Informatics1-2019/Lab08

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A lap korábbi változatát látod, amilyen Gaebor (vitalap | szerkesztései) 2019. november 18., 00:13-kor történt szerkesztése után volt.
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Tartalomjegyzék

MatLab

MatLab is used for various numerical calculations including:

  • solving linear (matrix) equations
  • solving differential-equations
  • control systems

It's name comes from Matrix Laboratory, Cleve Moler started to develop it in the 70's on the University of New Mexico. Since then it is owned and maintained by mathworks and it is proprietary.

There is a free alternative: Octave.

First steps

Getting the program

Calculator

MatLab can be used as an advanced calculator. Try the following (in MatLab):

2 + 3

press Enter:

> 2 + 3
ans =  5

Try these as well:

2 - 3
2 * 3
2 / 3
floor (2 / 3)
mod (2, 3)
2^3
sqrt (2)
log (2)
log (3)
log (8) / log (2)
exp (1)
pi
cos (pi / 2)
(180 / pi) * acos (0.5)

This is how you quit

exit

Data types

Every number is a floating point number, even if it happens to be an integer:

1000 / 9
ans = 111.11

We can however force it to use integers:

int32 (1000) / int32 (9)
ans = 111

In MatLab every number is real as long as it does not turn out to be complex:

sqrt (2)
sqrt (-2)

Number representation

  • double: double precision float, 64 bit (8 byte)
    • real: 8 byte
    • complex: 16 byte
  • single: single precision float, 32 bit (4 byte)
    • real: 4 byte
    • complex: 8 byte
  • int32: 32 bites two's complement integer (4 byte)
  • int8: 8 bites two's complement integer -128..127 (1 byte)
  • uint32: 32 bites unsigned integer (4 byte)
  • uint8: 8 bites unsigned integer: 0..255 (1 byte)

Size does matter:

log(single(1.0001))
log(double(1.0001))
int32(100+100)
int8(100+100)

Matrices

In MatLab every number is a matrix

  • numbers: 1x1
  • vectors:
    • row vector: 1xn
    • column vector: nx1
  • matrix: nxm

About.

Row vector:

[1 2 3 4]
[1, 2, 3, 4]

Column vector:

[1; 2; 3; 4]

This is not a column vector:

[[1], [2], [3], [4]]

Matrix:

[1 2; 3 4]
[1, 2; 3, 4]

Special matrices:

  • zeros: all 0
  • ones: all 1
  • eye: 1 in diagonal, 0 elsewhere
  • diag: square diagonal, with the speciafied diagonal
zeros (2, 3)
eye (2, 3)
ones (3, 1)
diag ([1, 2, 3, 4])

Also try:

size (5)
size ([1, 2, 3])
size ([1; 2; 3])

Ranges

Try the following:

1:10

We can specify a step as well:

1:0.1:2
1:2:10

Does not work for complex numbers, since there's no order on complex numbers!
The result will always be double but we can convert it:

int32 (1:0.5:10)

Decreasing range:

4:-1:1

Empty range:

4:1:1
4:1

Diagonal matrix:

> diag(1:4)
ans =
  1 0 0 0
  0 2 0 0
  0 0 3 0
  0 0 0 4

typeinfo

You can find out the type of an item with typeinfo.

typeinfo (1)
typeinfo (int32(1))
typeinfo (i)
typeinfo ([1 2 3 4])
typeinfo ([1 i -1 -i])
typeinfo (1:4)
typeinfo ([1:4])
typeinfo (1 >= 0)
typeinfo ("sometext")
typeinfo ([1 + 2i 1 - 1i])

Operations with matrices

Since every number is a 1x1 matrix, this can be applied to numbers as well.

Transpose

Use an accent ('):

> [1 2; 3 4]'
ans =
 1 3
 2 4
> _

Or

> (1:4)'
ans =
  1
  2
  3
  4

For complex matrices the accent means andjugate

in other words, the conjugate of the transpose:
> [1 2i; 3i 4]'
ans =
  1 - 0i   0 - 3i
  0 - 2i   4 - 0i

Conjugate: (2 + 1i)'

Addition

1 + (1:4)
eye (2, 2) + ones (2, 2)
[1; 2; 3; 4] - [4; 3; 2; 1]

Multiplication

Every multiplication is matrix multiplication:

> [1 2; 3 4]*[1 2; 3 4]
ans =
   7   10
  15   22

Power as well, so is the inverse:

[1 2; 3 4]^2
[1 2; 3 4]^-1

For multiplications the dimensions must be compatible:

ones (2, 3) * ones (3, 5)

Multiplying a row vector and a column vector results in a scalar, while the reverse is a dyiadic multiplication:

[1,2,3]*[1;2;3]
[1;2;3]*[1,2,3]

Every element or the whole

If power is a matrix operator, then what is this?

[1 2; 3 4]^0.5

And this?

sqrt([1 2; 3 4])

Some operators apply to every element of a matrix, while others apply to the matrix itself.

> (1:4)^2
error: for A^b, A must be a square matrix

This results in an error since multiplying a 1x4 matrix with itself is not possible. But:

> (1:4).^2
ans =
  1  4  9 16

Putting a dot in front of an operator makes it apply to every element of a matrix instead of apply to the matrix itself. For addition this does not matter.

> [1 2; 3 4] + [1 2; 3 4]
ans =
  2   4
  6   8
> [1 2; 3 4] .+ [1 2; 3 4]
ans =
  2   4
  6   8

But for multiplication it matters:

> [1 2; 3 4] * [1 2; 3 4]
ans =
   7   10
  15   22
> [1 2; 3 4] .* [1 2; 3 4]
ans =
   1    4
   9   16

Power:

> [1 2; 3 4]^-1
ans =
 -2.00000   1.00000
  1.50000  -0.50000
> [1 2; 3 4].^-1
ans =
  1.00000   0.50000
  0.33333   0.25000

Elementry functions apply to every element usually:

sin (0:0.1:2*pi)
exp ([0,-1;1,0])

While elementry operators apply to the matrix (*, ^, /, \)

Variables

We use variables to store values.

a = 2
b = 3
a + b

The variable ans always exists. It stores the last calculated value.
If a variable doesn't exist we can't use it in a calculation:

> a / q
error: `q' undefined

Using a semicolon (;) we can do silent calculations:

a = 2;
b = 3;
a + b

The whos commands shows the currently stored variables.

> whos
Variables in the current scope:
  Attr Name        Size                     Bytes  Class
  ==== ====        ====                     =====  =====
       a           1x1                          8  double
       ans         1x1                          8  double
       b           1x1                          8  double
Total is 3 elements using 24 bytes

Variables can be overwritten anytime:

> a = 2;
> a = [1 2; 3 4];
> whos
Variables in the current scope:
  Attr Name        Size                     Bytes  Class
  ==== ====        ====                     =====  =====
       a           2x2                         32  double

Indexes

Let M be a 3x3 matrix. The jth element of the ith row of this matrix is:

M = rand(3,3);
i = 1;
j = 3;
M(i,j)

Concatenating matrices:

[M M]
[M; M]

Partial series:

l = 0:0.1:1;
l (:)
l (1:11)
l (1:5)
l (5:end)
l (1:2:11)

Also:

l (1:2:11) = 0

Part of a matrix:

A = [1 2 3; 4 5 6; 7 8 9];
A (1:3, 1:2)
A (1:2, 1:3)

Skipping one line:

A ([1, 3], :)

Or selecting a specific part of a matrix:

S = ones (8, 8);
S (3:6, 3:6) = -1

Or a given index:

S = ones (8, 8);
S ([1 2 8], [2 6]) = 8

Flattening a matrix:

A = eye (3, 3);
A (:)

Vectorization

In MatLab we usually work with multiple numbers not just one. For example let us calculate the norm of every row in the matrix X:

X = rand (10, 3);
sqrt (sum (X.^2, 2))
ans =
  0.99105
  0.86977
  1.29362
  0.91697
  1.26149
  0.84024
  1.45410
  1.19791
  1.01153
  1.07420

Let us look at the functions:

  • X.^2: calculates the square of the items
  • sum(●, 2): sums up the values into a column vector (the meaning of the 2 is to sum up along the second dimension)
  • sqrt: square root of every item

Calculate the function value 2x^2-3x+1 where x is a row vector:

x=0:0.1:1;
fx=2.*x.^2 - 3.*x + 1

Functions

Try the following (multiple lines)!

> function fx = f (x)
>   fx = 1 / (x^2 + 1);
> endfunction

Test it:

> f(3)
ans = 0.10000

Defining functions:

function <<result>> = <<function name>> (<<variables>>)
  ...
endfunction

Any command can be inside the function, but at the end the <<result>> variable has to contain the result of the function. We usually use silent calculation (;) inside the function.

Another function:

function R = remove_last (x)
  R = x (1:end-1);
endfunction

Example:

> remove_last (1:5)
ans =
  1   2   3   4

Tasks

What is this?

What does the following do?

A=[1 2 3; 4 5 6; 7 8 9];
B=[9 8 7; 6 5 4; 3 2 1];
trace (A*B')
A(:)' * B(:)

What is trace(A*B')?

SOE

Calculate the solution of the following system of equations:

 x + 2y = 3
4x + 5y = 6

Solution 1:

A = [1 2; 4 5]
b = [3; 6]
x = A^-1 * b

Solution 2:

x = A \ b

More SOE

Solve the following systems of equations:

 x + 5y = 1
2x + 4y = 2
 x + 5y = 1
2x + 4y = 2
5x - 6y = -1
 x + 2y + 5z = 1
5x + 4y + 6z = 2

Large matrix

  • Create the following matrix using the least characters you can:
1 2 2 2 2 2 2 2 2 2
2 2 2 2 2 2 2 2 2 2
2 2 3 2 2 2 2 2 2 2
2 2 2 4 2 2 2 2 2 2
2 2 2 2 5 2 2 2 2 2
2 2 2 2 2 6 2 2 2 2
2 2 2 2 2 2 7 2 2 2
2 2 2 2 2 2 2 8 2 2
2 2 2 2 2 2 2 2 9 2
2 2 2 2 2 2 2 2 2 10
  • Same as before:
 0  1  0  0  0  0  0  0  0
-1  0  1  0  0  0  0  0  0
 0 -1  0  1  0  0  0  0  0
 0  0 -1  0  1  0  0  0  0
 0  0  0 -1  0  1  0  0  0
 0  0  0  0 -1  0  1  0  0
 0  0  0  0  0 -1  0  1  0
 0  0  0  0  0  0 -1  0  1
 0  0  0  0  0  0  0 -1  0
  • Chessboard
  1  -1   1  -1   1
 -1   1  -1   1  -1
  1  -1   1  -1   1
 -1   1  -1   1  -1
  1  -1   1  -1   1

Function on matrices

Write a function that applies the function 2sin2x + 1 to every element of a given matrix.

Part of a matrix

Write a function that separates a 5x5 matrix into a 2x5 matrix with the 2nd and 4th row, and a 3x5 with the 1st, 3rd, 5th.

Part of a matrix function

Combine the previous two so that the function applies the 2sin2x + 1 function to both new matrices.

Every second column

Write a function that outputs the matrix with every second column skipped in the input matrix. (Help, size can help with the dimension of the matrix.)

Applying a function to elements

Write a function that applies the function 2sin2x + 1 to every second column of a matrix.

Help yourself

Numerical differentiation

Differentiate the function f(x)=2x^2-3x+1 numerically! Given a row vector x, and fx with the corresponding function values.

x = 0:0.1:1
fx = 2.*x.^2 - 3.*x + 1

The numerical differentiation:

df = (fx(2:end) - fx(1:end-1)) ./ 0.1

With non-linear steps:

df = (fx(2:end) - fx(1:end-1)) ./ (x(2:end) - x(1:end-1))
Személyes eszközök