Informatics1-2017/Practice9
(→Indexes) |
(→typeinfo) |
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| (egy szerkesztő 7 közbeeső változata nincs mutatva) | |||
| 133. sor: | 133. sor: | ||
typeinfo ([1:4]) | typeinfo ([1:4]) | ||
typeinfo (1 >= 0) | typeinfo (1 >= 0) | ||
| − | typeinfo (" | + | typeinfo ("sometext") |
typeinfo ([1 + 2i 1 - 1i]) | typeinfo ([1 + 2i 1 - 1i]) | ||
| 298. sor: | 298. sor: | ||
A (:) | A (:) | ||
| − | == | + | == Vectorization == |
| − | + | In Octave we usually work with multiple numbers not just one. | |
| − | + | For example let us calculate the norm of every row in the matrix <tt>X</tt>: | |
X = rand (10, 3); | X = rand (10, 3); | ||
sqrt (sum (X.^2, 2)) | sqrt (sum (X.^2, 2)) | ||
| 314. sor: | 314. sor: | ||
1.01153 | 1.01153 | ||
1.07420 | 1.07420 | ||
| − | + | Let us look at the functions: | |
| − | * <tt>X.^2</tt>: | + | * <tt>X.^2</tt>: calculates the square of the items |
| − | * <tt>sum(●, 2)</tt>: | + | * <tt>sum(●, 2)</tt>: sums up the values into a column vector (the meaning of the 2 is to sum up along the second dimension) |
| − | * <tt>sqrt</tt>: | + | * <tt>sqrt</tt>: square root of every item |
| − | + | Calculate the function value <tt>2x^2-3x+1</tt> where <tt>x</tt> is a row vector: | |
x=0:0.1:1; | x=0:0.1:1; | ||
fx=2.*x.^2 - 3.*x + 1 | fx=2.*x.^2 - 3.*x + 1 | ||
| − | + | == Functions == | |
| − | + | Try the following (multiple lines)! | |
| − | == | + | |
| − | + | ||
> function fx = f (x) | > function fx = f (x) | ||
> fx = 1 / (x^2 + 1); | > fx = 1 / (x^2 + 1); | ||
> endfunction | > endfunction | ||
| − | + | Test it: | |
> f(3) | > f(3) | ||
ans = 0.10000 | ans = 0.10000 | ||
| − | + | Defining functions: | |
| − | function << | + | function <<result>> = <<function name>> (<<variables>>) |
... | ... | ||
endfunction | endfunction | ||
| − | + | Any command can be inside the function, but at the end the <tt><<result>></tt> variable has to contain the result of the function. We usually use silent calculation ('''<tt>;</tt>''') inside the function. | |
| − | + | Another function: | |
function R = remove_last (x) | function R = remove_last (x) | ||
R = x (1:end-1); | R = x (1:end-1); | ||
endfunction | endfunction | ||
| − | + | Example: | |
> remove_last (1:5) | > remove_last (1:5) | ||
ans = | ans = | ||
1 2 3 4 | 1 2 3 4 | ||
| − | == | + | == Tasks == |
| − | + | ||
| − | + | ||
| − | + | ||
| − | === | + | === What is this? === |
| − | + | What does the following do? | |
A=[1 2 3; 4 5 6; 7 8 9]; | A=[1 2 3; 4 5 6; 7 8 9]; | ||
B=[9 8 7; 6 5 4; 3 2 1]; | B=[9 8 7; 6 5 4; 3 2 1]; | ||
trace (A*B') | trace (A*B') | ||
A(:)' * B(:) | A(:)' * B(:) | ||
| − | + | What is <tt>trace(A*B')</tt>? | |
| − | === | + | === SOE === |
| − | + | Calculate the solution of the following system of equations: | |
x + 2y = 3 | x + 2y = 3 | ||
4x + 5y = 6 | 4x + 5y = 6 | ||
| − | 1 | + | Solution 1: |
A = [1 2; 4 5] | A = [1 2; 4 5] | ||
b = [3; 6] | b = [3; 6] | ||
x = A^-1 * b | x = A^-1 * b | ||
| − | 2 | + | Solution 2: |
x = A \ b | x = A \ b | ||
| − | === | + | === More SOE === |
| − | + | Solve the following systems of equations: | |
x + 5y = 1 | x + 5y = 1 | ||
| 386. sor: | 381. sor: | ||
5x + 4y + 6z = 2 | 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 | 1 2 2 2 2 2 2 2 2 2 | ||
2 2 2 2 2 2 2 2 2 2 | 2 2 2 2 2 2 2 2 2 2 | ||
| 400. sor: | 395. sor: | ||
2 2 2 2 2 2 2 2 2 10 | 2 2 2 2 2 2 2 2 2 10 | ||
| − | * | + | * Same as before: |
0 1 0 0 0 0 0 0 0 | 0 1 0 0 0 0 0 0 0 | ||
-1 0 1 0 0 0 0 0 0 | -1 0 1 0 0 0 0 0 0 | ||
| 411. sor: | 406. sor: | ||
0 0 0 0 0 0 0 -1 0 | 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 | ||
| 418. sor: | 413. sor: | ||
1 -1 1 -1 1 | 1 -1 1 -1 1 | ||
| − | === | + | === Function on matrices === |
| − | + | Write a function that applies the function <tt>2sin<sup>2</sup>x + 1</tt> 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 <tt>2sin<sup>2</sup>x + 1</tt> 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, <tt>size</tt> can help with the dimension of the matrix.) | |
| − | === | + | === Applying a function to elements === |
| − | + | Write a function that applies the function <tt>2sin<sup>2</sup>x + 1</tt> to every second column of a matrix. | |
| − | === | + | === Help yourself === |
| − | + | The [https://www.gnu.org/software/octave/doc/v4.0.3/ octave documentation], [http://stackoverflow.com/search?q=octave stackoverflow] or Google. | |
| − | + | ||
| − | === | + | === Numerical differentiation === |
| − | + | Differentiate the function <tt>f(x)=2x^2-3x+1</tt> numerically! Given a row vector <tt>x</tt>, and <tt>fx</tt> with the corresponding function values. | |
x = 0:0.1:1 | x = 0:0.1:1 | ||
fx = 2.*x.^2 - 3.*x + 1 | fx = 2.*x.^2 - 3.*x + 1 | ||
| − | + | The [https://en.wikipedia.org/wiki/Numerical_differentiation numerical differentiation]: | |
df = (fx(2:end) - fx(1:end-1)) ./ 0.1 | 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)) | df = (fx(2:end) - fx(1:end-1)) ./ (x(2:end) - x(1:end-1)) | ||
A lap jelenlegi, 2017. november 6., 15:50-kori változata
Tartalomjegyzék |
Octave
Octave is a numerical calculator, its the free (opensource) version of MatLab.
First steps
Getting the program
Use one of the following from home:
- install Octave, free for all platforms
- using putty log in to leibniz, start octave with the octave command
- run octave from the computer lab
Calculator
Octave can be used as an advanced calculator. Try the following (after entering octave):
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 Octave 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 octave every number is a matrix
- numbers: 1x1
- vectors:
- row vector: 1xn
- column vector: nx1
- matrix: nxm
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 Octave 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
The octave documentation, stackoverflow or Google.
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))
