INF040 - MathLab-Tranpar. 1

Introdução ao MATLAB

M

ATLAB

Basico

Introduction to M

ATLAB

Online Help

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The help command

_{>> help}

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The help window

_{>> helpwin}

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The lookfor command

_{>> lookfor}

»

CD Change current working directory.

CD directory-spec sets the current directory to the one specified. CD .. moves to the directory above the current one.

CD, by itself, prints out the current directory. WD = CD returns the current directory as a string.

Use the functional form of CD, such as CD('directory-spec'), when the directory specification is stored in a string.

See also PWD.

»

CD Change current working directory.

CD directory-spec sets the current directory to the one specified. CD .. moves to the directory above the current one.

CD, by itself, prints out the current directory. WD = CD returns the current directory as a string.

Use the functional form of CD, such as CD('directory-spec'), when the directory specification is stored in a string.

Introduction to M

ATLAB

Calculations at the Command Line

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»cmd_line

MATLAB as a calculator

Assigning Variables

Numbers stored in double-precision

floating point format

Introduction to M

ATLAB

Working with Files & Variables

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CD / PWD, LS / DIR - navigating directories

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WHAT - displays the files within a directory

(grouped by type)

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! - invoke operating system

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WHICH - identifies the object referenced by given

name (function / variable)

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CLEAR - remove function / variable from memory

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WHOS - lists workspace variables and details

(size, memory usage, data type)

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SIZE - returns the size of matrix

Introduction to M

ATLAB

»workspace

Command line

variables saved in

MATLAB workspace

Workspace Browser

Introduction to M

ATLAB

»openvar ans

Array Editor

double-click /

Open

For editing 2-D

numeric arrays

Introduction to M

ATLAB

Working with Matrices

MATLAB == MATrix LABoratory

»load durer

»image(X);

»colormap(map)

»load detail

»image(X);

»colormap(map)

Introduction to M

ATLAB

The Matrix in MATLAB

4

10

1

6

2

8

1.2

9

4

25

7.2

5

7

1

11

0

0.5

4

5

56

23

83

13

0

10

1

2

Rows (m) 3

4

5

Columns

(n)

1 2 3 4 5

A =

_{A (2,4)}

A (17)

Rectangular Matrix:

Scalar: 1-by-1 array

Vector: m-by-1 array

1-by-n array

Matrix: m-by-n array

Matrix elements can be EITHER

Introduction to M

ATLAB

Any MATLAB expression can

be entered as a matrix element

Entering Numeric Arrays

» a=[1 2;3 4] a = 1 2 3 4 » b=[-2.8, sqrt(-7), (3+5+6)*3/4] b = -2.8000 0 + 2.6458i 10.5000 » b(2,5) = 23 b = -2.8000 0 + 2.6458i 10.5000 0 0 0 0 0 0 23.0000 » a=[1 2;3 4] a = 1 2 3 4 » b=[-2.8, sqrt(-7), (3+5+6)*3/4] b = -2.8000 0 + 2.6458i 10.5000 » b(2,5) = 23 b = -2.8000 0 + 2.6458i 10.5000 0 0 0 0 0 0 23.0000

Row separator:

semicolon (;)

Column separator:

space / comma (,)

»num_array1

Use square

brackets [ ]

Matrices must

be rectangular.

Introduction to M

ATLAB

Entering Numeric Arrays - cont.

» w=[1 2;3 4] + 5 w = 6 7 8 9 » x = 1:5 x = 1 2 3 4 5 » y = 2:-0.5:0 y = 2.0000 1.5000 1.0000 0.5000 0 » z = rand(2,4) z = 0.9501 0.6068 0.8913 0.4565 0.2311 0.4860 0.7621 0.0185 » w=[1 2;3 4] + 5 w = 6 7 8 9 » x = 1:5 x = 1 2 3 4 5 » y = 2:-0.5:0 y = 2.0000 1.5000 1.0000 0.5000 0 » z = rand(2,4) z = 0.9501 0.6068 0.8913 0.4565 0.2311 0.4860 0.7621 0.0185

Scalar expansion

Creating

sequences:

colon operator (:)

Utility functions for

creating matrices.

Introduction to M

ATLAB

Numerical Array Concatenation - [ ]

»num_cat

» a=[1 2;3 4]

a =

1 2 3 4

» cat_a=[a, 2*a; 3*a, 4*a; 5*a, 6*a]

cat_a = 1 2 2 4 3 4 6 8 3 6 4 8 9 12 12 16 5 10 6 12 15 20 18 24 » a=[1 2;3 4] a = 1 2 3 4

» cat_a=[a, 2*a; 3*a, 4*a; 5*a, 6*a]

cat_a = 1 2 2 4 3 4 6 8 3 6 4 8 9 12 12 16 5 10 6 12 15 20 18 24

Use [ ] to combine

existing arrays as

matrix “elements”

Row separator:

semicolon (;)

Column separator:

space / comma (,)

Use square

brackets [ ]

The resulting

matrix must

be rectangular.

4*a

Introduction to M

ATLAB

Array Subscripting / Indexing

4

10

1

6

2

8

1.2

9

4

25

7.2

5

7

1

11

0

0.5

4

5

56

23

83

13

0

10

1

2

3

4

5

1 2 3 4 5

A =

A(3,1)

A(3)

A(1:5,5)

A(:,5)

A(21:25)

A(4:5,2:3)

A([9 14;10 15])

• Use () parentheses to specify index

• colon operator (:) specifies range / ALL

• [ ] to create matrix of index subscripts

• ‘end’ specifies maximum index value

A(1:end,end)

A(:,end)

Introduction to M

ATLAB

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Create a 5x5 Pascal matrix ("P_mat")

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Extract elements to create row vectors

for the first 5 rows of Pascal’s Triangle

(Look for patterns in the element locations in "P_mat")

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Combine the vectors into a single

matrix with zeros above the diagonal

"P_mat"

Exercise: Creating Matrices

1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 . . . Pascal’s Triangle 1 0 0 0 0 1 1 0 0 0 1 2 1 0 0 1 3 3 1 0 1 4 6 4 1 1 0 0 0 0 1 1 0 0 0 1 2 1 0 0 1 3 3 1 0 1 4 6 4 1 1 1 1 1 1 1 2 3 4 5 1 3 6 10 15 1 4 10 20 35 1 5 15 35 70 1 1 1 1 1 1 2 3 4 5 1 3 6 10 15 1 4 10 20 35 1 5 15 35 70 row1 = 1 row2 = 1 1 row3 = 1 2 1 row4 = 1 3 3 1 row5 = 1 4 6 4 1 row1 = 1 row2 = 1 1 row3 = 1 2 1 row4 = 1 3 3 1 row5 = 1 4 6 4 1

Introduction to M

ATLAB

Solution: Creating Matrices (1)

» P_mat = pascal(5) % Create Pascal's Matrix

» row1 = P_mat(1) % Extract Rows

» row2 = P_mat(2:4:6) ... » row5 = P_mat(5:4:21)

» new_mat = row1 % Create New Matrix

» new_mat(2,1:2) = row2 ... » new_mat(5,1:5) = row5

% NOTE: GENERALIZED FORM: % Nmax = length(P_mat);

% rowN = P_mat(N:Nmax-1:(N-1)*Nmax+1) % new_mat(N,1:N) = rowN

» P_mat = pascal(5) % Create Pascal's Matrix

» row1 = P_mat(1) % Extract Rows

» row2 = P_mat(2:4:6) ... » row5 = P_mat(5:4:21)

» new_mat = row1 % Create New Matrix

» new_mat(2,1:2) = row2 ... » new_mat(5,1:5) = row5

% NOTE: GENERALIZED FORM: % Nmax = length(P_mat);

% rowN = P_mat(N:Nmax-1:(N-1)*Nmax+1) % new_mat(N,1:N) = rowN

Introduction to M

ATLAB

Solution: Creating Matrices (2)

» P_mat = pascal(5) % Create Pascal's Matrix % Rearrange Matrix - Pascal Triangle in upper triangle % Flip Matrix Horizontally (LEFT-RIGHT):

% - Could also use FLIPLR():

» temp = P_mat(1:5, 5:-1:1)

% Extract diagonals (rows of Pascal’s Triangle)

» row1 = diag(temp,4)' » row2 = diag(temp,3)' » row3 = diag(temp,2)' » row4 = diag(temp,1)' » row5 = diag(temp)'

% Create New Matrix as before

» P_mat = pascal(5) % Create Pascal's Matrix

% Rearrange Matrix - Pascal Triangle in upper triangle % Flip Matrix Horizontally (LEFT-RIGHT):

% - Could also use FLIPLR():

» temp = P_mat(1:5, 5:-1:1)

% Extract diagonals (rows of Pascal’s Triangle)

» row1 = diag(temp,4)' » row2 = diag(temp,3)' » row3 = diag(temp,2)' » row4 = diag(temp,1)' » row5 = diag(temp)'

% Create New Matrix as before

Introduction to M

ATLAB

Solution: Creating Matrices (3)

» P_mat = pascal(5) % Create Pascal's Matrix % Rearrange Matrix - Pascal Triangle in lower triangle % Flip Matrix Vertically (UP-DOWN):

% - Could also use FLIPUD():

» temp1 = P_mat(5:-1:1, 1:5)

% Extract lower triangular matrix:

» temp2 = tril(temp1)

% Create New Matrix by sorting columns of "temp2":

» new_mat = sort(temp2)

» P_mat = pascal(5) % Create Pascal's Matrix

% Rearrange Matrix - Pascal Triangle in lower triangle % Flip Matrix Vertically (UP-DOWN):

% - Could also use FLIPUD():

» temp1 = P_mat(5:-1:1, 1:5)

% Extract lower triangular matrix:

» temp2 = tril(temp1)

% Create New Matrix by sorting columns of "temp2":

» new_mat = sort(temp2)

Introduction to M

ATLAB

Matrices & Linear Algebra

Matrix Operators

Array operators

‘ comp. transpose .’ array transpose ^ power .^ array power * multiplication .* array mult. / division ./ array division \ left division

+ addition - subtraction

»help ops

»help matfun

Common Matrix Functions

inv() matrix inverse det() determinant rank() matrix rank

eig() eigenvectors & values svd() singular value dec. norm() matrix / vector norm

Introduction to M

ATLAB

Matrix Multiplication

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Inner dimensions must be equal

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Dimension of resulting matrix = outermost

dimensions of multiplied matrices

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Resulting elements = dot product of the rows of

the 1st matrix with the columns of the 2nd matrix

» a = [1 2 3 4; 5 6 7 8]; » b = ones(4,3); » c = a*b c = 10 10 10 26 26 26 » a = [1 2 3 4; 5 6 7 8]; » b = ones(4,3); » c = a*b c = 10 10 10 26 26 26 [2x4] [4x3] [2x4]*[4x3] [2x3]

a(2nd row).b(3rd column)

Introduction to M

ATLAB

Solving Simultaneous Equations

using “Left Division”

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If [A] is square matrix (m = n):

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For overdetermined system (m>n):

Warning if rank deficient (dependent columns) - solution not unique

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For undetermined system (m<n):

[A]{x} = {b}

{x} = ?

{x} = [A]

{b}

[A] = mxn {x} = nx1 {b} = mx1 » x = A\b; » x = A\b; » x = A\b; » x = A\b;

Introduction to M

ATLAB

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Solve this set of simultaneous equations:

Example: Solving Equations

» A = [-1 1 2; 3 -1 1;-1 3 4]; » b = [2;6;4]; » x = inv(A)*b x = 1.0000 -1.0000 2.0000 » x = A\b x = 1.0000 -1.0000 2.0000 » A = [-1 1 2; 3 -1 1;-1 3 4]; » b = [2;6;4]; » x = inv(A)*b x = 1.0000 -1.0000 2.0000 » x = A\b x = 1.0000 -1.0000 2.0000

-x

₁

+ x

₂

+ 2x

₃

= 2

3x

₁

- x

₂

+ x

₃

= 6

-x

₁

+ 3x

₂

+ 4x

₃

= 4

»solve_examp

Introduction to M

ATLAB

Array Multiplication

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Matrices must have the same dimensions

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Dimensions of resulting matrix = dimensions of

multiplied matrices

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Resulting elements = product of corresponding

elements from the original matrices

Same rules apply for other array operations

»array_mult

Introduction to M

ATLAB

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In most languages - use loops:

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In MATLAB - use Array Operations:

Example: Array Operations

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»array_examp

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0

»

0

Use TIC and TOC to

measure elapsed time

Vectorized code is

Introduction to M

ATLAB

Boolean Operations

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1 = TRUE

0 = FALSE

»bool_ops

Introduction to M

ATLAB

Exercise: Matrix Operations

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You are given measured data of current vs. applied DC

voltage for a “black box” circuit

Introduction to M

ATLAB

Solution: Matrix Operations

»mat_ops

(uses: >>mat_ops_setup.m & >>mat_ops.mat)

% Assign loaded data:

» A = voltage; b = current;

% Using left division

» x = A\b

% Using Pseudoinverse

» pseudo_inv_A = inv(A'*A)*A’; » x = pseudo_inv_A*b

% Repeat using linear data

» indices = find(voltage<8);

» A = voltage(indices); b = current(indices); ...

% Assign loaded data:

» A = voltage; b = current;

% Using left division

» x = A\b

% Using Pseudoinverse

» pseudo_inv_A = inv(A'*A)*A’; » x = pseudo_inv_A*b

% Repeat using linear data

» indices = find(voltage<8);

» A = voltage(indices); b = current(indices); ...

Initial Fit

Introduction to M

ATLAB

String Arrays

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Created using single quote delimiter (')

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Each character is a separate matrix element

(16 bits of memory per character)

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Indexing same as for numeric arrays

» str = 'Hi there,'

str =

Hi there,

» str2 = 'Isn''t MATLAB great?'

str2 =

Isn't MATLAB great?

» str = 'Hi there,'

str =

Hi there,

» str2 = 'Isn''t MATLAB great?'

str2 =

Isn't MATLAB great?

»string_array

1x9 vector

Introduction to M

ATLAB

» str2 = 'Isn''t MATLAB great?'; » new_str2=[new_str; str2]

new_str2 =

Hi there, Everyone! Isn't MATLAB great?

» str ='Hi there,'; » str1='Everyone!';

» new_str=[str, ' ', str1]

new_str =

Hi there, Everyone!

» str2 = 'Isn''t MATLAB great?'; » new_str2=[new_str; str2]

new_str2 =

Hi there, Everyone! Isn't MATLAB great?

1x19 vector

1x9 vectors

String Array Concatenation

»string_cat

Using [ ] operator:

Each row must be same length

Row separator:

semicolon (;)

Column separator:

space / comma (,)

For strings of different length:

• STRVCAT

• STR2MAT

new_str3 =

Hi there, Isn't MATLAB great?

» new_str3 = strvcat(str, str2)

new_str3 =

Hi there, Isn't MATLAB great?

2x19 matrix

2x19 matrix

(zero padded)

1x19 vectors

Introduction to M

ATLAB

Working with String Arrays

String Comparisons

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STRCMP - compare whole strings

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STRNCMP - compare first ‘N’ characters

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FINDSTR - finds substring within a larger string

Converting between numeric & string arrays:

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NUM2STR - convert from numeric to string array

Introduction to M

ATLAB

Exercise: String Arrays

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Create a 5x5 magic matrix ("M")

•

Use "M" to create a string array which will

display as follows:

Tab character

New line character

HINT:

Create rows

separately &

concatenate

Introduction to M

ATLAB

Solution: String Arrays

» M = magic(5)

» M_string = num2str(M)

» heading = ['5x5 Magic Matrix:', char(10)]

» row0 = [char(9), '''A'' ''B'' ''C'' ''D'' ''E'''] » row1 = ['1', char(9), M_string(1,:)]

» row2 = ['2', char(9), M_string(2,:)] » row3 = ['3', char(9), M_string(3,:)] » row4 = ['4', char(9), M_string(4,:)] » row5 = ['5', char(9), M_string(5,:)]

» string_array = strvcat(heading,row0,row1,row2,row3,row4,row5) » M = magic(5)

» M_string = num2str(M)

» heading = ['5x5 Magic Matrix:', char(10)]

» row0 = [char(9), '''A'' ''B'' ''C'' ''D'' ''E'''] » row1 = ['1', char(9), M_string(1,:)]

» row2 = ['2', char(9), M_string(2,:)] » row3 = ['3', char(9), M_string(3,:)] » row4 = ['4', char(9), M_string(4,:)] » row5 = ['5', char(9), M_string(5,:)]

» string_array = strvcat(heading,row0,row1,row2,row3,row4,row5)

Introduction to M

ATLAB

Multidimensional Arrays

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