Command Line Terminal

help

format

format dataType指定数字的格式

who

显示当前有哪些变量

whos

显示当前变量的详细信息

clear

清除所有变量，后加变量名则清除单个变量

;

句末加分号，不显示计算结果

方向键

显示历史命令

clc

清屏

time

tic
command
toc
% count the time of runing command

Matrix

矩阵的输入

Row vector:
	a = [1 2 3]
Column vector:	
	b = [1;2;3]

Array Indexing

1. A(row.column)
   • A([row1 row2],[column1 column2]) 得到交集
2. A(index)
   • A([index1 index2 index3])
   • A([index1 index2;index3 index4])

Replacing Entries

delete row: A(row,:) = []

Colon Operator

A = begin:end  => A=[begin begin+1 .... end]
A = begin:i:end => A=[begin begin+i begin+2i ...]

Colon opertator 用作index时代表全部

Array Concatenation

F = [A B]
F = [A;B]

Array Manipulation

with Matirx

1. ‘+’ : 同位置相加
2. ‘-’ : 同位置相减
3. ‘*’ : 前一个矩阵的行与后一个矩阵的列相乘
4. ‘/’ : 相当于与逆矩阵做乘法
5. .*: 同位置相乘
6. ./: 同位置相除
7. A’: A的转置

with number

1. ‘+’ : 每个entry都加number
2. ‘-’ : 每个entry都减number
3. ‘*’ : 每个entry都乘number
4. ‘/’ : 每个entry都除number
5. .*: 同 *
6. ./: 同 /
7. ^ : n个矩阵相乘
8. .^: 每个entry都求n次幂

Special Matrix

1. eye(n): n*n identity matrix
2. zeros(n1,n2): n1*n2 zero matrix
3. ones(n1,n2): n1*n2 matirx with every entry as 1
4. diag(): diagonal matrix
5. linspace(): linearly spaced vectors
6. rand(): uniformly distributed random numbers

Matrix Related Functions

1. max(A): find max entry in every column
2. max(max(A)):find max entry
3. min(A):find min entry in every column
4. sum(A):get sum in every column
5. mean(A):get average in every column
6. sort(A):sort every column of A
7. sortrows(A):sort A by first column and exchange rows
8. size(A): return row and column
9. length(A): vector’s length
10. find(A == num): return num’s index

Structured programming

if elseif else

if condition1
	statement1
elseif condition2
	statement2
else
	statement3
end

switch

switch expression
case value1
	statement1
case value2
	statement2
otherwise
	statement3
end

while

while condition
	statement
end

for

for variable = start : increment : end
	commands
end

break

terminates the execution of for or while loops

Tips

1. At the beginning of the script, use command
   • clear all to remove previous varibles
   • close all to close all figures
   • clc to clear history
2. Use semicolon ; at the end of commands to inhibit unwanted output
3. Use ellipsis … to make scripts more readable
4. Ctrl+C to terminate the running script

Function

example

% function name = file name 
function y = freebody(x0, v0, t)
x = x0 + v0.*t + 1/2*9.8*t.*t;

function [a F] = acc(v2, v1, t2, t1, m)
a = (v2 - v1)./(t2 - t1);
F = m.*a;

[Acc Force] = acc(20, 10, 5, 4, 1)

Function default variables

1. inputname

   Variable name of function input

2. mfilename

   File name of currently running function

3. nargin

   Number of function input arguments

4. nargout

   Number of function out arguments

5. varargin

   Variable length input argument list

6. varargout

   Variable length out argument list

Function handles

A handle is a pointer to a function

A way to create anonymous function

f = @(x) exp(-2*x);
x = 0:0.1:2;
% draw figure
plot(x, f(x))

Variables

1. numeric

   1. int8, int16, int32, int64
   2. uint8, uint16, uint32, uint64
   3. single
   4. double

2. char

   % char is represented in ASCII using a numeric code between 0 to 255
   a = 'c'
   % declare string variable
   str = 'abcdeabc'
   
   str(3)
   % print 'c' , 3 as index
   
   str == 'a'
   % print 1 0 0 0 1 0 0
   
   str(str == 'a') = 'Z'
   % a -> Z  print 'ZbcdZbc'
   

3. logical

4. cell

5. structure

6. function handle

Structure

declare

% 1
student.id = 'first';
student.name = 'first';
student.number = 123456;
student.grade = [100, 75, 73;...
				95, 91, 85.5;...
				100, 98, 72];
student

student(2).id = 'second';
student(2).name = 'second';
student(2).number = 123456;
student(2).grade = [100, 75, 73;...
                   95, 91, 85.5;...
                   100, 98, 72];

% 2
A = struct('data', [3 4 5;8 0 1],...
			'nest', struct('x', [1 2 3], ...
						   'y', [3 4 5]));
% A.data = [3 4 5;8 0 1]
% A.nest.x = [1 2 3]
% A.nest.y = [3 4 5]

Structure Function

1. fieldnames(S) : get all the fildnames of S
2. rmfield(S, ‘id’) : remove the field ‘id’ of S

Cell

delare

delared using {}

% 1
A(1,1) = {[1 2;3 4]};
A(1,2) = {'string'};
A(2,1) = {3+7i};
A(2,2) = {-pi:pi:pi};
A

% 2
A{1,1} = [1 2;3 4];
A{1,2} = 'string';
A{2,1} = 3+7i;
A{2,2} = -pi:pi:pi;
A

Accessing cell array

A(1,1)
% return a cell data

A{1,1}
% return the content in the cell

Cell Array Function

1. cell(row,column) : create cell array
2. cell2struct : convert cell array to structure
3. mat2cell : convert array to cell array with different sized cells
4. num2cell : convert array to cell array with consistently sized cells
5. cat(dimension,A,B) : concat cell array in different dimensions
6. reshape(A,row,column) : reshape cell array

Type Convertion

1. double()
2. single()
3. int8() int16() int32() int64()
4. uint8() uint16() uint32() uint64()

File Access

save

1. work space

   save filename.mat
   % save workspace data in file
   
   save filename.mat -ascii
   % save readable workspace data in file 
   

2. xls

   xlswrite('filename.xlsx', variable, sheet_page, location)
   
   % example
   xlswrite('filename.xlsx', M, 1, 'E2:E4') % M is a matrix
   xlswrite('filename.xlsx', {'Mean'}, 1, 'E1')
   

load

1. work space

   load ('filename.mat')
   
   load ('filename.mat', '-ascii')
   

2. xls

   score = xlsread('filename.xlsx')
   
   score = xlsread('filename.xlsx', 'B2:D4')
   
   [score header] = xlsread('filename.xlsx')
   

Low-level File I/O Functions

1. fopen
2. fclose
3. fscanf
4. fprintf
5. feof

% example
x = 0:pi/10:pi;y = sin(x);
fid = fopen('filename.txt', 'w');
for i=1:11
	fprintf(fid, '%5.3f %8.4f\n', x(i), y(i));
end
fclose(fid);

plot

plot()

1. plot(x, y) : plot each vector pairs (x,y)
2. plot(y) : plot each vector pairs (x,y) where x = [1 … n] ,n=length(y)

hold on/off

use hold on to have both plots in one figure

hold on
plot(cos(0:pi/20:2*pi))
plot(sin(0:pi/20:2*pi))
hold off

plot style

plot(x, y, 'str')

LineSpec (MATLAB Functions) (northwestern.edu)

1. Data markers
   • dot : .
   • cross : X
   • circle : O
   • plus sign : +
2. Line types
   • solid line : -
   • dashed line : –
   • dash-dotted line : -.
   • dotted line : :
3. Colors
   • black : k
   • $\color{#0000FF}{蓝}$ : b
   • cyan : c
   • green : g
   • magenta : m
   • red : r
   • white : w
   • yellow : y

legend()

add legend to graph

plot(x,y,'bd-', x,h,'gp:')
legend('L1', 'L2')

title & label

1. title()
2. xlabel()
3. ylabel()
4. zlabel()

text & annotation

str = '$$ \int_{0}^{2} x^2\sin(x) dx $$';
text(x, y, str, 'Interpreter', 'latex');
annotation('arrow', 'X', [0.32,0.5], 'Y', [0.6,0.4]);

figure adjustment

Identifying the handle of an object

1. h = plot(x,y);
2. Utility functions
   1. gca ： return the handle of the current axes
   2. gcf ： return the handle of the current figure
   3. allchild : Find all children of specified objects
   4. ancestor : Find ancestor of graphics object
   5. delete : delete an object
   6. findall : find all graphics objects

Fetching or Mordifying properties

1. fetch properties : get()

   h = plot(x,y);
   % get line properties
   get(h);
   
   % get axes properties
   get(gca)
   
   % get figure properties
   get(gcf)
   

2. modify properties : set()

   set(handle, property, value)
   

properties

1. Axes
   • FontSize
   • XTick
   • XTickLabel
   • FontName
2. Line
   • LineStyle
   • LineWidth
   • Color
3. Figure
   • Position : [left, bottom, width, height]

Mutiple plots

1. serval plots in different figures

   figure, plot(x, y1);
   figure, plot(x, y2);
   

2. serval plots in **a figure **

   % subplot(row, column, num)
   subplot(2,2,1); plot(x,y); axis normal
   subplot(2,2,2); plot(x,y); axis square
   subplot(2,2,3); plot(x,y); axis equal
   subplot(2,2,4); plot(x,y); axis equal tight
   

Control of Grid, Box and Axis

1. grid on/off
2. box on/off
3. axis on/off
4. axis normal
5. axis square
6. axis equal
7. axis equal tight

Save figures

1. saveas(gcf, 'filename', 'formattype')
   

2. print

Option

• Bitmap
  • jpeg
  • png
  • tiff
  • bmpmono
  • bmp
• Vector
  • pdf
  • eps
  • epsc
  • meta
  • svg
  • ps
  • psc

Advanced 2D plots

Logarithm Plots

Take the logarithm of the axis

1. semilogx
2. semilogy
3. loglog

plotyy

two y axis

[AX, H1, H2] = plotyy(x,y1,x,y2);
set(get(AX(1), 'Ylabel'), 'String','Left Y-axis');
set(get(AX(2), 'Ylabel'), 'String','Right Y-axis');
set(H1, 'LineStyle', '--'); set(H2, 'LineStyle', ':');

graph

1. Histogram

   1. hist(y, bins)

   

2. bar chart

   1. bar(y) : create a bar chart of y
   2. bar3(y) : create a 3D bar chart of y
   3. bar(y, ‘stacked’) : create a stacked bar chart of y
   4. barh(y) : create a hrizontal bar chart of y

   

3. pie chart

   1. pie(a) : create a pie chart
   2. pie(a,[0,0,0,1]) : create a pie chart and ‘1’ is a separated section
   3. pie3(a) : create a 3d pie chart

   

4. polar chart

   1. polar(theta, r)

   

5. stairs and stem chart

   1. staris(y)
   2. stem(y)

   

6. boxplot

   1. boxplot(MPG, Origin)

   

7. error bar

   1. errorbar(x,y,e)

   

color

fill

t = (1:2:15)*pi/8; x = sin(t); y = cos(t)
fill(x,y,'r'); axis square off;
text(0,0,'STOP','Color','w','FontSize',80 ...
	'FontWeight','bold','HorizontalAlignment','center');

color space

[R G B]

color bar

colorbar;

Display the color of the gradation

imagesc

Convert the element values in matrix A to different colors by size : imagesc(A)

clormap

set the current colormap colormap([name])

3D plots

plot3

plot3(x,y1,z1,'r',x,y2,z2,'b');

meshgrid

create two matrices that satisfy the following conditions

- the same for each row of x
- the same for each column of y
- The length of vector x is the number of **columns** of the new matrix
- The length of vector y is the number of **rows** of the new matrix

[X,Y] = meshgrid(x,y);
% x = [1 2 3]; y = [4 5]
% X = [1 2 3;1 2 3] y = [4 4 4;5 5 5]

Surface Plots

1. mesh()
2. surf()
3. meshc() : add contour at the bottom
4. surfc()

x = -3.5:0.2:3.5; y = -3.5:0.2:3.5;
[X,Y] = meshgrid(x,y);
Z = X.*exp(-X.^2-Y.^2)；
subplot(1,2,1); mesh(X,Y,Z);
subplot(1,2,2); surf(X,Y,Z);

contour

Draw contour lines of 3D graphics

1. contour()
2. Clabel() : Add elevation labels to contour plots
3. contourf() : fill color

view

vary the view angle

view(Azimuth, Elevation)

light

% example

% provide a light
L1 = light('Position',[-1 -1 -1]);

% set the light that have been provided
set(L1, 'Position', [-1 -1 1]);

%set the color of light
set(L1, 'Color', 'g');

Digital Image

Type

1. Binary Image : 0 is balck and 1 is white
2. Greyscale Image : one channel , 0 to 255, black to white
3. Color Image : three channel , R G B

Read and Show Image

• Read an image : imread()

• Show an image : imshow()

  I = imread('file'); % I is an image matrix	
  imshow(I);
  

Image info

imageinfo(file)

Write Image

imwrite(I, 'filename.png')

Image Processing

Image Arithmetic

1. imadd
2. imsubtract
3. immultiply
4. imdivide
5. imcoplement

Image Histogram

1. imhist

   imhist(I) Display the grayscale histogram of image i

2. histeq

   I1 = histeq(I) histeq can produce an image with an average gray level distribution probability

Geometric Transformation

Transform	Example	Transfomation Matirx	command
Translation			imtranslate()
Scale			imresize()
Shear			-
Rotation			imrotate()

Image thresholding

if pixel < threshold ,

pixel = black,

else pixel = white

1. graythresh() : computes an optimal threshold

2. im2bw() : converts an images into binary images

   I = imread('file.png');
   level = graythresh(I);
   bw = im2bw(I,level);
   

Background Estimation

Estimation for the gray level of the background

BG = imopen(I,strel('disk',15));
imshow(BG);

Background subtraction

If thresholding the inital image, some item will be subtracted and some background will be remained as dot

So, subtract background to optimize the image thresholding

I2 = imsubtract(I,BG);

Connected-component labeling : bwlabel()

[labeled, numObjects] = bwlabel(BW);
[labeled, numObjects] = bwlabel(BW,n);
% n is the Connected search area, default: 8

Color-coding Objects : label2rgb()

Convert a label matrix into an RGB color image

Visualize the labeled regions

RGB_label = label2rgb(labeled);
imshow(RGB_label);

Object Properties : regionprops()

Provides a set of properties for each connected component

data = regionprops(labeled,'basic');

Interactive Selection : bwselect()

Select objects using the mouse

objI = bwselect(BW);
imshow(objI);

Differentiation and integration

Polynomial

$$ f(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}, $$

$$ differentiation: f^{\prime}(x)=a_{n} n x^{n-1}+a_{n-1}(n-1) x^{n-2}+\cdots+a_{1} $$

$$ integration: \int f(x)=\frac{1}{n+1} a_{n} x^{n+1}+\frac{1}{n} a_{n-1} x^{n}+\cdots+a_{0} x+k $$

% enter a polynomial to Matlab
p = [an, an-1, .... ,a0];

polyval()

return the value of polynomial in the x

p = [1,2,3,4]; x = -2:0.01:5;
f = polyval(p, x);
plot(x,f);

polyder()

return the polynomial differentiation

p = [1,2,3,4];
polyder(p);

polyint()

return the polynomial integration

polyint(p, k);

Numerical

$$ Differentiation: f^{\prime}\left(x_{0}\right)=\lim _{h \rightarrow 0} \frac{f(x0+h)-f(x0)}{h} $$

$$ integration: s=\int_{a}^{b} f(x) d(x) \approx \sum_{i=0}^{n} f\left(x_{i}\right) \int_{a}^{b} L_{i}(x) d x $$

diff()

calculates the differences between adjacent elements of a vector

x = [1 2 5 2 1];
diff(x);
% return 1 3 -3 -1

% Numerical differentiation using diff()
% at a point
x0 = pi/2; h = 0.1;
x = [x0,x0+h];
y = [sin(x0), sin(x0+h)];
m = diff(y)./diff(x);

% within a range
h = 0.1;
x = [0:h:2*pi];

integral()

caculate the numerical integration

y = @(x) 1./(x.^3 - 2*x);
integral(y,0,2);
% caculate the integration from 0 to 2

% Double integral
integral2(f,begin1,end1,begin2,end2);
% two param: x from begin1 to end1 ; y from begin2 to end2

% Triple integral
integral3(f,begin1,end1,begin2,end2,begin3,end3);

integration

1. Midpoint rule

   $$ \int_{x_{0}}^{x_{3}} f(x) d x \approx h f_{0}+h f_{1}+h f_{2}=h \sum_{i=0}^{n-1} f_{i} $$

$$ f_{0} = f\left(\frac{x_{0}+x_{1}}{2}\right) $$

Midpoint Rule Using sum()

h = 0.05; x = 0:h:2;
midpoint = (x(1:end-1) + x(2:end))./2;
y = f(midpoint);
s = sum(h*y);

2. Trapezoid Rule

   $$ \int_{x_{0}}^{x_{3}} f(x) d x \approx h \frac{f_{0}+f_{1}}{2}+h \frac{f_{1}+f_{2}}{2}+h \frac{f_{2}+f_{3}}{2} $$

$$ f_{0}=\frac{f\left(x_{0}\right)+f\left(x_{1}\right)}{2} $$

Trapezoid Rule Using trapz()

h = 0.05; x = 0:h:2;
y = f(x);
s = h*trapz(y);

3. Second-order Rule : 1/3 Simpson’s

   $$ \int_{x_{0}}^{x_{2}} f(x) d x \approx \frac{h}{3}\left(f_{0}+{4 f_{1}}+{{f_{2}}}\right) $$

Equation rooting

Symbolic Root Finding Approach

Using sym or syms to create symbolic variables

syms x
x = sym('x')

Function sovle finds roots for equation

syms x
y = x*sin(x)-x;
solve(y,x)
% return the ans of y = 0

% Solving Multiple Equations
syms x y
eq1 = x - 2*y + 5;
eq2 = x + y - 6;
A = solve(eq1,eq2,x,y);

% Solving Equations Expressed in Symbols
syms x a b
solve('a*x^2-b') % express x in terms of a and b
solve('a*x^2-b', 'b') % express b in terms of a and x

Symbolic Differentation and Integration

1. Symbolic Differentation : diff()

   syms x
   y = 4*x^5;
   yprime = diff(y) % 20*x^4
   

2. Symbolic Integration : int()

   syms x;
   y = x^2*exp(x);
   z = int(y);
   z = z - subs(z, x, 0);
   % subs is substitue 0 for x in z
   

Numeric root solver

fsolve()

f = @(x) (2*x + sin(x));
fsolve(f,0);
% f is a function handle
% 0 is initial guess

fzero()

it can’t solve tangent curve question

options

set params for fsolve or fzero

f = @(x) x.^2
options = optimset('MaxIter', 1e3, 'TolFun', 1e-10);
% MaxIter is number of iterations
% TolFun is Tolerance
fsolve(f,0.1,options);
fzero(f,0.1,options);

Finding Roots of Polynomials

Using roots()

roots only work for Polynomials

roots([1,2,3,4,5]);

Linear equation

$$ \left{\begin{array}{l} 3 x-2 y=5 \ x+4 y=11 \end{array}\right. $$

$$ \underbrace{\left[\begin{array}{cc} 3 & -2 \ 1 & 4 \end{array}\right]}{A}\underbrace{\left[\begin{array}{c} x \ y \end{array}\right]}{x} =\underbrace{\left[\begin{array}{c} 5 \ 11 \end{array}\right]}_{b} $$

Matrix Decomposition Factorization

1. qr : Orthogonal-triangular decomposition
2. ldl : Block LDL’factorization for Hermitian indefinite matrices
3. ilu : Sparse incompelete LU factorization
4. lu : LU matrix factorization
5. chol : Cholesky factorization
6. gsvd : Generalized singular value decomposition
7. svd : Singular value decomposition

Matrix Left Division

Solving systems of linear equations Ax = b using factorization methods

Using / or mldivide()

A = [1 2 1;2 6 1;1 1 4];
b = [2; 7 ; 3];
x = A/b;

Cramer’s(Inverse) Method

$$ A A^{-1}=A^{-1} A=I $$

$$ x=A^{-1} b $$

Inverse Matrix

Using inv()

A = [1 2 1;2 6 1;1 1 4];
b = [2; 7 ; 3];
x = inv(A)*b;

Functions to Check Matrix Condition

Inverse matrix maybe not exist

Using cond() : Matrix Condition number

A = [1 2 3;2 4.0001 6;9 8 7];
cond(A);

Linear System

$$ \left{\begin{array}{l} 2 \cdot 2-12 \cdot 4=x \ 1 \cdot 2-5 \cdot 4=y \end{array}\right. \newline \newline $$

Eigenvalues and Eigenvectors

$$ \boldsymbol{A} \boldsymbol{v}{i}=\lambda{i} \boldsymbol{v}{i},\lambda{i} \in \Re $$

$$ \boldsymbol{b}=\sum \alpha_{i} \boldsymbol{v}{i}, \alpha{i} \in \Re $$

$$ \boldsymbol{A} \boldsymbol{b}=\sum \alpha_{i} \boldsymbol{A} \boldsymbol{v}{i}=\sum \alpha{i} \lambda_{i} \boldsymbol{v}_{i} $$

Using eig() to get v and λ

[v, d] = eig(A);
% A = v*d*inv(v)

Statistics

Central Tendency

1. mean : Average or mean value of array
2. median : Median value of array
3. mode : Most frequent values in array
4. prctile : Percentiles of a data set

Range

1. max : Largest elements in array
2. min : Smallest elements in array

Variance

$$ Variance : s=\frac{\sum\left(x_{i}-\bar{x}\right)^{2}}{n-1} $$

$$ Standard-deviation:S=\sqrt{\frac{\sum\left(x_{i}-\bar{x}\right)^{2}}{n-1}} $$

1. sld : Standard deviation
2. var : variance

Skewness

A measure of distribution skewness

Using skewness() to get skewness

y = skewness(X)

Kurtosis

A measure of distribution flatness

A Kurtosis of a normal distribution is zero

Hypothesis Test

Two-tailed Significance Test

One-tailed Significance Test

Common Hypothesis Test

1. ranksum() : Wilcoxon rank sum test
2. signrank() : Wilcoxon signed rank test
3. ttest() : One-sample and paired-sample t-test
4. ttest2() : Two-sample t-test
5. ztest() : z-test

Polynomial Curve Fitting

Using polyfit()

x = [1 2 3 4 5];
y = [1 2 3 4 5];
fit = polyfit(x,y,1);
% return the confficients of f(x) = ax + b (order = 1)

% plot the f(x) and (x,y)
xfit = [x(1):0.1:x(end)]; yfit = fit(1)*xfit + fit(2);
plot(x,y,'bo',xfit,yfit);

Linearly Correlated

1. scatter() : scatte plot
2. corrcoef() : correlation , [-1,1]

x = [1 2 3 4 5];
y = [1 2 3 4 5];
scatter(x,y); box on; axis square;
corrcoef(x,y)

Multiple Linear Regression

Using regress()

X = [ones(length(x1),1) x1 x2];
b = regress(y,x);
% y = a + bx1 + cx2

Curve Fitting Toolbox

Using cftool() to open Toolbox

Interpolation

1. Linear

   

2. cosine

   

3. Cubic

   

4. Hermite

   

Common Interpolation approaches

1. interp1() : 1-D data inerpolation
2. pchip() : Piecewise Cubic Hermite Interpolation Polymial
3. spline() : Cubic spline data interpolation
4. mkpp() : Make piecewise polynomial

x_m = 0:0.5:2*pi;
y_m = cos(x);
x = 0:0.1:2*pi;
y = interp1(x_m,y_m,x);
plot(x_m,y_m,'.','color','g');
hold on;
plot(x,y,'.','color','r');
hold off;

(to be continued…)