluate at

𝑥 = 3,2,1

3𝑝𝑖.

𝑓 𝑥 = 𝑥5 + log𝑠𝑖𝑛 𝑥

𝑡𝑎𝑛 𝑥+ 𝑒

𝑥2

2

• Plot the function from 0 to 1

• In fstscript

ADVANCED TOPIC

Way to be a nerd

Logical values

• It’s all about 0 and 1– Logical TRUE is treated as 1 in computation

– Logical FLASE is treated as 0 in computation

• The expression 3>1 equals 1

• The expression 1>3 equals 0

• Why do I care?– Indexing matrices

– Defining piecewise function / if-statement alternative

A(A>5)=0;

Exercise: Make a vectorized piecewise function

• The absolute value function is defined as:

𝑓 𝑥 = ቊ−𝑥, 𝑥 < 0𝑥, 𝑥 ≥ 0

• How can we define a vectorized anonymous function that implements the absolute value function in one single line?

Condition Value of (x<0) Value of 𝑓 𝑥 Value of (x<0)*x

𝑥 < 0 1 −1 × 𝑥 x

𝑥 ≥ 0 0 1 × 𝑥 0

Condition Value of (x>=0) Value of 𝑓 𝑥 Value of (x>=0)*x

𝑥 < 0 0 −1 × 𝑥 0

𝑥 ≥ 0 1 1 × 𝑥 x

Exercise: Make a vectorized piecewise function

• The absolute value function is defined as:

𝑓 𝑥 = ቊ−𝑥, 𝑥 < 0𝑥, 𝑥 ≥ 0

• How can we define a vectorized anonymous function that implements the absolute value function in one single line?

• myabs = @(x) (x>=0) .* x - (x<0) .* x;

• myabs2 = @(x) (2*(x>=0)-1) .* x;

Function fminsearch

• The built-in function fminsearch performs an unconstrained search for a minimum value of a function

• Syntax:– [x, fval] = fminsearch(fun, x0)

– fun: function handle that links to a function with one input and one output. This function can be a built-in function, a user defined anonymous function or a function that’s defined in a function m-file. You usually put an “@” sign before function name to pass your function as function handle

– x0: initial guess

– x: the value at which the minimum value is found

– fval: optional, the minimum value that is found

Function fminsearch

• Example:– [x1 fmin1]= fminsearch(@(x) x^2 +3*x + 4, 1)

– Alternatively, we can write things in two lines:

– qufx = @(x) x^2 +3*x + 4;

– [x1 fmin1]= fminsearch(qufx , 1)

• Example 2: Least Square

– min𝒃

𝑆𝐸 = σ𝑖 𝑦𝑖 − ො𝑦𝑖2 = σ𝑖 𝑦𝑖 − 𝒙 ∙ 𝒃 2

– In Matlab, we can define SE as:

– sefx = @(b) sum((Y – X*b).^2);

– Now can you use fminsearch to find the b that minimizes sefx?

– Set x0 = [2;3;4]

Exercise: Numerical OLS

• Load the data as before

• Use fminsearch to solve the ordinary least square coefficient

clear;

data = xlsread('book1.xlsx','Sheet2');

X = data(:,1:2); %extract first 2 columns into X

Y = data(:,end); %grab the last column

clear data; %destroy variable data

X = [X, ones(size(X,1),1)]; %add a column of 1’s to the

end of X

M-files from today

• Code introduced in this tutorial can be downloaded at the following link:

– https://z.umn.edu/mfm17-matlab-full