Patozone

French

 

logiciel = software

les boucles = loops

 

Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point

 

Strain rate tensor

In continuum mechanics, the strain rate tensor is a physical quantity that describes the rate of change of the deformation of a material in the neighborhood of a certain point, at a certain moment of time. It can be defined as the derivative of the strain tensor with respect to time, or as the symmetric component of the gradient (derivative with respect to position) of the flow velocity.

 

 

 

 

 

 

Introduction

 

Intro to matlab (ml) commands:

demo - gives ml examples of features

- help - explains any function; help help explains how help works

- lookfor searches help

- doc is documentation page

- quit or exit to exit ml

 

Files saved under the current folder section

- semicolon at end of statement suppresses the output

 

namelengthmax gives the max variable length

- who shows the defined variables

- whos shows the vars and other info about them

- clear clears all vars

- clear varname clears a specific var

 

vars stored as types

- groups of types are called classes

- floats are numbers c decimal places, and can be single or double (precision), where double can store larger numbers than single

- integers can be type: int8, int32 or int 64, which vary on the amt of info each int var stores (int8 uses 8 bits)

- one bit is used for the sign (pos or neg)

-- unsigned int types (uint8, uint16, uint32, uint64) does not store a sign and can only be pos

- numbers stores as type double by default

 

- the range of a type shows the smallest and largest numbers that can be stored (uint8 can store 0-255 bc has 2^8 or 256 integers

- to find the range of a type, use: intmin() or intmax()

 

- type char stores characters or strings

- logical stores T/F

- type casting is converting from one type to another, ie:

>> van = 3 + 3

>> vani = int8(van)

 

- class() shows the var type

- numbers displayed c 4 decimal pts by default, which can be made longer c long (15 decimals) or short (the default of 4); using format: >> format long

- format can also change spacing from loose (default) or compact: >> format compact

- long expressions carried onto next line by the ellipsis or continuation operator, which is 3 or more dots: ...

 

Unary operators work a single val or operand and binary operators which operate 2+ vals / operands

- ie., "-" is unary for negation, binary for subtraction

- e used for exponential notation: >>2e4 ans = 20000

(same as >>2 * 10^4)

- ml uses standard order of operations

- elfun is another way to show help topics

- rem and mod return remainder from division

-- ie >>rem(13.5) ans = 3

- sign returns 1 if ans pos, 0 if neg

- fix rounds elements of var to nearest int towards 0

- floor rounds towards minus infinity

- ceil rounds towards plus infinity

- round goes to nearest integer

- sqrt for the squareroot, nthroot to find the nth root

- sin is sine function in radians, arcsine is asin, sinh is the hyperbolic sine, and sind is degrees not radians

- i or j is the sqrt(-1)

- inf is infinity; NaN is not a number, like 0/0

- e or e^1 is equivalent to exp(1)

 

Random Numbers

- seed is predetermined val or from the comps clock that determines random numbers

- rand generates uniformly distributed random real numbers

- rng used to set the initial seed, ie >>rng('shuffle')

- rng('default') gets back to original seed ( 0 )

- >>rand*10 generates random int in interval 0-10

- to make rand int from 0-10: >> round(rand*10)

- make a rand int in range 20-35:

>> low = 20

>> high = 35

>> round(rand*(high-low)+low)

# ie. round(rand*range+lowerLimit)

 

Characters

- must be in quotes, or interpreted as var (ie 'x')

- can convert char to ASCII code (>> int32('a') = 97

- char() converts numbers to chars, ie char(97) = a

- numbers are displayed indented, chars are left align

- thus chars can be manipulated like numbers:

>>'a' +1 = 98   >> char('a' +2) = c

 

Relational / Boolean / logical expressions are either T/F

- relational operators are < > <= >= == ~=

- true is 1, false is 0 in the logical class

- can compare chars using their ASCII vals

- || or, && and, ~ not

-- not (~) is unary operator, others are binary

-- || and && are short-circuit operators (if the result of the expression can be determined based on the first part of the expression then the second part won't be eval'd

- xor(,) returns a logical true if one and only one of the arguments is true, and false if >1 is true

- relational operators are lower on order of operations than most other stuff

 

- examples:

>> 3==5+2   ans=0

>> 'b' < 'a' +1 ans = 0

>> 10>5+2 ans = 1

>> (10>5)+2 ans = 3

>> 'c' == 'd' - 1 && 2 < 4 ans = 1

>> 'c' == 'd' - 1 || 2 > 4 ans =1

>> xor('c' == 'd' - 1, 2 > 4)   ans = 1

>> xor('c' == 'd' - 1, 2 < 4)   ans = 0

>> 10 > 5 > 2   ans = 0

Vectors and Matrices

 

Store data of the same type; MATLAB short for MATrix LABoratory

- vector can be row (1 x n) or column (n x 1) vector

-- vectors same as 1-D array in other languages (thus can call its use an array operation)

-- scalars have 1 x 1 dimensions (one value)

- matrix has r x c dimensions

 

Create a Row Vector

- put in [ ] square brackets separated by space or comma

>> k = [4 5 6 7]

>> k = [4,5,6,7]    ## same thing

 

- colon mark : or colon operator used to iterate stuff

>> itr = 1:5    itr = 1 2 3 4 5

-- adding another colon mark changes to step value

>> itr = 1:3:10   itr = 1 4 7 10

>> vec = 9:-2:1    vec =  9     7     5     3     1

 

- linespace(x,y,n) makes linearly spaced vector c n values in range x to y

>> ls = linspace(1,49,7)    ls = 1  9  17  25  33   41  49

-- similarly, logspace(x,y,n) makes logarithmically spaced vector from 10^x to 10^y c n values

>> ls = logspace(1,6,6)      ls =  10    100   1000     ... 10000      100000     1000000

- transpose operator ( ' )

>> x=x'

x =

    1.0000

    3.2500

    5.5000

    7.7500

   10.0000

 

can create automatically as column vector by transposition: for column vector starting at 5 ending at 9, spaced by 2 elements:

>> x=(5:2:9)'

x =

     5

     7

     9

 

5x5 matrix of random numbers:

> x=rand(5)

x =

    0.8147    0.0975    0.1576    0.1419    0.6557

    0.9058    0.2785    0.9706    0.4218    0.0357

    0.1270    0.5469    0.9572    0.9157    0.8491

    0.9134    0.9575    0.4854    0.7922    0.9340

    0.6324    0.9649    0.8003    0.9595    0.6787

 

5 row, 1 column of rand numbers:

>> x=rand(5,1)

x =

    0.7577

    0.7431

    0.3922

    0.6555

    0.1712

zeros function:

>> x=zeros(6,3)

x =

     0     0     0

     0     0     0

     0     0     0

     0     0     0

     0     0     0

     0     0     0

 

- vector variables can be created using existing vars, called vector concatenation:

>> newVec = [vec ls]

newVec =

  Columns 1 through 10

           9           7           5           3           1          10         100        1000       10000      100000

  Column 11

     1000000

- each element number called index or subscript.

-- start at 1, can access index by using vector name c ( )

>> newVec(2)     ans =    7

-- above pronounced: "newVec sub 2"

 

Saving variable data to file datafile.mat

>> save datafile.mat data

 

clear

 

> load datafile.mat

 

get value from 6th row 3rd column, data is the name of a 7x4 matrix:

>> v=data(6,3)

v =

    9.0698

end can be used to get to the last row or column:

>> v=data(end, 3)

v =

    5.3002

can use arithmetic c end:

>> p=data(end-1,3)

p =

    9.0698

to get the last 2 columns of matrix data:

>> volumes=data(:,3:4)

volumes =

    4.0753       NaN

    6.6678    2.1328

    1.5177    3.6852

    3.6375    8.5389

    4.7243   10.1570

    9.0698    2.8739

    5.3002    4.4508

to get the 6th element from the vector density:

>> p=density(6)

p =

    6.1100

can select a subset of a vector:

>> p=density(2:5)

p =

    1.7800

    0.8600

    1.6000

    3.0000

to change a variable combining indexing and assignment:

>> v2(1)=0.5

v2 =

    0.5000

    2.1328

    3.6852

    8.5389

   10.1570

    2.8739

    4.4508

 

Performing Array Ops on Vectors

- add 1 to each element of v1

>> r=v1+1

r =

    5.0753

    7.6678

    2.5177

    4.6375

    5.7243

   10.0698

    6.3002

- can combine vectors of the same size

>> vs = v1+v2

vs =

    4.5753

    8.8006

    5.2029

   12.1764

   14.8813

   11.9437

    9.7510

- can multiply array by a scalar

> va = vs/2

va =

    2.2877

    4.4003

    2.6014

    6.0882

    7.4406

    5.9718

    4.8755

- max() can extract the max val of a vector:

>> vm = max(va)

vm =

    7.4406

- can perform math ops on vecs:

>> vr=round(va)

vr =

     2

     4

     3

     6

     7

     6

     5

- the * does matrix mult, while the .* does elementwise multiplication

> mass = density.*va

mass =

    1.2125

    7.8325

    2.2372

    9.7411

   22.3220

   36.4880

   12.3838

 

Multiple Outputs from Function Calls

- dsize has the size of the data variable

>> dsize = size(data)

dsize =

     7     4

- can use [ ] to extract single or double output vars:

>> [dr,dc] = size(data)

dr =

     7

dc =

     4

- can use max() to get the max val c corresponding index val and assign it to vars

>> [vMax, ivMax] = max(v2)

vMax =

   10.1570

ivMax =

     5

- to create matrix c 5 rows, 7 columns c random ints from 1 to 20:

>> x=randi(20,5,7)

x =

    14     5     5     2     1     1    18

     3     1    15    18    20    18    19

    20     8     8    20     7    14     1

    10     5     4    19    15    16     2

    18    17    15     9     8    12     5

 

Plotting Vectors

>>plot(x, y)

- can changes plot styles, such as to change to red star makers with no line can use:

> plot(sample, mass2, 'r*')

- can plot graphs on top of each other c the hold on command:

>> hold on

- use hold off to go back to default of writing over (and deleting) graphs if made sequentially

- close alll command closes the figure window

- if plot single vector by itself, matlab uses the nth element as the x-axis in the plot

- plot can accept vars that have assoc vals:

>> plot(v1,'Linewidth',3)

- can also plot after other stuff:

> plot(sample, v1, 'ro', 'LineWidth', 4)

- give it a title() !

>> title('Sample Densities')

- labelling the y-axis

>> ylabel('Density (g/cm^3)')

- can select the var names in the workspace to plot against one another

 

Project: Electricity Usage

 

Will plot e use in diff sectors (res, com, ind)

- residential highest, then commercial and industrial

>> load electricity.mat

>> usage #shows the values in usage

 

## change an NaN val in row 2 col 3

>> usage(2,3)=2.74

 

## create variable res that has the first column of usage

>> res = usage(:,1)

 

## create variable comm that has the 2nd column of usage

>> comm = usage(:,2)

 

## for industrial

ind = usage(:,3)

 

## Create column vector for years 1991 to 2013

>> yrs = (1991:2013)'

 

## plot res against yrs c blue dashed line

>> plot(yrs, res, 'b--')

>> hold on ## place hold on to add graph on top

>> plot(yrs, comm, 'k:')  ## commercial in blk dotted ln

>> plot(yrs, ind, 'm-.') ## industrial, magenta dash-dot

>> title('July Electricity Usage') ## Add title

>> legend('res', 'comm', 'ind')  ## Add legend

 

Project: Audio Frequency

 

Will create a signal that has the beat phenomenon (2 signals close together causes amplitude pulse) by analyzing the signal's freq content

 

% create var fs for sampling freq of audio signal

>> fs = 10

>> t=0:1/fs:20;  ## t is times audio sampled, is a vector from 0 to 20 in interval of 1/fs

 

% y is sum of 2 sine waves

>> y=sin(1.8*2*pi*t)+sin(2.1*2*pi*t);

>>plot(t,y)

 

% the Fourier transform, which gives info about the freq content of the signal

>> yfft = fft(y)

 

% makes var with numel() function, which returns # elements in array

>> n = numel(y)

>> f=0:fs/n:fs*(n-1)/n ## makes vec f

 

% need abs() to plot Fourier transform values which are complex numbers

>> plot(f, abs(yfft))

 

% the spikes on the left side of plot are freqs of 2 sine wives creating the beat phenomenon when close together

 

% there are 4 spikes b/c of the Nyquist freq, which is 5, or fs/2, here. When input vals are real numbers, fft function always returns data c magnitude symmetric around the Nyquist freq (the 2nd half of plot is a mirror image of 1st half)

 

Go to new script to create new script. Running programs will save it before execution

 

Can run MATLAB scripts from command line by typing in the name of the script

 

Logical Ops and Vars

 

%Is pi greater than 3? (1=true)

 

- can compare vectors to each other

 

% Create logical array as array index, where matlab extracts array elements where index is true

>>v = v1(v1<4)

 

% can use logical indexing to replace vals in array, to replaces vals in v1 that are greater than 5 with 10:

>> v1(v1>5) = 10

 

 

Flow Control

 

Control whether code is executed c if statement, that is finished with the end statement, ex:

% Calculate A

A = randn(1)

 

% Calculate B only if A is greater than 0

if A>0

    B = sqrt(A)

else

    B=0

end

 

% for Loop, inc idx by 1 from 1 to 5

% Calculate x

x = (11:15).^2;

 

% TODO - wrap the line below in a for loop

for idx=1:5

    disp(x(idx))

end

 

Stellar Motion

 

Use ML to find observed wavelength of star and compare to canonical value to see if blue or red shifted (Doppler effect)

 

>> load starData

>> nObs = size(spectra,1);

>> lambdaStart = 630.02;

>> lambdaDelta = 0.14;

 

>> lambdaEnd = lambdaStart+(nObs-1)*lambdaDelta

lambdaEnd =

  679.8600

 

% Column vector from lambda start to end in steps of lambda delta, inversed to make column vector

>> lambda = (lambdaStart:lambdaDelta:lambdaEnd)'

% each column of spectra is a different star, the 6th star is HD 94028, extract its spectra

s = spectra(:,6)

 

% plot spectra s as function of wavelength lambda c plot scales on both axes, with some style

>> loglog(lambda, s, '.-')

 

% creates 2 vars for min val of s and index where min value occurred

>> [sHa, idx] = min(s)

 

% use idx to index into lambda to find wavelength of Hydrogen-alpha line, stored as lambda Ha

>> lambdaHa = lambda(idx)

 

% plot single point as red square

>> loglog(lambdaHa, sHa, 'rs', 'Markersize', 8)

 

% Redshift factor, to determine speed of star relative to Earth

>> z = (lambdaHa/656.28)-1

 

% multiplying redshift factor by speed of light, can calculate speed of star relative to Earth

>> speed = z*3e5

 

Building a Discrete LTI System

1. Solve for most recent sample [y]n

2. Add Delay blocks

3. Construct right-hand side of eqn

4. Set initial conditions and define constants

 

dspstartup in MATLAB command lets you automatically configure settings for the ML session