course: astm22 – computational astrophysics alexey bobrick...

Short introduction to MATLAB

Course: ASTM22 – Computational Astrophysics

Teachers:Alexey Bobrick – [email protected] Hobbs – [email protected]

MATLAB introduction lecture notes by:Daniel Carrera (doktorand)

Alexey Bobrick Short MATLAB introduction slide 1 of 26

mailto:[email protected]

Introduction to MATLAB

● Why MATLAB?● Lightning tour of MATLAB

– Variables and arrays– Conditionals– Loops– Functions– Importing data– Plots– Videos

● Machine precision– How computers store numbers– Arithmetic error– Implications for your programs


Why MATLAB?

Array-based programming language

1. The basic object is the array (e.g. vector, matrix).

2. Perform operations on entire arrays (e.g. vector addition, matrix product, QR factorization, etc).

3. Compact, clear code.

for (i = 0; i < n; i++) { for (j = 0; j < m; j++) { C = A*B C[i,j] = 0; for (k = 0; k < q; k++) { C[i,j] += A[i,k]*B[k,j]; } }}


Why MATLAB?

Array-based programming language

1. The basic object is the array (e.g. vector, matrix).

2. Perform operations on entire arrays (e.g. vector addition, matrix product, QR factorization, etc).

3. Compact, clear code.

4. Good balance of performance and ease of use.

5. Array languages are a key tool in scientific computing.


How to get MATLAB

MATLAB download for Lund University students:

http://program.ddg.lth.se/

Popular alternatives to MATLAB

- Octave http://www.gnu.org/software/octave/

Extremely similar language to MATLAB. Get the latestversion to get a GUI.

- Python + SciPyhttp://www.enthought.com/products/epd.php

General purpose programming language witha module for scientific computing.


Lightning tour of MATLAB – Fancy calculator

>> 2 + 3 + 7 + 12 + 31 55

>> pi 3.1416

>> sqrt(2) >> help sqrt 1.4142

>> cos(2*pi) % Cosine uses radians. 1

>> (2+3i)^2 % Complex numbers. -5 + 12i

>> exp(123) 2.6195e+53

>> log(exp(123)) 123


Lightning tour of MATLAB – Variables and arrays

>> a = 7 7

>> b = 2*a 14

>> u = [6 5 4 3 2] 6 5 4 3 2

>> u + [1 1 1 1 1] 7 6 5 4 3

>> u * 2 12 10 8 6 4

>> sqrt(u) 2.4495 2.2361 2.0000 1.7321 1.4142



>> uu = 6 5 4 3 2

>> max(u)ans = 6

>> min(u)ans = 2

>> sum(u)ans = 20

>> mean(u)ans = 4

>> length(u)ans = 5



>> A = [ 1 2 3 ; 4 5 6 ] % 2x3 Matrix.A =

1 2 3 4 5 6

>> size(A) % Matrix dimensions.ans =

2 3

>> uu = 6 5 4 3 2

>> size(u) % MATLAB sees 'u' as a 1x5 matrix.ans =

1 5



>> u' % Matrix transpose. 6 5 4 3 2

>> u' * u % Matrix product. 36 30 24 18 12 30 25 20 15 10 24 20 16 12 8 18 15 12 9 6 12 10 8 6 4

>> u .* u 36 25 16 9 4

>> u' .* uerror



>> A = rand(3) % Random matrix

0.573291 0.663319 0.265920 0.531253 0.481396 0.361992 0.203692 0.745661 0.053230

>> b = [1;2;3] % Column vector 1 2 3

>> x = A\b % Solve the linear system Ax = b -13.6425 6.5480 16.8387



a = [4 5 6]b = [7 8 6]

if a == b

disp(“Drive a car”)

elseif any(a >= b)

disp(“Take the bus”)

else

disp(“Take the train”)

end

if ( condition ) ...elseif ( condition ) ...else



for i = start:end ...end

while ( condition ) ...end

r = [];for i = 1:30 r(i) = i*iend

y = 100;while (y > 10) y = sqrt(y)*3end


Lightning tour of MATLAB – Loops vs arrays

Rule #1: Aim to replace loops with array operations

Speed Loops are run by the MATLAB interpreter. Arrayops are done by the backend, written in C.

Clarity More often than not, your code will be moreconcise and clear.

Education MATLAB was designed around arrays. Learnanother way to approach programming.


Lightning tour of MATLAB – Functions

Each function is stored in a separate file, the file name must match the function name.

File name: function_name.m

File contents:

function [output vals] = function_name(input vals)

... instructions ...


Lightning tour of MATLAB – Functions

Example:

File name: lnorms.m

File contents:

%% Return the L1, L2 and L-infinity norms of a vector%

function [l1 l2 linf] = lnorms(vals)

l1 = sum( abs(vals) )l2 = sqrt( sum(vals .* vals) )linf = max( abs(vals) )


Lightning tour of MATLAB – Importing data


Suppose you have a data file called:

data_file.dat

If it stores arranged numeric data, it can be imported by:

>> M = dlmread(data_file.dat)

… and the data will be saved into matrix M

dlmread automatically infers the format of the data in the file

Lightning tour of MATLAB – Plots

Example:

Making a plot for arrays x,y: >> plot(x,y) %plot(x,y,'o') for different markers

Setting the options (after 'plot' command): xh=xlabel('x (m)'); yh=ylabel('internal energy (J/Kg)'); th=title('Internal energy vs distance'); %labels and title Axis([-0.4 0.4 1.8 2.6]) %plotrange: xmin,xmax,ymin,ymax set(gca,... 'linewidth',2,... %thickness of the line 'xcolor',[0,0,0],... %plot color 'fontsize',18,... 'fontname','arial'); %main plot style set([xh,yh,th],... 'fontweight','bold',... 'fontsize',18,... 'color',[0,0,0]); %plot label and title style


Lightning tour of MATLAB – Plots

Example:

To make M-row N-column multi-panel plot:

>> subplot(M,N,plot number)>> … commands for making a plot of a given number>> subplot(M,N,next plot number)>> … commands for the next plot

>> pause(0.1) %Add if the plots are made in a loop


Lightning tour of MATLAB – Videos


Example:

To make a video consisting of several plots:

vidObj = VideoWriter('peaks.avi');open(vidObj);Z = peaks; surf(Z); axis tightset(gca,'nextplot','replacechildren');for k = 1:20 surf(sin(2*pi*k/20)*Z,Z) currFrame = getframe; writeVideo(vidObj,currFrame);endclose(vidObj);

Lightning tour of MATLAB – Machine precision

Single precision (32 bit)

1.b1b

2b

3b

4b

5b

6b

7b

8b

9…b

23

(or about 7 – 8 decimal digits of precision)0 = positive1 = negative e – 127

(goes from -127 to +128)

Largest value: ~ 2128 ≈ 3 x 1038




Double precision (64 bit)

1.b1b

2b

3b

4b

5b

6b

7b

8b

9…b

52

(or about ~17 decimal digits of precision)0 = positive1 = negative e – 1023

(goes from -1023 to +1024)

Largest value: ~ 21024 ≈ 10309


- Adding or subtracting very different numbers:

- Subtracting very similar numbers.

>> a = 1e18a = 1.0000e+18

>> b = 327b = 327

>> c = a + bc = 1.0000e+18

>> c - aans = 384

!!!



Some fractions cannot be represented in binary.

In decimal: 1/3 = 0.33333333333...1/10 = 0.1

In trinary: 1/3 = 0.1 In binary: 1/2 = 0.1

1/5 = 0.0011001100110011...1/10 = 0.0001100110011001...

Numbers like 0.1 and 0.2 cannot be represented exactly in binary.




MATLAB tries to do some magic to detect and correct these problems:

>> a = 0.2a = 0.20000

>> a = a + aa = 0.40000

>> a = a + a + 0.2a = 1

But that also means that you may have a nasty surprise when the magic fails...

Lightning tour of MATLAB – Defensive programming

Make your code resilient to machine error.

% Suppose 'a' and 'b' are not integers.

if (a == b)

Option 1: if (a >= b)

Option 2: epsilon = 1e-5...if ( abs(a - b) < epsilon )

You have to make a judgement and decide which (if any) of these alternatives is appropriate.


course: astm22 – computational astrophysics alexey bobrick...

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