computer simulation lab
DESCRIPTION
Computer Simulation Lab. “Lecture 4”. Electrical and Computer Engineering Department SUNY – New Paltz. Logical vectors. objectives L ogical operators Logical vectors Logical functions. Logical Operators. r = 0; rTRANSCRIPT
SUNY-New Paltz
Computer Simulation Lab
Electrical and Computer Engineering Department
SUNY – New Paltz
“Lecture 4”
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Logical vectors
objectives
1. Logical operators2. Logical vectors3. Logical functions
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Logical Operators
r = 1;
r <= 0.5
ans= 0
r = 0;
r <= 0.5
ans= 1
r = .5;
r <= 0.5
ans= 1
r = 1:5;
r <= 3
ans= 1 1 1 0 0
a = 1:5;
b = [0 2 3 5 6];
a == b
ans= 0 1 1 0 0
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Application of Logical Vectors
x = 0 : pi/20 : 3 * pi;y = sin(x);y = y .* (y > 0); plot(x, y)
1-form vector 0<x<3*pi2- form a vector such that if y > 0 then it is one Multiply y by this vector
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Avoiding Division by Zero
x = -4*pi : pi/20 : 4*pi;x = x + (x == 0)*eps;y = sin(x) ./ x;plot(x, y) Sin(ε)/ε=1
1-form vector x -4*pi <x< 4*pi2- form a new vector x such that if x=0, then it is eps3- Divide sin(x) by this vector
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Logical Operators
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Operator Precedence
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Be Careful!
0 < r < 1 Wrong!
(0 < r) & (r < 1)
Correct!
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Subscripting with logical vectors
a = [-2 0 1 5 9];
a([5 1 3])
ans: 9 -2 1
a(logical([0 1 0 1 0]))
ans: 0 5
a = a(a > 0)
ans: 1 5 9
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Verifying Logical Vectors
a = [ 1 –1 2 0 3]b= a > 0islogical(a)ans: 0islogical(b)ans: 1
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Logical functions
any(x)
all(x)
exist(’a’)
find(x)
returns the scalar 1 (true) if any element of x is non-zero (true).
returns the scalar 1 if all the elements of x are non-zero.
returns 1 if a is a workspace variable.
returns a vector containing the subscripts of the non-zero (true) elements of x
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Examples
a=[0 2 1 2 3 4]
a = a( find(a) )
x = [8 1 -4 8 6];
find(x >= max(x))
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Logical functions
isempty(x)
isinf(x)
isnan(x)
returns 1 if x is an empty array and 0 otherwise.
returns 1’s for the elements of x which are +Inf or −Inf, and 0’s otherwise
returns 1’s where the elements of x are NaN and 0’s otherwise.Example: x(isnan(x)) = [ ]
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Using ‘any’ and ‘all’
if any(a ~= b)
statementend
if a ~= b
statement
end
if all(a >= 1)
do something
end
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Logical vectors instead of ‘elseif’ ladders
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Matlab Code Using if/elseif
inc =[5000 10000 15000 30000 50000];
for ti = inc
if ti < 10000
tax = 0.1 * ti;
elseif ti < 20000
tax = 1000 + 0.2 * (ti - 10000);
else
tax = 3000 + 0.5 * (ti - 20000);
end;
disp( [ti tax] )
end;
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Logical Way
inc = [5000 10000 15000 30000 50000];
tax = 0.1 * inc .* (inc <= 10000);
tax = tax+(inc > 10000 & inc <= 20000) ...
.* (0.2 * (inc-10000) + 1000);
tax = tax + (inc > 20000) .* (0.5*(inc-20000)+ 3000);
disp( [inc’ tax’] );
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Matrices
Objectives
• ways of creating and manipulating matrices;
• matrix operations.
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Creating matrices
a = [1 2; 3 4];x = [5 6];a = [a; x]
a =
1 2
3 4
5 6
b = a’
b=1 3 5
2 4 6
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Subscripts
a(3,3)
a(2:3,1:2)
a(3,:)
a(1:2,2:3) = ones(2)
x = [0:30:180]’;
trig(:,1) = x;
trig(:,2) = sin(pi/180*x);
trig(:,3) = cos(pi/180*x);
a =1 2 3
4 5 6
7 8 9
trig = 0 0 1.0000 30.0000 0.5000 0.8660 60.0000 0.8660 0.5000 90.0000 1.0000 0.0000 120.0000 0.8660 -0.5000 150.0000 0.5000 -0.8660 180.0000 0.0000 -1.0000
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‘:’ Operator
a =1 23 4
b = a(:)
b=1234
b =1 2 34 5 6
a =0 00 00 0
a(:) = b
a =1 54 32 6
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Duplicating rows and columns: tiling
a =
1 2 3
repmat(a, [3 1]) or repmat(a, 3, 1)
ans =
1 2 3
1 2 3
1 2 3
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Exercise
1 2 3 1 2 3 1 2 3
4 5 6 4 5 6 4 5 6
1 2 3 1 2 3 1 2 3
4 5 6 4 5 6 4 5 6
1 2 3 1 2 3 1 2 3
4 5 6 4 5 6 4 5 6
4 3 4 3 4 3
1 2 1 2 1 2
4 3 4 3 4 3
1 2 1 2 1 2
1 3 5 7 9 11 13 15 17 19
10 9 8 7 6 5 4 3 2 1
1 3 5 7 9 11 13 15 17 19
10 9 8 7 6 5 4 3 2 1
1 3 5 7 9 11 13 15 17 19
10 9 8 7 6 5 4 3 2 1
1 3 5 7 9 11 13 15 17 19
10 9 8 7 6 5 4 3 2 1
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Deleting rows and columns
a(:,2) = [ ]
a(1,2) = [ ]
a(2:2:6) = [ ]
a(:, logical([1 0 1]))ans =
1 34 67 9
a =1 2 34 5 67 8 9
a =
1 7 5 3 6 9
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Elementary matrices
eye(3)ans =
1 0 00 1 00 0 1
pascal(4)ans =
1 1 1 11 2 3 41 3 6 101 4 10 20
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Generating Specialized Matrices
compan - Companion matrix. gallery - Higham test matrices. hadamard - Hadamard matrix. hankel - Hankel matrix. hilb - Hilbert matrix. invhilb - Inverse Hilbert matrix. magic - Magic square. pascal - Pascal matrix. rosser - Classic symmetric eigenvalue toeplitz - Toeplitz matrix. vander - Vandermonde matrix. wilkinson - Wilkinson's eigenvalue test
matrix.
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Using MATLAB functions with matrices
a =1 0 11 1 10 0 1
all(a)
any(a) 1 1 1
0 0 1
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Manipulating matrices
diag
fliplr
flipud
rot90
tril
extracts or creates a diagonal.
flips from left to right.
flips from top to bottom.
Rotate 90 egrees
Extract Lower Triangle
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Array (element-by-element) operations on matrices
a =1 2 34 5 6
a .^ 2ans =1 4 9 16 25 36
a =1 2 34 5 67 8 9for v = adisp(v’)end
1 4 72 5 83 6 9
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Visualization of matrices
A = 1000; % amount borrowed
n = 12; % number of payments per year
for r = 0.1 : 0.01 : 0.2
fprintf( ’%4.0f%’, 100 * r );
for k = 15 : 5 : 25
temp = (1 + r/n) ^ (n*k);
P = r * A * temp / n / (temp - 1);
fprintf( ’%10.2f’, P );
end;
fprintf( ’\n’ ); % new line
end;
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Vectorize the Problem
for r = 0.1 : 0.01 : 0.2
k = 15 : 5 : 25;
temp = (1 + r/n) .^ (n*k);
P = r * A * temp / n ./ (temp - 1);
disp( [100 * r, P] );
end;
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Eliminate for loops!
r = [0.1:0.01:0.2]’
r = repmat(r, [1 3])
k = 15:5:25
k = repmat(k, [11 1])
temp = (1 + r/n) .^ (n * k);
P = r * A .* temp / n ./ (temp - 1)
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Multi-dimensional arrays
a = [1:2; 3:4]
a(:,:,2) = [5:6; 7:8]
a(:,:,1) =1 2
3 4
a(:,:,2) =5 6
7 8
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Matrix operations
Matrix multiplication
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Other matrix functions
det determinant.eig eigenvalue decomposition.expm matrix exponential, i.e. eA, where A
is a matrix.inv inverse.lu LU factorization (into lower and upper
triangular matrices).qr orthogonal factorization.svd singular value decomposition