Array and Matrix
Operations
Dr. Marco A. Arocha
INGE3016-MATLAB
Sep 11, 2007, Dic 7, 2012
1
Array and Matrix Operations
OPERATION
Commands
Comments
Array addition
a+b
array addition and matrix addition are identical
Array subtraction
a-b
array subtraction and matrix subtraction are identical
Array
multiplication
a .* b
element-by-element multiplication of a and b; both arrays must
be the same shape, or one of them must be a scalar
Array right
division
a ./ b
element-by-element division of a by b: a(i,j)/b(i,j); both arrays
must be the same shape, or one of them must be a scalar
Array left division
a .\ b
element-by-element division of b by a: b(i,j)/a(i,j); both arrays
must be the same shape, or one of them must be a scalar
Array
exponentiation
a .^ b
e-by-e exponentiation of a to b exponents: a(i,j)^b(i,j); both
arrays must be the same shape, or one of them must be a
scalar
Matrix
Multiplication
a*b
the number of columns in a must equal the number of rows in b
Matrix right
division
a/b
a * inv(b), where inv(b) is the inverse of matrix b
Matrix left division
a\b
inv(a) * b, where inv(a) is the inverse of matrix a
Matrix
exponentiation
a^b
matrix multiplication of a: a*a*a*...a, b times
2
Array Operations
 Assume the following arrays of length n:
A  [a1 a2 a3 ...a n ]
B  [b1 b2 b3 ...bn ]
 Addition of arrays of the same length is defined as:
A  B  [a1  b1 , a2  b2 ,..., an  bn ]
 Subtraction of arrays of the same length:
A  B  [a1  b1 , a2  b2 ,..., an  bn ]
 Multiplication of arrays of the same length: ‘.*’
A . * B  [a1 * b1 , a2 * b2 ,..., an * bn ]
 Division of arrays of the same length: ‘./’
A. / B  [a1 / b1, a2 / b2 ,..., an / bn ]
 Exponentiation of arrays of the same length ‘ .^’
A.^B=[a1^b1, a2^b2, a3^b3,…,an^bn]
3
Array Addition
With 1D arrays:
With 2D arrays:
>> A=[1 3 5 ];
>> B=[2 4 6];
>> A+B
>> A=[1 3 5; 2 4 6]
A=
1 3 5
2 4 6
ans =
3
7 11
>> B=[-5 6 10; 2 0 9]
B=
-5 6 10
2 0 9
>> A+B
ans =
-4 9 15
4 4 15
4
Array Multiplication
>> A=[1,3,5;2,4,6]
A=
1 3 5
2 4 6
>> B=[2,3,4;-1,-2,-3]
B=
2 3 4
-1 -2 -3
>> A.*B
ans =
2 9 20
-2 -8 -18
Arrays must be of the same size
5
Array Division
>> A=[2,4,6]
A=
2 4 6
>> B=[2,2,2]
B=
2 2 2
>> A./B
% num/den
ans =
1 2 3
>> A.\B
% den\num
ans =
1.0000 0.5000 0.3333
Right division
Left division
6
Array Exponentiation
>> A=[2,4,6]
A=
2 4 6
>> B=[2,2,2]
B=
2 2 2
>> B.^A
ans =
4 16 64
7
Special Cases:
array <operator> scalar
scalar<operator> array
If one of the arrays is a scalar the following are valid
expressions. Given:
>> A=[1 2 3];
>> A.*5
>> A+2
ans =
3 4
5
>> A-1
ans =
0 1
ans =
5 10 15
>> A./2
2
ans
0.5 1.0 1.5
8
Dot is optional in the above
two examples
Special Cases:
array <operator> scalar
scalar<operator> array
Given:
a=[1 2 3]
If one of the arrays is a scalar the following are valid expressions:
>> a*2
ans =
2 4
>> a.*2
ans =
2 4
>> a/2
6
ans =
0.5000 1.0000 1.5000
>> a./2
ans =
0.5000 1.0000 1.5000
6
9
Special Cases
>> A=[5]
A=
5
>> B=[2,4,6]
B=
2 4 6
Period is optional here
>> A.*B
ans =
10 20 30
10
The basic data element in the MATLAB
language is the array
• Scalar
• 1x1 array
• Vectors: 1-D arrays
• Column-vector: m x 1 array
• Row-vector: 1 x n array
• Multidimensional arrays
• m x n arrays
11
MATRIX
• Special case of an array:
n, columns
m, rows
Rectangular
array
12
Square Matrix
• m=n
Square matrix of order three
2 4 6


A  3 5 7 
1 2 3
Can reference individual elements
Z=3*A(2,3)
Main diagonal:
[2,5,3], i.e, Ai,j where i=j
13
Self-dimensioning
Upon initialization, MATLAB automatically allocates the correct
amount of memory space for the array—no declaration
needed, e.g.,
a=[1 2 3];
% creates a 1 x 3 array
% without previously separate memory for storage
14
Self-dimensioning
Upon appending one more element to an array, MATLAB
automatically resizes the array to handle the new element
>> a=[2 3 4]
a=
2 3 4
>> a(4)=6
a=
2 3 4 6
>> a(5)=7
a=
2 3 4 6
% a contains 3 elements
% now a contains 4 elements
% now a contains 5 elements
7
15
More on appending elements to an array:
>> a=[1 2 3]
a=
1 2 3
>> a=[a 4]
a=
1 2 3
4
>> b =[a; a]
b=
1 2 3
1 2 3
>> c=[a; 2*b]
c=
1 2 3
2 4 6
2 4 6
4
4
4
8
8
16
Self-dimensioning is a
MATLAB key feature
This MATLAB key feature is different from most
programming languages, where memory allocation and
array sizing takes a considerable amount of
programming effort
Due to this feature alone MATLAB is years ahead, such
high level languages as: C-language, FORTRAN, and
Visual Basic for handling Matrix Operations
17
Deleting array elements
>>A=[3 5 7]
A=
3 5 7
>> A(2)=[ ]
A=
3 7
>> B=[1 3 5; 2 4 6]
B=
1 3 5
2 4 6
>> B(2,:)=[ ]
B=
1 3 5
Deletes row-2, all column elements
18
Storage Mechanism
for Arrays
19
Storage mechanism for arrays
Two common ways of storage mechanism,
depending on language:
A=
1
2
3
• One row at a time: row-major order (*)
1 3 5 2 4 6 3 5 7
3
4
5
5
6
7
Row-1
Row-2
Row-3
• One column at a time: column-major order
1 2 3 3 4 5 5 6 7
col-1
col-2
col-3
Last one is the MATLAB way of array storage
20
(*) C Language uses row-major order
Accessing Individual Elements
of an Array
>> A=[1 3 5; 2 4 6; 3 5 7]
A=
1
2
3
3
4
5
>> A(2,3)
ans =
6
5
6
7
% row 2, column 3
Two indices is
the usual way
to access an
element
21
Accessing elements of an Array by a
single subscript
>> A=[1 3 5; 2 4 6; 3 5 7]
A=
1
2
3
3
4
5
5
6
7
In memory they are
arranged as:
123345567
If we try to access them with
only one index, e.g.:
>> A(1)
Recall:
ans =
column1
major
>> A(4)
order in
ans =
memory
3
>> A(8)
ans =
6
22
Accessing Elements of an Array by a
Single Subscript
>> A=[1 3 5; 2 4 6; 3 5 7]
A=
1
2
3
3
4
5
5
6
7
In memory
With one index & colon operator:
>> A(1:2:9)
ans =
1 3 4 5 7
The index goes from 1 up to 9 in
increments of 2, therefore the
indices referenced are:
1, 3, 5, 7, 9,
and the referenced elements are:
A(1), A(3), A(5), A(7),and A(9)
A(1)=1 A(4)=3 A(7)=5
A(2)=2 A(5)=4 A(8)=6
A(3)=3 A(6)=5 A(9)=7
23
Example
Add one unit to each
element in A:
1 1
𝑎= 1 1
1 1
1
1
1
Given:
A(1:1:3;1:1:3)=1
Answer-1:
for ii=1:1:9
A(ii)=A(ii)+1;
end
24
Example, continuation
Answer-2:
Answer-4:
• A(1:1:9)=A(1:1:9)+1;
• A=A.*2;
Answer-3:
Answer-5:
• A=A(1:1:9)+1;
• % one index
• A=A+1;
25
Exercise
With one index,
Referencing is OK,
Initializing is not.
A=
Initialize this Matrix with one index:
for k =1:1:25
if mod(k,6)==1
A(k)='F';
else
A(k)='M';
end
end
F
M
M
M
M
M
F
M
M
M
M
M
F
M
M
M
M
M
F
M
M
M
M
M
F
% ‘F’ elements are in indices: 1, 7, 13, 19, and 25
% looks beautiful but doesn’t work at all, elements are not distributed as desired
% We can make reference to the elements of a 2-D array with one index
% however we can’t initialize a 2-D array with only one index.
26
Accessing Elements of an Array
>> A=[1 3 5; 2 4 6; 3 5 7]
A=
1
2
3
3
4
5
5
6
7
>> A(2,:)
ans =
2 4 6
(2, :) means row 2, all
columns
A colon alone “ : “ means all the elements of that
dimension
27
Accessing Elements of an Array
>> A=[1 3 5; 2 4 6; 3 5 7]
row,column
A=
1
2
3
>> A(2:3, 1:2)
3
4
5
5
6
7
ans =
2 4
3 5
Means:
rows from 2 to 3, and
columns from 1 to 2,
referenced indices are:
(2,1)
(2,2)
(3,1)
(3,2)
28
Vectorization
29
Vectorization
• The term “vectorization” is frequently associated
with MATLAB.
• Means to rewrite code so that, instead of using a
loop iterating over each scalar-element in an
array, one takes advantage of MATLAB’s
vectorization capabilities and does everything in
one go.
• It is equivalent to change a Yaris for a Ferrari
30
Vectorization
Operations executed one by one:
Vectorized code:
x = [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ];
y = zeros(size(x)); % to speed code
x = [ 1 :1:10 ];
y = x.^3;
for k = 1:1:size(x)
y(k) = x(k)^3;
end
31
Vectorization
Operations executed one by one:
Vectorized code:
x = [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ];
y = zeros(size(x));
x = [ 1 :1:10 ];
y = sin(x);
for ii = 1:1:size(x)
y(ii) = sin(x(ii));
end
32
Vectorization
Operations executed one by one:
Vectorized code:
x = [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ];
y = zeros(size(x));
x = [ 1 :1:10 ];
y = sin(x)./x;
for ii = 1:1:size(x)
y(ii) = sin(x(ii))/x(ii);
end
33
Vectorization
Operations executed one by one:
WRONG Vectorization:
% 10th Fibonacci number (n=10)
% 10th Fibonacci number (n=10)
f(1)=0;
f(2)=1;
f(1)=0;
f(2)=1;
for k = 3:1:n
f(k) = f(k-1)+f(k-2);
end
k= [ 3 :1:n];
f(k) = f(k-1)+f(k-2);
CAN’T
34
Vectorization
Operations executed one by one:
% Find factorial of 5: 5!
x=[1:1:5];
p=1;
for ii = 1:1:length(x)
p=p*x(ii);
end
Wrong Vectorization: Why this
code doesn’t work?:
x=[1:1:5];
p(1)=1;
ii=2:1:length(x);
p(ii)=p(ii-1)*x(ii);
35
Vectorization-Exercise:
Vectorize the following loop:
Answer:
for ii=1:1:n+1
tn(ii)=(to(ii-1)+to(ii+1))/2;
end
ii=[1:1:n+1];
tn(ii)=(to(ii-1)+to(ii+1))/2;
Note: to, the old temperatures
array has been initialized
previously, i.e., all elements
already exist in memory
36
Matrix Operations
Follows linear algebra rules
37
Vector Multiplication
• Dot product (or inner product or scalar product)
•
•
•
•
Adding the product of each pair of respective elements in A and B
A must be a row vector
B must be a column vector
A and B must have same number of elements
38
Vector Multiplication
>> A=[1,5,6]
A=
1 5 6
>> B=[-2;-4;0]
B=
-2
-4
0
>> A*B
ans =
-22
1*(-2)+5*(-4)+6*0=-22
~ No period before the asterisk *
~ The result is a scalar
~ Compare this with array multiplication
39
Matrix Multiplication
• Compute the dot
products of each row
in A with each
column in B
• Each result becomes
a row in the resulting
matrix
A
B
A*B
m
AB  pij   aik bkj
k 1
( n x m) * ( m x p )  n x p
40
No commutative: AB≠BA
Matrix Multiplication
Math Syntax: AB
MATLAB Syntax: A*B (NO DOT)
>> A=[1 3 5; 2 4 6]
A=
1 3 5
2 4 6
>> B=[-2 4; 3 8; 12 -2]
B=
-2 4
3 8
12 -2
>> A*B
ans =
67
80
18
28
Sample calculation:
The dot product of row one of A and column one of B:
(1*-2)+(3*3)+(5*12)=67
41
Transpose
Columns become rows
42
Transpose
MATLAB:
>> A=[1,2,3;4,5,6;7,8,9]
1

A  4
7
1

T
A  2
3
2 3

5 6
8 9
4 7

5 8
6 9 
A=
1
4
7
2
5
8
3
6
9
4
5
6
7
8
9
>> A'
ans =
1
2
3
43
Determinant
• Transformation of a square matrix that results in a scalar
• Determinant of A: |A| or det A
• If matrix has single entry:
A=[3] det A = 3
44
Determinant
Example with matrix of order 2:
 a11 a12 
det 
 a11  a22  a21  a12

a21 a22 
>> A=[2,3;6,4]
A=
2 3
6 4
>> det(A)
ans =
-10
MATLAB instructions
45
Matrix Exponentiation
• A must be square:
A2=AA (matrix multiplication)
A3=AAA
MATLAB
>> A=[1,2;3,4]
A=
1 2
3 4
>> A^2
ans =
7 10
15 22
>> A^3
ans =
37 54
81 118
46
Operators Comparison
Array Operations
.*
./
.^

Matrix Operations
*
/
^
“+” and “-”
apply to both array and matrix operations and produce
same results
47
Operators Comparison
Array Operations
a=[1,2,3,4,5];
b=[5,4,3,2,1];
c=a.*b
Matrix Operations
a=[1,2,3,4,5];
b=[5,4,3,2,1];
c=a*b
 Find the results of the two statements
above, discuss the results
48
Operators Comparison
Array Operations
a=[1,2,3,4,5];
b=[5,4,3,2,1]’;
c=a.*b
Matrix Operations
a=[1,2,3,4,5];
b=[5,4,3,2,1]’;
c=a*b
 Find the results of the two statements
above, discuss the results
49
Operator Precedence
You can build expressions that use any combination of arithmetic, relational, and
logical operators. Precedence levels determine the order in which MATLAB
evaluates an expression. Within each precedence level, operators have equal
precedence and are evaluated from left to right. The precedence rules for MATLAB
operators are shown in this list, ordered from highest precedence level to lowest
precedence level:
•
•
•
•
•
•
•
•
•
•
•
Parentheses ()
Transpose (.'), power (.^), complex conjugate transpose ('), matrix power (^)
Unary plus (+), unary minus (-), logical negation (~)
Multiplication (.*), right division (./), left division (.\), matrix multiplication (*), matrix
right division (/), matrix left division (\)
Addition (+), subtraction (-)
Colon operator (:)
Less than (<), less than or equal to (<=), greater than (>), greater than or equal to (>=),
equal to (==), not equal to (~=)
Element-wise AND (&)
Element-wise OR (|)
Short-circuit AND (&&)
Short-circuit OR (||)
50
Built-in Matrix
Generators
To cop with arrays that are used very frequently
51
Zero Matrix
>> A=zeros(2)
If you specify one parameter,
it returns a square matrix of order 2
A=
0
0
0
0
>> A=zeros(2,4)
A=
0
0
0
0
0
0
0
0
If you specify 2 parameters,
It returns a 2 x 4 matrix
52
Ones Matrix
>> A=ones(3)
A=
1
1
1
>> A=ones(3,2)
column
A=
1
1
1
1
1
1
1
1
1
row
1
1
1
Same syntax as zeros matrix
53
Quiz
n=10;
ones(1,n+1)
output
ans =
54
Random function
• Generates an array
of pseudorandom
numbers whose
elements are
distributed in the
range [0,1]
A 2x3 matrix of random numbers:
>> A=rand(2,3)
A=
0.9501 0.6068 0.8913
0.2311 0.4860 0.7621
55
Identity Matrix: eye function
>> eye(3)
ans =
1
0
0
0
1
0
>> eye(2,3)
ans =
0
0
1
1
0
0
1
0
0
0
0
>> eye(4,3)
ans =
>> eye(4)
ans =
1
0
0
0
0
1
0
0
1
0
0
0
0
1
1
0
0
0
0
1
0
0
0
0
1
0
56
Useful Array
Functions
Better knowing their existance
57
Number of dimensions
>> A=[1,2;3,4;5,6]
A=
1
3
5
2
4
6
>> ndims(A)
ans =
2
>> B=ones(2,3,2)
B(:,:,1) =
1
1
1
1
1
1
B(:,:,2) =
1 1 1
1 1 1
>> ndims(B)
ans =
3
58
Size
Returns the length of each
dimensions of its argument
>> A=[1,2;3,4;5,6]
A=
1
3
5
2
4
6
>> size(A)
ans =
3
2
>> B=zeros(2,3,2,4)
>> size(B)
ans =
2 3 2 4
>> [m,n,s,t]=size(B)
m=
2
n=
3
s=
2
t=
4
59
Diagonal
Returns the
elements of the
main diagonal
Elements with
equal row and
column indices:
(1,1), (2,2), (3,3),
etc.
>> A=[1 3 5; 2 4 6; 0 2 4]
A=
1
2
0
3
4
2
5
6
4
>> diag(A)
ans =
1
4
4
60
Length
• Returns the length of the
largest dimension of an
array
Array is 3x2:
>> A=[1 3; 2 4; 0 2]
A=
1
2
0
3
4
2
>> length(A)
ans =
3
61
Sort
If a vector, the sort is in
ascending order
>> A=[4 2 3 9 1 2]
A=
4 2 3 9 1 2
>> sort(A)
ans =
1 2 2
3
4
9
If a 2-D array, it sorts each
column
>> A=[4 5 6; 7 8 9; 1 2 3]
A=
4 5 6
7 8 9
1 2 3
>> sort(A)
ans =
1 2 3
4 5 6
7 8 9
62
sort by
column
Sort
>> A=[4 6 5; 8 7 9; 1 3 2]
A=
4
8
1
6
7
3
5
9
2
>> sort(A,1)
ans =
1
4
8
3
6
7
2
5
9
>> sort(A,2)
ans =
4
7
1
5
8
2
sort by
row
6
9
3
63
Linear Systems of
Equations
Two ways
Matrix Division
Matrix Inverse
64
In general a system of m equations in n
unknowns can be written as:
a11 x1  a12 x2  ...  a1n xn  b1
a21 x1  a22 x2  ...  a2 n xn  b2

...
AX  B
am1 x1  am 2 x2  ...  amn xn  bm
In matrix form:
 a11 a12 ... a1n
 a a ... a
2n
A   21 22
 ...

am1 am 2 ... amn






 x1 
x 
X   2
 ... 
 
 xn 
 b1 
b 
B 2
 ... 
 
bm 
65
In general a system of m equations in n
unknowns can be written as:
a11x1  a12 x2  a1n xn  b1
a21x1  a22 x2  a2 n xn  b2

...
AX  B
am1 x1  am 2 x2 amn xn  bm
 a11 a12 a1n   x1 
 b1 
a a a  x 
b 
 21 22 2 n   2    2 
 ...
  ... 
 ... 

  
 
am1 am 2 amn   xn 
bm 
The solution to the linear system:
X=A\B (matrix left division)
66
Example:
3x1
x1
 5 x1
 2 x2
 2 x2
 10 x2
 x3  5
 3x3  13  AX  B
 5 x3  0
2
1
3
1

2
3


 5  10  5
A
 x1 
5
 x   13
 2
 
 x3 
 0 
X
B
67
X=A\B, the MATLAB solution
>> A=[3 2 1; 1 2 3; -5 -10 -5]
A=
>> X=A\B
X=
2.5000
-4.5000
6.5000
3 2 1
1 2 3
-5 -10 -5
>> B=[5;13;0]
B=
Verify the answer:
5
13
0
>> B= A*X <E>
B=
5
13
0
68
Matrix Inverse
IF
• A is the coefficient matrix
• X is the solution vector
• m = n, A is square matrix
i.e., number of rows equal the number of columns
• det A is non-zero,
Then
A-1 exist
69
Inverse
• Inverse is a square matrix such that
A-1A= I , the identity matrix
• The solution of the system is given by
A-1AX
= IX =
X=A-1B
• If A is order 3, the
identity matrix is also
order 3:
1 0 0


I  0 1 0 
0 0 1
70
Example
A system of 2 equations and 2 unknowns:
2x1- x2 = 2
>> A=[2 -1; 1 1]
x1+ x2 = 5
A=
2 -1
1 1
>> B=[2;5]
B=
2
5
>> X=inv(A)*B
X=
2.3333
2.6667
71
72
Logical Arrays and
Masks
Section 4.3 Textbook
73
Two possible values
Logical Data Type
True—(1)
False—(0)
74
Logical Arrays
Example:
n=10;
ii=[1:1:n+1];
c= mod(ii,2)==0
c=
0 1 0 1
0
1
0
1
Memory:
Memory:
ii(1)=1
c(1)=0
ii(2)=2
c(2)=1
ii(3)=3
c(3)=0
…
…
ii(11)=11
c(11)=0
0
1
0
% Produces a n+1-element Logical Array named c in which
elements are true (1) if ii is even and false (0) otherwise
75
Application-Midpoint Rule
𝑏=3 −𝑥
𝑒 2 (2𝑥
𝑎=0
−
𝑥2
)dx
2
clc, clear;
a = 0; b = 3; n = 100; h = (b-a)/n;
x = [a:h:b];
f=exp(-x./2).* (2.*x-x.^2./2);
ii=[1:1:n+1];
c = mod(ii,2)==0;
t=c.*f; I=2*h*sum(t);
Only f’s with coefficients equal to 1 survive
76
Logical Arrays
Example:
>> n = 12;
>> iia= [1:1:n+1];
>> coeff = 2*(mod(iia,3)==1)
coeff =
2
0
0
2
0
0
2
0
0
2 0 0 2
% Produces the Logical Array coeff in which their n+1 elements are true (1) if the
reminder of iia divided by 3 is one and false (0) otherwise
% This result could be adapted to solve Simpson 1/3 integration rule
77
Masks
• Logical arrays have a very important special property—they
serve as a mask for arithmetic operations.
• A mask is an array that selects particular elements of another
array for use in an operation
• The specified operation will be applied to the selected
elements, and not to the remaining elements
Mascaras sirven para “enmascarar” los elementos que no
queremos que entren en efecto
78
mask, example:
sqrt(a(b)) will take the square root of all
elements for which the logical array b is
true.
>> a=[4, 5, 6]
a=
4 5 6
>> b= a > 5
b=
0 0 1
b is a
logical
array
>> a(b)=sqrt(a(b))
a=
4.0000 5.0000 2.4495
a(b) in the LHS will affect only those
elements of a for which b is true.
a(b)=sqrt(a(b)) will replace in the original
a array only those elements that has
been square rooted. The instruction
doesnot affect the rest of the elements
(i.e., 4, and 5)
To understand these instructions after
defining a and b=a>5, run sequentially
sqrt(a(b))
and a(b)=sqrt(a(b))
79
mask, example:
>> a=[1 2 3; 4 5 6; 7 8 9]
a=
1 2 3
4 5 6
7 8 9
>> b=a>5
b=
b is a
0 0 0
logical
0 0 1
array
1 1 1
>> a(b)=sqrt(a(b))
sqrt(a(b)) will take the square root of all
elements for which the logical array b is
true.
a(b) in the LHS will affect only those
elements of a for which b is true.
a(b)=sqrt(a(b)) will replace in the original
a array only those elements that has
been square rooted. The instruction
doesnot affect the rest of the elements
(i.e., 1, 2, 3, 4, and 5)
a=
1.0000 2.0000 3.0000
4.0000 5.0000 2.4495
2.6458 2.8284 3.0000
To understand these instructions after
defining a,
and b=a>5, run sequentially
sqrt(a(b))
and a(b)=sqrt(a(b))
80
mask, example:
>> a=[1 2 3; 4 5 6; 7 8 9]
a=
1 2 3
4 5 6
7 8 9
>> b=a>5
b=
0 0 0
b is a
0 0 1
logical
1 1 1
To understand these instructions after
defining a,
and b=a>5, run sequentially
sqrt(a(b))
and a(b)=sqrt(a(b))
array
>> sqrt(a(b))
• ans =
•
•
•
•
2.6458
2.8284
2.4495
3.0000
81
With loops and if statement
for ii=1:1:3
for jj=1:1:3
if a(ii,jj)>5
a(ii,jj)= sqrt(a(ii,jj));
end
end
end
Please don’t be confused this is not a method by
logical arrays and masks. This just to show you
how difficult it is without logical arrays.
82
masks:
simpson rule example
Produce: c=[1 4 2 4 2 4 2 … 4 1]
SOLUTION-1
c=ones(1,n+1);
ii=[1:1:n+1];
b=mod(ii,2)==0;
c(b)=4*c(b);
bb=mod(ii,2)~=0;
c(bb)=2*c(bb);
c(1)=1;
c(n+1)=1;
(8 lines)
SOLUTION-2
c=ones(1,n+1);
ii=[1:1:n+1];
b=mod(ii,2)==0
c(b)=4*c(b);
c(~b)=2*c(~b);
c(1)=1;
c(n+1)=1;
(7 lines)
83
mask: simpson rule example
Produce: c=[1 4 2 4 2 4 2 … 4 1]
SOLUTION-3
c=ones(1,n+1);
b=mod(1:1:n+1,2)==0;
c(b)=4*c(b);
bb=mod(1:1:n+1,2)~=0;
c(bb)=2*c(bb);
c(1)=1;
c(n+1)=1;
(7 lines)
SOLUTION-4
c=ones(1,n+1);
b=mod(1:1:n+1,2)==0
c(b)=4*c(b);
c(~b)=2*c(~b);
c(1)=1;
c(n+1)=1;
(6 lines)
84
mask: simpson rule example
Produce: c=[1 4 2 4 2 4 2 … 4 1]
SOLUTION-5
c=ones(1,n+1);
b=mod(1:1:n+1,2)==0;
c(b)=4;
bb=mod(1:1:n+1,2)~=0;
c(bb)=2;
c(1)=1;
c(n+1)=1;
(7 lines)
SOLUTION-6
c=ones(1,n+1);
b=mod(1:1:n+1,2)==0
c(b)=4;
c(~b)=2;
c(1)=1;
c(n+1)=1;
(6 lines)
85
mask: simpson rule example
Produce: c=[1 4 2 4 2 4 2 … 4 1]
SOLUTION-7:
c=2*ones(1,n+1);
b=mod(1:1:n+1,2)==0
c(b)=4;
c(1)=1;
c(n+1)=1;
(5 lines)
SOLUTION-8:
c(3:2:n-1)=2;
c(2:2:n)=4
c(1)=1;
c(n+1)=1;
(4 lines)
86
Produce: c=[1 3 3 2 3 3 2… 3 3 1]
n=12;
c=3*ones(1,n+1); % also c(1:1:n+1)=3 % initially all are 3
ii=[1:1:n+1];
b=mod(ii,3)==1;
% also b=mod(1:1:n+1,3)==1
c(b)=(2/3)*c(b); % also c(b)=2/3
c(1)=1;
c(n+1)=1;
c % to print the results
87
Example
1. Create a 1000-elements array containing the
values, 1, 2,…, 1000. Then take the square
root of all elements whose value is greater
than 5,000 using a for loop and if construct
2. Create a 1000-elements array containing the
values, 1, 2,…, 1000. Then take the square
root of all elements whose value are smaller
than 5000 using a logical array & masks
88
Quiz-Solution
• Create a 10,000elements array
containing the
values, 1, 2,…,
10,000. Then take
the square root of all
elements whose
value is greater than
5,000 using a logical
array
• x=[1:1:10000];
• b=x>5000;
• x(b)=sqrt(x(b));
89
Quiz
Create a 100-elements array containing the values, 1, 2,…, 100.
Then take the square of all elements whose values are
between 50 and 75 using logical arrays
Solution
ii=[1:1:100];
b=ii>50 & ii<75;
ii(b)=ii(b).^2;
90
Example
We want to take the square root of any element in a twodimensional array “a” whose value is greater than 5 , and to
square the remaining elements in the array. The code for this
operation using loops and branches is shown in the following
slide
1 3
• 𝑎= 7 8
5 2
4 7
2 3
9 6
91
Solution
% Square rooted and squared elements
% Using loops and if constructs
clc; clear;
a=[1,3,4,7;7,8,2,3;5,2,9,6]
Please don’t be confused this is not a method by
for ii=1:1:3
logical arrays and masks. This just to show you
for jj=1:1:4
how difficult it is without logical arrays.
if a(ii,jj)>5
a(ii,jj)=sqrt(a(ii,jj))
else
a(ii,jj)=a(ii,jj)^2
end
end
end
92
Solution
% Square rooted and squared elements
% Using logical arrays and masks
clc; clear;
a=[1,3,4,7;7,8,2,3;5,2,9,6]
b= a>5;
a(b)=sqrt(a(b));
a(~b)=a(~b).^2;
a
93