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2.1 Introduction
1) MATLAB
: Data analysis and visualization tool designed with powerful support for matrices and matrix operations
2) MATLAB's standard data type : Matrix
- All data are considered to be matrices of some sort
- Images are matrices whose elements are the grey values (or possibly the RGB values) of its pixels
- Single values are considered to be 1x1 matrices, while a string is a 1xn matrix of char (n : string's length)
3) MATLAB's function
: A keyword which accepts various parameters, and produces some sort of output (matrix, string, graph, figure)
2.2 Basic use of Matlab
1) Command line driven : all commands are entered by typing them after the prompt symbol (>>)
2) format function
- Matlab 계산은 내부적으로 double precision BUT default display format은 8 decimal places
- format function 이용해서 display format 바꿀 수 있음
>> format long
>> 11/7
ans =
1.57142857142857
2.3 Variables and the workspace
1) Variables : Names to store values
2) Workspace : listing all currently defined variables, their numeric data type and sizes in byte
① [View] - [Workspace]
② whos function : ans is variable (to store result of last calculation)
>> whos
Name Size Bytes Class
a 1x1 8 double array
ans 1x1 8 double array
b 1x1 8 double array
Grand total is 3 elements using 24 bytescf) who function : ans is a list of variable names only
>> who
Your variables are:
a ans b
2.4 Dealing with matrices
>> a=[4 -2 -4 7;1 5 -3 2;6 -8 -5 -6;-7 3 0 1]2.4.1 Matrix element
1) (row, column) Indexing
>> a(2,3)
ans =
-3
2) A single number Indexing
>> a(10)
ans =
-3>> a([1 6 11 16])
ans =
4 5 -5 1
3) Colon operator ( : )
→ A vector of values
a:b → All integers from integer a to b
a:i:b → All values from a by increment i up to b
a(:) → All the matrix elements as a single column (세로로 한줄)
>> 2:3:16
ans =
2 5 8 11 14
>> a(2,1:3)
ans =
1 5 -3
>> a(2:4,3)
ans =
-3
-5
0
>> a(2:3,3:4)
ans =
-3 2
-5 -6
>> a(3,:)
ans =
6 -8 -5 -6
>> a(:,2)
ans =
-2
5
-8
3
2.4.2 Matrix operations
1) Standard Arithmetic operations
: add(+), subtract(-), multiply(*), invert(inv(a)), power(^), transpose(')
2) Geometric operations
(1) flipud, fliplr : flipping a matrix up/down and left/right respectively
>> flipud(a)
ans =
-7 3 0 1
6 -8 -5 -6
1 5 -3 2
4 -2 -4 7
>> fliplr(a)
ans =
7 -4 -2 4
2 -3 5 1
-6 -5 -8 6
1 0 3 -7
(2) rot90 : rotating a matrix by 90 degrees
>> rot90(a)
ans =
7 2 -6 1
-4 -3 -5 0
-2 5 -8 3
4 1 6 -7
(3) reshape : producing a matrix with elements taken column by column from the given matrix
>> c=[1 2 3 4 5; 6 7 8 9 10; 11 12 13 14 15; 16 17 18 19 20]
c =
1 2 3 4 5
6 7 8 9 10
11 12 13 14 15
16 17 18 19 20
>> reshape(c,2,10)
ans =
1 11 2 12 3 13 4 14 5 15
6 16 7 17 8 18 9 19 10 20
>> reshape(c,5,4)
ans =
1 7 13 19
6 12 18 5
11 17 4 10
16 3 9 15
2 8 14 20※ Error : product of two values ≠ number of elements of matrix
3) Dot operators
>> a=[4 -2 -4 7; 1 5 -3 2;6 -8 -5 -6; -7 3 0 1]
>> b=[2 4 -7 -4; 5 6 3 -2;1 -8 -5 -3; 0 -6 7 -1]
a =
4 -2 -4 7
1 5 -3 2
6 -8 -5 -6
-7 3 0 1
b =
2 4 -7 -4
5 6 3 -2
1 -8 -5 -3
0 -6 7 -1
(0) Usual matrix multiplication
>> a*b
ans =
-6 -6 35 -7
24 46 37 -7
-33 52 -83 13
1 -16 65 21
(1) Dot multiplication : Element-wise operation
>> a.*b
ans =
8 -8 28 -28
5 30 -9 -4
6 64 25 18
0 -18 0 -1
(2) Dot power : Element-wise operation
>> a.^2
ans =
16 4 16 49
1 25 9 4
36 64 25 36
49 9 0 1
(3) Reciprocals (역수)
>> 1./a
ans =
0.2500 -0.5000 -0.2500 0.1429
1.0000 0.2000 -0.3333 0.5000
0.1667 -0.1250 -0.2000 -0.1667
-0.1429 0.3333 Inf 1.0000
4) Operators on matrices
- Matlab Functions work by applying the function to each element in turn
Ex) Trigonometric and Exponential functions, and Logarithms
- Many iterations and repetitions can be done with vectorization rather than by using loops
2.4.3 Constructing matrices
1) zeros / ones
zeros(n) : a zeros matrix of size n x n
zeros(m,n) : a zeros matrix of size m x n
zeros(m,n,p,...) : an m x n x p x ... multidimensional array of zeros
zeros(a) : a zeros matrix of the same size as matrix a
m, n, p : number / a : matrix
2) rand / randn
a random numbers matrix (Size 옵션은 zeros 위 3개 형태와 같음)
rand : numbers are taken from a uniform distribution on interval [0,1]
randn : numbers are taken from a normal distribution N~(0,1)
>> floor(10*rand(3))
ans =
0 9 9
2 9 9
5 1 4
>> floor(100*randn(3,5))
ans =
71 148 67 163 72
-21 140 -121 48 -31
-13 141 71 103 29
3) A matrix every element of which is a function of one of its indices - using dot operators
>> rows = (1:10)'*ones(1,10)
rows =
1 1 1 1 1 1 1 1 1 1
2 2 2 2 2 2 2 2 2 2
3 3 3 3 3 3 3 3 3 3
4 4 4 4 4 4 4 4 4 4
5 5 5 5 5 5 5 5 5 5
6 6 6 6 6 6 6 6 6 6
7 7 7 7 7 7 7 7 7 7
8 8 8 8 8 8 8 8 8 8
9 9 9 9 9 9 9 9 9 9
10 10 10 10 10 10 10 10 10 10>> cols = ones(10,1)*(1:10)
cols =
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10>> A = rows + cols -1
A =
1 2 3 4 5 6 7 8 9 10
2 3 4 5 6 7 8 9 10 11
3 4 5 6 7 8 9 10 11 12
4 5 6 7 8 9 10 11 12 13
5 6 7 8 9 10 11 12 13 14
6 7 8 9 10 11 12 13 14 15
7 8 9 10 11 12 13 14 15 16
8 9 10 11 12 13 14 15 16 17
9 10 11 12 13 14 15 16 17 18
10 11 12 13 14 15 16 17 18 19>> [cols,rows] = meshgrid(1:10,1:10)
cols =
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
rows =
1 1 1 1 1 1 1 1 1 1
2 2 2 2 2 2 2 2 2 2
3 3 3 3 3 3 3 3 3 3
4 4 4 4 4 4 4 4 4 4
5 5 5 5 5 5 5 5 5 5
6 6 6 6 6 6 6 6 6 6
7 7 7 7 7 7 7 7 7 7
8 8 8 8 8 8 8 8 8 8
9 9 9 9 9 9 9 9 9 9
10 10 10 10 10 10 10 10 10 102.4.4 Vectorizations
Vectorization : an operation carried out over an entire matrix or vector
Matlab is designed to perform vectorized commands very quickly !
(1) Very efficient replacement for Loops
>> tic, for i=1:10^6,sin(i); end,toc
경과 시간은 0.119952초입니다.
>> tic, i=1:10^6;sin(i); toc
경과 시간은 0.045064초입니다.(2) To generate the first 10 Square numbers
>> [1:10].^2
ans =
1 4 9 16 25 36 49 64 81 100(3) Logical Operators : To obtain all positive elements of the matrix
>> a>0
ans =
1 0 0 1
1 1 0 1
1 0 0 0
0 1 0 1
2.5 Plots
(1) plot(x, y)
>> x=[0:0.1:2*pi];
>> plot(x,sin(x))
(2) plot(x, y1, 'symbol1', x, y2, 'symbol2')
>> plot(x,sin(x),'.', x,cos(x),'o')
2.6 Help in Matlab
(1) help / doc + <function>
>> help for
for - 지정된 횟수를 반복하는 for 루프
지정된 횟수만큼 루프에서 명령문 그룹을 실행합니다.
for index = values, statements, end
참고 항목 break, colon, continue, end, if, parfor, return, switch
for에 대한 문서>> doc for
(2) lookfor + <topic>
Use 'lookfor' when you want to find help on a particular topic, but don’t know the function to use
>> doc help
>> doc for
>> lookfor exponential
exp - Exponential.
expm - Matrix exponential.
expint - Exponential integral function.
expmdemo1 - Matrix exponential via Pade approximation.
expmdemo2 - Matrix exponential via Taylor series.
expmdemo3 - Matrix exponential via eigenvalues and eigenvectors.
quatexp - Calculate the exponential of a quaternion.
vivaldi - Creates a vivaldi notch antenna on a ground plane with exponential
sweeptone - Exponential swept sine signal
exp - Overload exponential for DataMatrix object.
ewcov - Asset covariance from return series with exponential weighting.
simByQuadExp - Simulate Bates sample paths by Quadratic-Exponential
simByQuadExp - SIMBYTRANSITION Simulate CIR sample paths with Quadratic-Exponential
simByQuadExp - Simulate Heston sample paths by Quadratic-Exponential
exp - @FINTS/EXP exponential of the values in a FINTS object.
cordiccexp - CORDIC-based approximation of complex exponential e^(j*THETA).
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