This document discusses vectorized code in MATLAB. It covers operations on vectors and matrices such as addition, subtraction, and multiplication. It discusses using vectors and matrices as function arguments and performing element-wise operations. It also covers logical vectors and examples of vectorizing code using loops to perform operations such as finding maximum/minimum values, counting elements, and comparing elements in less than 3 sentences.

1. CSCI 101
Vectorized Code

2. Outline
• Operations on Vectors and Matrices
• Vectors and Matrices as Function Arguments
• Logical Vectors

3. Operations on Vectors and Matrices
v=[3 7 2 1]
for i = 1:length(v)
v(i) =v(i) * 3;
end
v= [3 7 2 1];
>> v=v*3
9 27 6 3
mat = [4:6; 3:-1:1]
mat =
4 5 6
3 2 1
>> mat * 2
ans =
8 10 12
6 4 2

4. Scalar Operations
Operation Algebraic Form MATLAB Form
Addition a + b a + b
Subtraction a – b a – b
Multiplication a × b a * b
Division
a
b
a / b
Exponentiation ab a ^ b
Same operators can be applied to arrays following linear algebra rules.
These rules are specific to each operation
A+B (A and B should have same size)
A*B (A columns should equal B rows)
A^B (A should be squared matrix)
A/B = A* B-1 (B should be invertible and A columns should be equal B-1 rows )

5. Element Operations
Both arrays must have the same number of rows and columns
A + B = [1 2 3] + [3 4 5] = [4 6 8]
Element and Algebraic operations are the same
A .* B = [1 2 3] .* [3 4 5] = [3 8 15]
A .^ B = [4 9 6] .^ [2 2 2] = [16 81 36]
A ./ B =[4 9 6] ./ [2 3 2] = [2 3 3]

6. Array Operations
Operation MATLAB form Comments
Addition a + b Adds each element in a to the element with the same
index in b
Subtraction a – b Subtracts from each element in a the element with the
same index in b
Multiplication a .* b Multiplies each element in a with the element in b
having the same index
Either a or b can be a scalar
Right Division a ./ b Element by element division of a and b:
a(i,j) / b(i,j) provided a, b have same shape
Either a or b can be a scalar
Left Division a . b Element by element division of a and b:
b(i,j) / a(i,j) provided a, b have same shape
Either a or b can be a scalar
Exponentiation a .^ b Element by element exponentiation of a and b:
a(i,j) ^ b(i,j) provided a, b have same shape
Either a or b can be a scalar

7. Operations Rules Examples
v=[3 7 2 1];
>> v ^ 2
??? Error using ==> mpower
Inputs must be a scalar and a square matrix.
To compute elementwise POWER, use POWER (.^)
instead.
>> v .^ 2
ans =
9 49 4 1

8. Operations Rules Examples
v1 = 2:5;
 v1 = [2 3 4 5]
v2 = [33 11 5 1];
>> v1 * v2 %Error why
>> v1*v2’ % what is the output
>> v1 .* v2 % v1 and v2 must have same size

9. Vectors and Matrices as Function
Arguments
• Many functions accept scalars as input
• Some functions work on arrays
• Most scalar functions accept arrays as well
– The function is performed on each element in the array
individually
• Try x = pi/2; y = sin(x) in MATLAB
• Now try
x = [0 pi/2 pi 3*pi/2 2*pi];
y = sin(x)

10. Functions Examples
v1=[1 3 2 7 4 -2]
v2=[5 3 4 1 2 -2]
[mx mxi]=max(v1)  7 4
v=v1-v2 -4 0 -2 6 2 0
v=sign(v1-v2) -1 0 -1 1 1 0
x=sum(v1)  15
v=find(v1>3)  4 5

11. Logical Vectors
v1=[1 3 2 7 4 -2]
v2=[5 3 -4 -1 2 -2]
v=v1>0  1 1 1 1 1 0
v=v2>0 1 1 0 0 1 0
a=v2(v2>0) 5 3 2
x=sum(v2>0) 3
v=true(1,5)  1 1 1 1 1
v=false(1,5)  0 0 0 0 0
x=all(v1>0)  1 (are all ones?)
y=any(v1>0)  0 (any one?)

12. Example
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B = [0 0 0 1]

13. Example
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12
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D

14. Example
Write a program using Matlab loop(s) to calculate the following
while assuming that the variable x has been initialized:
𝑆 =
𝑖=1
10
𝑥𝑖
2
N=10;
x = 1:N;
S=0;
for i = 1:N
S=S+ x(i)*x(i);
end
S
Vectorize your code S=sum(x.*x)

15. Getting Maximum and Minimum Value
v=[1 3 2 7 4 2]
maxv=0;
for =1:length(v)
if v(i)>maxv
maxv=v(i);
maxi=i;
end
end
% Repeat for the minimum
i v(i) maxv maxi
1 1 01 ?1
2 3 13 12
3 2 3 2
4 7 37 24
5 4 7 4
6 2 7 4
Vectorize your code: [maxv maxi]=max(v)
Example

16. Counting Elements
v=[1 3 2 -7 4 -2]
c=0;
for =1:length(v)
if v(i)>0
c=c+1;
end
end
% Repeat for the negative
i v(i) c
1 1 01
2 3 12
3 2 23
4 -7 3
5 4 34
6 -2 4
Vectorize your code c = sum(v>0)
Example

17. Comparing Elements
v1=[1 3 2 7 4 -2]
v2=[5 3 4 1 2 -2]
for i=1:length(v1)
if v1(i)>v2(i)
v(i)=1;
else
v(i)=0;
end
end
i v1(i) v2(i) v(i)
1 1 5 0
2 3 3 0
3 2 4 0
4 7 1 1
5 4 2 1
6 -2 -2 0
Vectorize your code: v= v1>v2
Example

18. Vertical Motion
% Vertical motion under gravity
g = 9.81; % acceleration due to gravity
u = 60; % initial velocity in metres/sec
t = 0 : 0.1 : 12.3; % time in seconds
s = u * t – g / 2 * t .ˆ 2; % vertical displacement in metres
plot(t, s), title( ’Vertical motion under gravity’ )
xlabel( ’time’ ), ylabel( ’vertical displacement’ )
grid
disp( [t’ s’] ) % display a table
Example