TEKNE - TECHNE: August 2018

Monday 27 August 2018

Neural Networks Simplified - 4

BASİTLEŞTİRİLMİŞ YAPAY SİNİR AĞLARI - 4
Bu eğitim metni Hinton'un iki kursunda kullanılan yapılar
için yapılmış basit örneklere dayanır. Bu kurslar sırasında
verilen örnekler internet üzerinde yaygın şekilde bulunabilir.

Bu kısımda Hinton'un kullanmış olduğu grafik fonksiyonları
örneklerle anlatılacak.

NEURAL NETWORKS SIMPLIFIED - 4
This is a tutorial based on simple examples made for the
structures used in Hinton's two courses. The exercises
given during these courses are widely available on the internet.

The graphics functions that Hinton has used will be studied
in this section.

Herhangi bir soru varsa beni aramakta tereddüt etmeyiniz.
Please do not hesitate to contact me if any questions.

Ali R+ SARAL
arsaral((at))yaho(o).com

GRAPHICS FUNCTIONS
*******************************************

x = -10:0.1:10;
plot (x, sin (x))
octave:3> x = -10:0.1:10; plot (x, sin (x),'+');
octave:4> x = -10:0.1:10; plot (x, sin (x),'-');
octave:5> x = -10:0.1:10; plot (x, sin (x),2)
octave:7> x = -10:0.1:10; plot (x, sin (x),'b*')
octave:8> x = -10:0.1:10; plot (x, sin (x),'ro')

scatter
x = randn (100, 1);
y = randn (100, 1);
scatter (x, y, [], sqrt(x.^2 + y.^2));

x =

0 1 2

octave:81> y
y =

0 1 2

octave:82> z
z =

0 0 0
0 1 2
0 2 4

x = 0:2; y = x;
z = x’ * y;
contour (x, y, z, 2:3)

octave:9> x=0:4;y=x;
octave:10> z=x'*y;
octave:11> contour(x,y,z,1:5)
octave:12> x
x =

0 1 2 3 4

octave:13> y
y =

0 1 2 3 4

octave:14> z
z =

0 0 0 0 0
0 1 2 3 4
0 2 4 6 8
0 3 6 9 12
0 4 8 12 16

x=linspace(-2*pi,2*pi);
y=linspace(0,4*pi);
[X,Y] = meshgrid(x,y);
Z=sin(X)+cos(Y);
figure
contour(X,Y,Z)

fplot ("cos", [0, 2*pi]);
fplot ("[cos(x), sin(x)]", [0, 2*pi]);

octave:2> tx=ty=linspace(-8, 8, 4)
tx =

-8.0000 -2.6667 2.6667 8.0000

octave:3> [xx, yy] = meshgrid(tx, ty)
xx =

-8.0000 -2.6667 2.6667 8.0000
-8.0000 -2.6667 2.6667 8.0000
-8.0000 -2.6667 2.6667 8.0000
-8.0000 -2.6667 2.6667 8.0000

yy =

-8.0000 -8.0000 -8.0000 -8.0000
-2.6667 -2.6667 -2.6667 -2.6667
2.6667 2.6667 2.6667 2.6667
8.0000 8.0000 8.0000 8.0000

tx = ty = linspace (-8, 8, 41)’;
[xx, yy] = meshgrid (tx, ty);
r = sqrt (xx .^ 2 + yy .^ 2) + eps;
tz = sin (r) ./ r;
mesh (tx, ty, tz);

[x, y, z] = peaks (20);
scatter3 (x(:), y(:), z(:), [], z(:));

subplot (2, 1, 1);
fplot (@sin, [-10, 10]);
subplot (2, 1, 2);
fplot (@cos, [-10, 10]);

imagesc(A);

title('Original');

imshow(to_show, [-extreme, extreme]);

title('hidden units of the RBM');

ADVANCED FUNCTIONS
************************************************
octave:2> a=[1,2,3;4,5;6]
error: vertical dimensions mismatch (1x3 vs 1x2)
octave:2> a=1:6
a =

1 2 3 4 5 6

octave:3> reshape(a,2,3)
ans =

1 3 5
2 4 6

octave:4> reshape(a,3,2)
ans =

1 4
2 5
3 6

octave:7> b=repmat(c,2)
b =

1 2 1 2
3 4 3 4
1 2 1 2
3 4 3 4

octave:8> b=repmat(c,2,3)
b =

1 2 1 2 1 2
3 4 3 4 3 4
1 2 1 2 1 2
3 4 3 4 3 4

FUNCTION CALLS
*******************************************************
[max_value, p] = max(h_of_x, [], 2);

octave:3> [max_value, p] = max(a,[],1)
max_value =

1 2 3 4 5 6 7 8 9 10

p =

1 1 1 1 1 1 1 1 1 1

octave:4> a=[1,2,3;4,5,6;7,9,2]
a =

1 2 3
4 5 6
7 9 2

octave:5> [max_value, p] = max(a,[],1)
max_value =

7 9 6

p =

3 3 2

octave:6> [max_value, p] = max(a,[],2)
max_value =

3
6
9

p =

3
3
2

octave:28> s="test";
octave:29> function testFuncWithPARM(s)
> fprintf(s)
> end

octave:31> testFuncWithPARM("aaaaaa\n")
aaaaaa

octave:32> i = 0;
octave:33> function testFuncWithIntPARM(i)
> fprintf('i =%d',i)
> end

octave:34> testFuncWithIntPARM(333);
i =333

octave:35> function [out1, out2] = testFuncWithMultiPARM()
> out1=1
> out2=2
> end

octave:36> testFuncWithMultiPARM()
out1 = 1
out2 = 2
ans = 1

extFunc.m external file
function [out3, out4] = testExtFunc()
out3=99
out4=100
end

octave:42> testExtFunc()
out3 = 99
out4 = 100
ans = 99

Friday 24 August 2018

Neural Networks Simplified - 3

BASİTLEŞTİRİLMİŞ YAPAY SİNİR AĞLARI - 3
Bu eğitim metni Hinton'un iki kursunda kullanılan yapılar
için yapılmış basit örneklere dayanır. Bu kurslar sırasında
verilen örnekler internet üzerinde yaygın şekilde bulunabilir.

Bu kısım ile hazır fonksiyonlar bitmiş oluyor. Bundan sonraki kısımlar
kullanıcı tarafından tanımlanan fonksiyonları ele alacak.
Daha sonra ise tek tek Hinton'un örnekleri basite indirgenerek
incelenecek.

NEURAL NETWORKS SIMPLIFIED - 3
This is a tutorial based on simple examples made for the
structures used in Hinton's two courses. The exercises
given during these courses are widely available on the internet.

Builtin functions is ending with this section. The next section will
study the user defined functions. After that Hinton's examples will
be studied with a very simplifying approach.

Herhangi bir soru varsa beni aramakta tereddüt etmeyiniz.
Please do not hesitate to contact me if any questions.

Ali R+ SARAL
arsaral((at))yaho(o).com
BASIC BUILTIN FUNCTIONS
*******************************************
octave:11> any([1,0,0])
ans = 1

octave:15> eye(2,4)
ans =

Diagonal Matrix

1 0 0 0
0 1 0 0
octave:12> any (eye (2, 4))
ans =

1 1 0 0

octave:13> ~any([1,0,0])
ans = 0

octave:14> ~any (eye (2, 4))
ans =

0 0 1 1

octave:2> isnan ([13, Inf, NA, NaN])
ans =

0 0 1 1

octave:3> isinf ([13, Inf, NA, NaN])
ans =

0 1 0 0

octave:6> eval ('error ("This is a bad example");', 'printf ("This error occurred:\n%s\n", lasterr ());');
This error occurred:
This is a bad example

octave:7> eval('error ("HATAAAAAAAAAAAA");')
error: HATAAAAAAAAAAAA

octave:7> eval('error ("ERRORRRRRRRRRRRRRRR");','fprintf("HATAAAAAAA");')
HATAAAAAAA

octave:11> any([1,0,0])
ans = 1

octave:15> eye(2,4)
ans =

Diagonal Matrix

1 0 0 0
0 1 0 0
octave:12> any (eye (2, 4))
ans =

1 1 0 0

octave:13> ~any([1,0,0])
ans = 0

octave:14> ~any (eye (2, 4))
ans =

0 0 1 1

ctave:19> resets dev environment -->clc;clear;close all

octave:19> checks environment values
octave:19> if ~exist('example_width', 'var') || isempty(example_width)
> end

octave:11> a=1:10
a =

1 2 3 4 5 6 7 8 9 10

octave:12> avg(a)
ans = 5.5000
octave:13> sum(a)
ans = 55
octave:14> avg(a) == sum(a)/10
ans = 1
octave:15> sum(a)./10
ans = 5.5000

INPUT OUTPUT FUNCTIONS
*********************************************
octave:16> pause

octave:17> fprintf("Press any key to continue");pause;
Press any key to continue

octave:18> % this is a line comment

octave:18> a=1; %This is a line comment

octave:19> %{
> This is a block comment
> 5}
> %}

fprintf('%s is a name, %d is a number\n',"Ali",58);
Ali is a name, 58 is a number

octave:22> a
a = 1
octave:23> a=1:5
a =

1 2 3 4 5

octave:24> b=6:10
b =

6 7 8 9 10

octave:25> disp([a b]);
1 2 3 4 5 6 7 8 9 10

octave:26> a=1:20
a =

Columns 1 through 15:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

octave:27> a=[a; a]
a =

Columns 1 through 15:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Columns 16 through 20:

16 17 18 19 20
16 17 18 19 20

octave:30> a=[a; a]
octave:30> a=[a; a]
'less' is not recognized as an internal or external command,
operable program or batch file.

octave:31> Above message is not valid for the diary recording

octave:31> more off
octave:32> a
a =

Columns 1 through 15:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Columns 16 through 20:

16 17 18 19 20
16 17 18 19 20
16 17 18 19 20
16 17 18 19 20
16 17 18 19 20
16 17 18 19 20
16 17 18 19 20
16 17 18 19 20

octave:33> diary off

FILE I/O
*******************************************
octave:22> S=load("dataset1.mat");
octave:23> fieldnames(S)
ans =
{
[1,1] = neg_examples_nobias
[2,1] = pos_examples_nobias
[3,1] = w_init
[4,1] = w_gen_feas
}

size(S.neg_examples_nobias)
size(S.pos_examples_nobias)

load("dataset1.mat");
size(neg_examples_nobias)
size(pos_examples_nobias)

data = csvread('ex1data2.txt');

% Load File

fid = fopen(filename);

if fid

file_contents = fscanf(fid, '%c', inf);

fclose(fid);

else

file_contents = '';

fprintf('Unable to open %s\n', filename);

end

movieList = cell(n, 1);

for i = 1:n

% Read line
line = fgets(fid);

% Word Index (can ignore since it will be = i)

[idx, movieName] = strtok(line, ' ');
% Actual Word

movieList{i} = strtrim(movieName);

end

fclose(fid);

STRING FUNCTIONS
**********************************************************
octave:4> strtok("aaa bbbb",' ')
ans = aaa

octave:5> strtrim(" aaa bbb ")
ans = aaa bbb
octave:6> length("12345")
ans = 5

octave:7> strcmp("aaa","aaa")
ans = 1

octave:9> strfind("dfbsdfbadddd","a")
ans = 8

octave:16> email_contents ="[ARS]"
email_contents = [ARS]
octave:17> email_contents = regexprep(email_contents, '<[^<>]+>', ' ');
octave:18> email_contents
email_contents = [ARS]
octave:19> diary off

Thursday 23 August 2018

NEURAL NETWORKS SIMPLIFIED - 2

BASİTLEŞTİRİLMİŞ YAPAY SİNİR AĞLARI - 2
Bu eğitim metni Hinton'un iki kursunda kullanılan yapılar
için yapılmış basit örneklere dayanır. Bu kurslar sırasında
verilen örnekler internet üzerinde yaygın şekilde bulunabilir.

Bu kısım daha ileri hazır fonksiyonları içerir.

NEURAL NETWORKS SIMPLIFIED - 2
This is a tutorial based on simple examples made for the
structures used in Hinton's two courses. The exercises
given during these courses are widely available on the internet.

This section includes more advanced built-in functions.

Herhangi bir soru varsa beni aramakta tereddüt etmeyiniz.
Please do not hesitate to contact me if any questions.

Ali R+ SARAL
arsaral((at))yaho(o).com

MATRIX UTILITY FUNCTIONS
*******************************************************

octave:2> a=[1,2,3;4,5,6]
a =

1 2 3
4 5 6

octave:3> log(a)
ans =

0.00000 0.69315 1.09861
1.38629 1.60944 1.79176

octave:4> a=[-1,-2,3;4,-5,-6]
a =

-1 -2 3
4 -5 -6

octave:5> abs(a)
ans =

1 2 3
4 5 6

octave:6> exp(a)
ans =

3.6788e-001 1.3534e-001 2.0086e+001
5.4598e+001 6.7379e-003 2.4788e-003

octave:7> sum(a)
ans =

3 -7 -3

octave:8> a
a =

-1 -2 3
4 -5 -6

octave:9> sumsq(a)
ans =

17 29 45

octave:10> a=[4,9]
a =

4 9

octave:11> sqrt(a)
ans =

2 3

octave:12> mod(a,2)
ans =

0 1

octave:13> mod(3,2)
ans = 1

octave:14> floor ([-2.7, 2.7])
ans =

-3 2

octave:15> floor(3.4)
ans = 3
octave:16> ceil(3.4)
ans = 4
octave:17> ceil([-2.7,2.7])
ans =

-2 3

octave:18> realmax()
ans = 1.7977e+308
octave:19> a=[1,2,3;4,5,6;7,8,9]
a =

1 2 3
4 5 6
7 8 9

octave:20> bsxfun(@minus, a,2)
ans =

-1 0 1
2 3 4
5 6 7

octave:21> a=[3,1,5;3,4,2;7,3,5]
a =

3 1 5
3 4 2
7 3 5

octave:22> sort(a)
ans =

3 1 2
3 3 5
7 4 5

octave:23> a
a =

3 1 5
3 4 2
7 3 5

octave:24> sort ([1, 2; 2, 3; 3, 1])
ans =

1 1
2 2
3 3

octave:25> [s, i] = sort ([1, 2; 2, 3; 3, 1])
s =

1 1
2 2
3 3

i =

1 3
2 1
3 2

octave:26> [s, i] = sort ([1, 2; 2, 3; 3, 1], 'descend');
octave:27> s
s =

3 3
2 2
1 1

octave:28> i
i =

3 2
2 1
1 3

octave:29> a
a =

3 1 5
3 4 2
7 3 5

octave:30> mean(a)
ans =

4.3333 2.6667 4.0000

octave:31> find(a==2)
ans = 8
octave:32> a(8)
ans = 2
octave:33> a=[1,1;1,1]
a =

1 1
1 1

octave:34> inv(a)
warning: inverse: matrix singular to machine precision, rcond = 0
ans =

Inf Inf
Inf Inf

octave:35> pinv(a)
ans =

0.25000 0.25000
0.25000 0.25000

octave:36> a=[1,2,3;4,5,6]
a =

1 2 3
4 5 6

octave:37> pinv(a)
ans =

-0.94444 0.44444
-0.11111 0.11111
0.72222 -0.22222

octave:38> a=[1,2,3;4,5,6;7,8,9]
a =

1 2 3
4 5 6
7 8 9

octave:39> X=[ones(3,1) a]
X =

1 1 2 3
1 4 5 6
1 7 8 9

octave:40> b=[1;2;3]
b =

1
2
3

octave:41> a(2,:) = b
a =

1 2 3
1 2 3
7 8 9

octave:42> a
a =

1 2 3
1 2 3
7 8 9

octave:43> a(2,:) = b'
a =

1 2 3
1 2 3
7 8 9

octave:44> Y=[a(:) ; b(:)]
Y =

1
1
7
2
2
8
3
3
9
1
2
3

octave:45> find(a>3)
ans =

3
6
9

octave:46> a
a =

1 2 3
1 2 3
7 8 9

octave:47> ndx=find(a>3)
ndx =

3
6
9

octave:48> for i=1:length(ndx)
> a(ndx(i)) = 1;
> end
octave:49> a
a =

1 2 3
1 2 3
1 1 1

octave:50>
octave:50> a=[1.2,3.4,5.0]
a =

1.2000 3.4000 5.0000

octave:51> round(a)
ans =

1 3 5

octave:52> randperm(3,4)
ans =

2 3 1
1 2 3
1 3 2
2 1 3

octave:53> randperm(3)
ans =

2 1 3

octave:54> a=cell(2,4)
a =
{
[1,1] = [](0x0)
[2,1] = [](0x0)
[1,2] = [](0x0)
[2,2] = [](0x0)
[1,3] = [](0x0)
[2,3] = [](0x0)
[1,4] = [](0x0)
[2,4] = [](0x0)
}
octave:55> a(1,1)=5
a =
{
[1,1] = 5
[2,1] = [](0x0)
[1,2] = [](0x0)
[2,2] = [](0x0)
[1,3] = [](0x0)
[2,3] = [](0x0)
[1,4] = [](0x0)
[2,4] = [](0x0)
}

octave:59> a=["a","b"]
a = ab
octave:60> a(1,1)
ans = a
octave:61> a(1,2)
ans = b
octave:62> a
a = ab
octave:63> a=cell(2)
a =
{
[1,1] = [](0x0)
[2,1] = [](0x0)
[1,2] = [](0x0)
[2,2] = [](0x0)
}
octave:64> a(1,1)="a"
a =
{
[1,1] = a
[2,1] = [](0x0)
[1,2] = [](0x0)
[2,2] = [](0x0)
}
octave:65> a(2,1)="b"
a =
{
[1,1] = a
[2,1] = b
[1,2] = [](0x0)
[2,2] = [](0x0)
}

diary off

Tuesday 21 August 2018

Neural Networks Simplified - 1

BASİTLEŞTİRİLMİŞ YAPAY SİNİR AĞLARI - 1
Bu eğitim metni Hinton'un iki kursunda kullanılan yapılar
için yapılmış basit örneklere dayanır. Bu kurslar sırasında
verilen örnekler internet üzerinde yaygın şekilde bulunabilir.

NEURAL NETWORKS SIMPLIFIED - 1
This is a tutorial based on simple examples made for the
structures used in Hinton's two courses. The exercises
given during these courses are widely available on the internet.

Herhangi bir soru varsa beni aramakta tereddüt etmeyiniz.
Please do not hesitate to contact me if any questions.

Ali R+ SARAL
arsaral((at))yaho(o).com

MATRIX DEFINITIONS
***************************************

a is a 2 x 3 matrix. a has 2 rows and 3 columns.
a 2 x 3 bir matristir. a'nın iki satırı ve 3 sütunu vardır.

octave:65> a=[1,2,3;4,5,6]
a =

1 2 3
4 5 6

octave:66> size(a)
ans =

2 3
An other matrix definition statement is: a(beg : end)
Bir başka matris tanımlama komutu: a(baş : son)
octave:73> a=(-1:3)
a =

-1 0 1 2 3

An other matrix definition statement is: a(beg :increment : end)
Bir başka matris tanımlama komutu: a(baş : arttırım adımı : son)
octave:75> a=(-1:0.5:2)
a =

-1.0000 -0.5000 0.0000 0.5000 1.0000 1.5000 2.0000

An other matrix definition statement is: linspace(beg : step count : end)
Bir başka matris tanımlama komutu: linspace(baş : adım sayısı : son)
octave:77> a=linspace(1,0.5,3)
a =

1.00000 0.75000 0.50000

octave:78> a=linspace(1,3,3)
a =

1 2 3

octave:79> a=linspace(1,3,4)
a =

1.0000 1.6667 2.3333 3.0000

MATRIX ELEMENT ADDRESSING
MATRİS ELEMAN ADRESLEME
***************************************

octave:84> a=[1,2,3;4,5,6]
a =

1 2 3
4 5 6

octave:85> a(1)
ans = 1
octave:86> a(1:4)
ans =

1 4 2 5

octave:87> a(1:6)
ans =

1 4 2 5 3 6

octave:88> a(1:2,1)
ans =

1
4

octave:89> a(1:2,3)
ans =

3
6

octave:90> a(1:2,[1 3])
ans =

1 3
4 6

octave:91> a(1:2,1:3)
ans =

1 2 3
4 5 6

octave:92> a(1:2,:)
ans =

1 2 3
4 5 6

octave:93> a(:,1:2)
ans =

1 2
4 5

octave:94> a(1,1:3)
ans =

1 2 3

octave:95> a(:,:)
ans =

1 2 3
4 5 6

octave:104> a=[1,2,3;4,5,6;7,8,9]
a =

1 2 3
4 5 6
7 8 9

octave:106> b=[1,2]
b =

1 2

octave:107> a(b,:)
ans =

1 2 3
4 5 6

octave:108> a(1,:)
ans =

1 2 3

BASIC MATRIX FUNCTIONS
***************************************

octave:2> ndims([1,2;3,4])
ans = 2

octave:10> ones(2,3)
ans =

1 1 1
1 1 1

octave:11> zeros(1,2)
ans =

0 0

octave:12> eye(1,3)
ans =

Diagonal Matrix

1 0 0

octave:13> eye(3)
ans =

Diagonal Matrix

1 0 0
0 1 0
0 0 1

octave:14> a = 13; a(ones (1, 4))
ans =

13 13 13 13

octave:17> rand(3)
ans =

0.61503 0.73559 0.16378
0.11622 0.89969 0.96928
0.16057 0.14347 0.84992

octave:14> a = 13;
octave:18> size(a)
ans =

1 1

octave:19> a=[1,2;3,4;5,6]
a =

1 2
3 4
5 6

octave:20> size(a)
ans =

3 2

octave:21> b=1
b = 1
octave:22> size(b)
ans =

1 1

octave:2> zeros(1)
ans = 0
octave:3> zeros(2,2)
ans =

0 0
0 0

octave:4> zeros(2,3)
ans =

0 0 0
0 0 0

octave:5> a=[1,2;3,4;5,6]
a =

1 2
3 4
5 6

octave:6> zeros(size(a))
ans =

0 0
0 0
0 0

octave:7> size(a)
ans =

3 2

octave:8> eye(0)
ans = [](0x0)

octave:9> eye(1)
ans = 1

octave:10> eye(2)
ans =

Diagonal Matrix

1 0
0 1

octave:11> eye(2,3)
ans =

Diagonal Matrix

1 0 0
0 1 0

octave:12> ones(0)
ans = [](0x0)

octave:13> ones(1)
ans = 1

octave:14> ones(2)
ans =

1 1
1 1

octave:15> ones(2,3)
ans =

1 1 1
1 1 1

octave:16> rand(0)
ans = [](0x0)

octave:17> rand(1)
ans = 0.76538

octave:18> rand(1)
ans = 0.74158

octave:19> rand(2)
ans =

0.62458 0.30045
0.48308 0.36512

octave:62> rand(2)
ans =

0.148336 0.481660
0.082268 0.182176

octave:20> rand(2,3)
ans =

0.056270 0.180314 0.794059
0.905070 0.366595 0.562126

octave:21> a=[1,2;3,4]
a =

1 2
3 4

octave:22> size(a)
ans =

2 2

octave:23> b=[1,2,3;4,5,6]
b =

1 2 3
4 5 6

octave:24> size(b)
ans =

2 3

octave:25> length(b)
ans = 3

octave:26> length(a)
ans = 2

octave:27> c=[1,2,3,4,5]
c =

1 2 3 4 5

octave:28> length(c)
ans = 5

octave:29> max(c)
ans = 5

octave:30> max(b)
ans =

4 5 6

octave:31> max(a)
ans =

3 4

octave:32> min(c)
ans = 1

octave:33> min(b)
ans =

1 2 3

octave:34> min(a)
ans =

1 2

octave:35> d=[2,3,1,5,6,4]
d =

2 3 1 5 6 4

octave:36> min(d)
ans = 1

octave:37> max(d)
ans = 6

octave:38> d=[2,3,1,5,6,4;1,2,3,4,5,6]
d =

2 3 1 5 6 4
1 2 3 4 5 6

octave:39> min(d)
ans =

1 2 1 4 5 4

octave:40> max(d)
ans =

2 3 3 5 6 6

octave:41> numel(d)
ans = 12

octave:42> numel(c)
ans = 5

octave:43> numel(b)
ans = 6

octave:44> numel(a)
ans = 4

norm (A, p, opt)
Compute the p-norm of the matrix A.
If the second argument is missing, p = 2 is assumed.
If A is a matrix (or sparse matrix):
p = 1 1-norm, the largest column sum of the absolute values of A.
p = 2 Largest singular value of A.

octave:45> norm(a)
ans = 5.4650 <===========

octave:46> a
a =

1 2
3 4

octave:49> [U,S,V]=svd(a)
Compute the singular value decomposition of A
A'nın tekil değer ayrıştırımını hesaplayınız.

U =

-0.40455 -0.91451
-0.91451 0.40455

S =

Diagonal Matrix

5.46499 0 <===========
0 0.36597

V =

-0.57605 0.81742
-0.81742 -0.57605

octave:50> b
b =

1 2 3
4 5 6

octave:51> [U,S,V]=svd(b)
U =

-0.38632 -0.92237
-0.92237 0.38632

S =

Diagonal Matrix

9.50803 0 0 <===========
0 0.77287 0

V =

-0.42867 0.80596 0.40825
-0.56631 0.11238 -0.81650
-0.70395 -0.58120 0.40825

octave:52> norm(b)
ans = 9.5080 <===========

octave:53> c
c =

1 2 3 4 5

octave:54> [U,S,V]=svd(c)
U = 1
S =

Diagonal Matrix

7.4162 0 0 0 0 <===========

V =

0.134840 -0.269680 -0.404520 -0.539360 -0.674200
0.269680 0.935914 -0.096129 -0.128172 -0.160215
0.404520 -0.096129 0.855807 -0.192258 -0.240322
0.539360 -0.128172 -0.192258 0.743656 -0.320430
0.674200 -0.160215 -0.240322 -0.320430 0.599463

octave:55> norm(c)
ans = 7.4162 <===========

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