Download eg 2+3

%matlab introduction course notes ---week 6, term 1
1. Matlab as calculator
1.1 basic arithmetic operator + - * / ^ ()
e.g. 2+3/4*5 =
3^2*4 =
3-4/4-2 =
1.2 extended arithmetic - accidental error
1/0 =
-1/0 =
0/0 =
1/Inf =
we know 1/(1/x) = x, then try 1/(1/0) = ?
complex number 1+i , -1+3*i
2. Numbers and formats
2.1 Different kind of numbers
Integer: e.g. 123, -218
Real : e.g. 1.234, - 10.9
complex : e.g. 3.21-3.4*i (i = sqrt(-1)).
Inf : Infinity (dividing by 0)
NaN: Not a number (0/0)
ex: notation for very large or small number, e.g. -1.34e+03 = ? , 1.34e-05 = ?
2.2 Calculation: 15 significant figures
The 'format' tells how matlab prints numbers. Type 'help format' in command window for
full list
e.g. pi = ?? usually 3.1416
format long
pi = ??
format short e
pi = ??
format short
pi = ??
If want to switch back to default format, type: format
2.3 finite accuracy consequences
Matlab limit accuracy (enough for most cases)
64 bits, store number as large as 2*10^308, as small as 2*10^(-308)
store any number 15 significant figures:
e.g. 1.23456789023456 (14 figures, can handle)
1.23456789023456789012 (20 digits, truncated to 15 figures)
round off cannnot be avoid.
e.g. what is sin (pi) = ?
try sin(pi) = ??---slight round-off error, take it as zero as long as small like 10^(-15).
3. Variables
3.1 combination of oth letter and number, case sensitive
e.g. a , x1, z2453, A, t = 2+3-9, 2*t-2
Not allowed: Net-c, 2p, %x, @sign
3.2 special names: eps (= 2^(-54)), pi --> avoid using
3.3 complex numbers : i, j = sqrt(-1), unless you change them
4. Suppressing output (don't want to show output)
hidden: x = -13; (semi-colon).
5. Build-in function
5.1 sin, cos, tan, sec = 1/sin, cosec = 1/cos, cotan
e.g. work out the coordinate of a point on a circle of radius 5 centred at origin, having an
elevation 30 degree = pi/6 radians.
so (x, y), where x = 5*cos(pi/6), y = 5*sin(pi/6).
5.2 inverse trig function
e.g. asin, acos, atan--> answer returned in radians, so asin(1) = pi/2
5.3 exponential
exp : exp(x) = e^x
logarithm: log: log to base e/ log10 to base 10
square root: sqrt().
e.g. x = 9; sqrt(x), exp(x), log(sqrt(x)), log10(x^2+6)
5.4 more
e.g. eigenvalue, eigenvector: eig
solving DE:
6. vectors
6.1 row vectors
a = [1 2 3] or a = [1, 2, 3]
e.g. V = [1 3 sqrt(5)], what is length(V)
space vitally important : e.g. v2 = [3+ 4 5], v3 = [3 +4 5];
add vector of the same length: e.g. V + v3, v4 = 3*v3, v5 = 2*V-3*v4, v6 = v+v2??? wrong!
since dimension must agree
build a row vector from existing ones: e.g. w = [1 2 3], z = [8, 9], cd = [2*z -w], sort(cd)
(ascending order)
look at value of particular entries: e.g. w(2) = ?
set w(3) = 100, then w = ??
6.2 column vector
e.g. c = [1; 3; sqrt(5)] or c2 = [3 return 4 return 5]
c3 = 2*c-5*c2
6.3 column notation : a shortcut for producing row vectors
e.g. 1:100
3:7
5:0.1:6
1:-1 --> []
0.32:0.1:0.6
-0.4:-0.3:-2
6.4 extracting bits of a vector
e.g. r5 = [1:2:6 -1:-2:-7]
get 3rd to 6th entries: r5(3:6) = , try r5(1:2:7) = ?, r5(6:-2:1) = ??
6.5 tranposing: row-> col, col-> row
e.g. w,w', c, c' (w = [1 2 -3], c = [1;2 ; 3.5])
t = w+2*c', T = 5*w'-2*c
7. vector operation
7.1 scalar product: u*v = sum (ui*vi)
u = [u1, ..., un]; v = [v1;...; vn]
e.g. u = [10 -11 12], v = [20; -21; -22]; prod = u*v
e.g. w = [2 1 3]; z = [7; 6; 5]; check: v*w, u*w', u*u', v'*z
norm of a vector: ||u|| = sqrt(sum(ui))
compute norm: sqrt(u*u') or norm(u)
7.2 dot product-vector of the same length times with each other
u.v = [u1v1,...,unvn]
e.g. u.*w, u.*v', u.*z, u'.*v
ex.: Tabulate y = x*sin(pi*x) for x = 0, 0.25, ... , 1
ans: x = 0:.25:1; y = x.*sin(pi*x);
7.3 dot divison of array-element by element division
e.g. a = 1:5, b = 6:10, check a./b = , a./a = , c = -2:2, a./c, a.*b-24, ans./c
ex: limit sin(pi*x)/x, as x Æ 0
ans: x = [.1 .01. .001 .0001], sin(pi*x)./x, format long , ans - pi
e.g. 1/x (wrong!), 1./x (correct)
7.4 dot power of array (.^) sqare all element of a vector
e.g. u.*u, u.^2, u.^4, u.*w.^(-2)
8. Matrix
8.1 generating matrix:
zeros, ones, rand, randn
e.g. Z = zeros(2,4)
F = 5*ones(3,3)
N = 10*rand(1,10); N = fix(N)---Æ fix: round toward zero (others, floor,
round, ceil)
R = randn(4,4)
8.2 load function
Read out file containing matrix generated earlier .
e.g. open new text file, input matrix (4 line) under Matlab directory
(C:\Users\jasmine\Documents\MATLAB\magik.txt). Save file in the current directory. Type
statement
load magik.txt
reads the file ad create a variable ‘magik’.
Or use Import Wizard.
5
9
8.3 m-file
e.g. creat a m-file named magik.m, containing
A = [16 3 2 13
10 11 8
6 7 12
4
15 14 1];
Statement ‘magik’ read the file and creastes a variable ‘A’.
8.4 concatenation
The process of joining small matrix to make bigger one.
e.g. B = [A A+32; A+48 A+16]; sum(B)
8.5 Deleting rows and columns
X = A; X(:,2) = []Ædeleting the second column
X(1,2) = []Æ error, since not a matrix anymore
X(2:2:10) = []Æremaining element into a row vector
8.6 Linear Algebra
e.g. A+A’Æ produce a symmetric matrix
A’*AÆ produce a symmetric matrix
det(A)Æ determinant is zero, mean matrix is singular, means no inverse
X = inv(A)Æ get warning
Poly(A) Æ return the coefficient in the characteristic polynomial det(A-lambda*I)
(order from lambda^4, …, lambda^1, constant).
A.*AÆ element-by-element multiplication (same as ./ .^)
8.7 building tables:
e.g. n = (0:9)’;Æ column vector
then, pows = [n n.^2 2.^n]Æ build a table of squares and powers of 2
e.g. format short g
x = (1:0.1:2)’; logs = [x log10(x)]Æ build a table of logariths
8.8 Multivariate data
e.g. heart rate; weight hours of exercise/week
D = [ 72
134
3.2
81
201
3.5
69
156
7.1
82
148
2.4
75
170
1.2]
mu = mean(D), sigma = std(D)
full list of data analysis: type ‘help datafun’, ‘help stats’
8.9 scalar expansion
e.g. subtract from a matrix from each element: B = A-8.5
e.g. Assign a scalar to all indices in a range: B(1:2,2:3) = 0
8.10 logical subscripting
e.g. remove missing data: x = [2.1 1.7 1.6 1.5 NaN 1.9 2.2 1.6 1.8 5.1 1.4 1.8];
use ‘isfinite(x)’----true for all finite number and false for NaN and Inf
x = x(isfinite(x))
e.g. remove outlier: x = x(abs(x-mean(x)) <= 2*std(x))
e.g. find location of prime number for magic matrix by setting the non-primes to 0:
A(~isprime(A)) = 0
8.11 the find function
e.g. k = find(isprime(A))’Æpick up location of primes in the magic square
A(k)Æ display those primes