array and matrix operations
DESCRIPTION
Array and Matrix Operations. Dr. Marco A. Arocha INGE3016-MATLAB Sep 11, 2007, Dic 7, 2012. Array Operations. Array Addition. With 1D arrays : >> A=[1 3 5 ]; >> B=[2 4 6]; >> A+B ans = 3 7 11. With 2D arrays : >> A=[1 3 5; 2 4 6] A = 1 3 5 - PowerPoint PPT PresentationTRANSCRIPT
Array and Matrix OperationsDr. Marco A. ArochaINGE3016-MATLABSep 11, 2007, Dic 7, 2012
1
2
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
Array Operations
3
Assume the following arrays of length n:
]...[ 321 naaaaA ]...[ 321 nbbbbB
Addition of arrays of the same length is defined as:
],...,,[ 2211 nn bababaBA
Subtraction of arrays of the same length:
],...,,[ 2211 nn bababaBA
Multiplication of arrays of the same length: ‘.*’
]*,...,*,*[*. 2211 nn bababaBA
Division of arrays of the same length: ‘./’
]/,...,/,/[/. 2211 nn bababaBA
Exponentiation of arrays of the same length ‘ .^’ A.^B=[a1^b1, a2^b2, a3^b3,…,an^bn]
Array AdditionWith 1D arrays:
>> A=[1 3 5 ];>> B=[2 4 6];>> A+B
ans =
3 7 11
With 2D arrays:
>> A=[1 3 5; 2 4 6]A = 1 3 5 2 4 6
>> B=[-5 6 10; 2 0 9]B = -5 6 10 2 0 9>> A+Bans = -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.*Bans = 2 9 20 -2 -8 -18
5
Arrays must be of the same size
Array Division>> A=[2,4,6]A = 2 4 6>> B=[2,2,2]B = 2 2 2>> A./B % num/den Right divisionans = 1 2 3>> A.\B % den\num Left divisionans = 1.0000 0.5000 0.3333
6
Array Exponentiation
>> A=[2,4,6]A = 2 4 6>> B=[2,2,2]B = 2 2 2>> B.^Aans = 4 16 64
7
Special Cases: array <operator> scalarscalar<operator> array
>> A+2ans = 3 4 5
>> A-1ans = 0 1 2
8
>> A.*5ans = 5 10 15
>> A./2ans 0.5 1.0 1.5
If one of the arrays is a scalar the following are valid expressions. Given: >> A=[1 2 3];
Dot is optional in the above two examples
Special Cases: array <operator> scalarscalar<operator> array
>> a*2ans = 2 4 6>> a.*2ans = 2 4 6
>> a/2ans = 0.5000 1.0000 1.5000
>> a./2ans = 0.5000 1.0000 1.5000
9
Given: a=[1 2 3]
If one of the arrays is a scalar the following are valid expressions:
Special Cases>> A=[5]A = 5
>> B=[2,4,6]B = 2 4 6
>> A.*Bans =
10 20 3010
Period is optional here
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:
12
Rectangulararray
m, rows
n, columns
Square Matrix
• m=n
321
753
642
A
13
Square matrix of order three
Z=3*A(2,3)
Can reference individual elements
Main diagonal:[2,5,3], i.e, Ai,j where i=j
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-dimensioningUpon appending one more element to an array, MATLAB
automatically resizes the array to handle the new element
>> a=[2 3 4] % a contains 3 elementsa = 2 3 4>> a(4)=6 % now a contains 4 elementsa = 2 3 4 6>> a(5)=7 % now a contains 5 elementsa = 2 3 4 6 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 4 1 2 3 4>> c=[a; 2*b]c = 1 2 3 4 2 4 6 8 2 4 6 8
16
Self-dimensioning is aMATLAB 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
18
Deletes row-2, all column elements
Storage Mechanism for Arrays
19
Storage mechanism for arrays
A = 1 3 5 2 4 6 3 5 7
20
Two common ways of storage mechanism,depending on language:
• One row at a time: row-major order (*)1 3 5 2 4 6 3 5 7
• One column at a time: column-major order1 2 3 3 4 5 5 6 7
Last one is the MATLAB way of array storage
(*) C Language uses row-major order
col-2
Row-2 Row-3
col-1 col-3
Row-1
Accessing Individual Elements of an Array
>> A=[1 3 5; 2 4 6; 3 5 7]
A =
1 3 5 2 4 6 3 5 7
>> A(2,3)% row 2, column 3
ans =
6 21
Two indices is the usual way to access an element
Accessing elements of an Array by a single subscript>> A=[1 3 5; 2 4 6; 3 5 7]
A =
1 3 5 2 4 6 3 5 7
In memory they are arranged as:
1 2 3 3 4 5 5 6 7
If we try to access them with only one index, e.g.:
>> A(1)ans = 1>> A(4)ans = 3>> A(8)ans = 6
22
Recall: column-major order in memory
Accessing Elements of an Array by a Single Subscript>> A=[1 3 5; 2 4 6; 3 5 7]
A =
1 3 5 2 4 6 3 5 7
With one index & colon operator:>> A(1:2:9)ans = 1 3 4 5 7The 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)
23
In memory
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
Example
Add one unit to each element in A:
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:
• A(1:1:9)=A(1:1:9)+1;
Answer-3:
• A=A(1:1:9)+1;• % one index
Answer-4:
• A=A.*2;
Answer-5:
• A=A+1;
25
Exercise
Initialize this Matrix with one index:
for k =1:1:25 if mod(k,6)==1 A(k)='F'; % ‘F’ elements are in indices: 1, 7, 13, 19, and 25 else A(k)='M'; endend
% 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
With one index,Referencing is OK, Initializing is not.
A= F M M M MM F M M MM M F M MM M M F MM M M M F
Accessing Elements of an Array
>> A=[1 3 5; 2 4 6; 3 5 7]
A =
1 3 5 2 4 6 3 5 7
>> A(2,:) ans = 2 4 6
(2, :) means row 2, all columns
27
A colon alone “ : “ means all the elements of that dimension
Accessing Elements of an Array
>> A=[1 3 5; 2 4 6; 3 5 7]
A =
1 3 5 2 4 6 3 5 7
>> A(2:3, 1:2)
ans = 2 4 3 5Means:
rows from 2 to 3, andcolumns from 1 to 2, referenced indices are:
(2,1)(2,2)(3,1)(3,2)
28
row,column
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 Ferrari30
VectorizationOperations executed one by one:
x = [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ];y = zeros(size(x)); % to speed code
for k = 1:1:size(x)y(k) = x(k)^3;
end
Vectorized code:
x = [ 1 :1:10 ];y = x.^3;
31
VectorizationOperations executed one by one:
x = [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ];y = zeros(size(x));
for ii = 1:1:size(x)y(ii) = sin(x(ii));
end
Vectorized code:
x = [ 1 :1:10 ];y = sin(x);
32
VectorizationOperations executed one by one:
x = [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ];y = zeros(size(x));
for ii = 1:1:size(x)y(ii) = sin(x(ii))/x(ii);
end
Vectorized code:
x = [ 1 :1:10 ];y = sin(x)./x;
33
VectorizationOperations executed one by one:
% 10th Fibonacci number (n=10)
f(1)=0;f(2)=1;
for k = 3:1:nf(k) = f(k-1)+f(k-2);
end
WRONG Vectorization:
% 10th Fibonacci number (n=10)
f(1)=0;f(2)=1; k= [ 3 :1:n];f(k) = f(k-1)+f(k-2);
CAN’T
34
VectorizationOperations 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:
for ii=1:1:n+1 tn(ii)=(to(ii-1)+to(ii+1))/2;end
Note: to, the old temperatures array has been initialized previously, i.e., all elements
already exist in memory
Answer:
ii=[1:1:n+1];tn(ii)=(to(ii-1)+to(ii+1))/2;
36
Matrix OperationsFollows 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*Bans = -22
39
~ No period before the asterisk *~ The result is a scalar~ Compare this with array multiplication
1*(-2)+5*(-4)+6*0=-22
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
m
kkjikij bapAB
1 pnpmmn x)x(*)x (
40
A B A*B
No commutative: AB≠BA
Matrix MultiplicationMath Syntax: ABMATLAB Syntax: A*B (NO DOT)
>> A=[1 3 5; 2 4 6]A = 1 3 5 2 4 6
41
Sample calculation:The dot product of row one of A and column one of B:(1*-2)+(3*3)+(5*12)=67
>> A*Bans =
67 18 80 28
>> B=[-2 4; 3 8; 12 -2]B = -2 4 3 8 12 -2
Transpose
42Columns become rows
Transpose
963
852
741
987
654
321
TA
A
MATLAB:>> A=[1,2,3;4,5,6;7,8,9]
A =
1 2 3 4 5 6 7 8 9
>> A'
ans =
1 4 7 2 5 8 3 6 9
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:
122122112221
1211det aaaaaa
aa
45
>> A=[2,3;6,4]A = 2 3 6 4
>> det(A)ans = -10
MATLAB instructions
Matrix Exponentiation
• A must be square:A2=AA (matrix multiplication)A3=AAA
MATLAB>> A=[1,2;3,4]A = 1 2 3 4>> A^2ans = 7 10 15 22>> A^3ans = 37 54 81 118
46
Operators Comparison
Array Operations
.*./.^
Matrix Operations
*/^
47
“+” and “-” apply to both array and matrix operations and produce same results
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
48
Find the results of the two statements above, discuss the results
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
49 Find the results of the two statements
above, discuss the results
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 GeneratorsTo cop with arrays that are used very frequently
51
Zero Matrix
>> A=zeros(2)
A =
0 0 0 0
>> A=zeros(2,4)
A =
0 0 0 0 0 0 0 0
52
If you specify one parameter,it returns a square matrix of order 2
If you specify 2 parameters,It returns a 2 x 4 matrix
Ones Matrix
>> A=ones(3)
A =
1 1 1 1 1 1 1 1 1
>> A=ones(3,2)
A =
1 1 1 1 1 1
53Same syntax as zeros matrix
row
column
Quiz
n=10;ones(1,n+1)
outputans =
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 0 0 1
>> eye(4)ans =
1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
>> eye(2,3)ans =
1 0 0 0 1 0
>> eye(4,3)ans =
1 0 0 0 1 0 0 0 1 0 0 0
56
Useful Array FunctionsBetter knowing their existance
57
Number of dimensions>> A=[1,2;3,4;5,6]A =
1 2 3 4 5 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
SizeReturns the length of each
dimensions of its argument
>> A=[1,2;3,4;5,6]A =
1 2 3 4 5 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 = 2n = 3s = 2t = 4
59
DiagonalReturns 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 3 5 2 4 6 0 2 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 3 2 4 0 2
>> length(A)ans =
361
SortIf 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>> A=[4 6 5; 8 7 9; 1 3 2]A =
4 6 5 8 7 9 1 3 2
>> sort(A,1)ans =
1 3 2 4 6 5 8 7 9
>> sort(A,2)ans =
4 5 6 7 8 9 1 2 3
63
sort by column
sort by row
Linear Systems of Equations
Matrix DivisionMatrix Inverse
64
Two ways
65
BAX2
1
2
1
22
222
212
11
121
111
...
...
...
...
mnmn
nn
nn
mm b
b
b
xa
xa
xa
xa
xa
xa
xa
xa
xa
mn
n
n
mm a
a
a
a
a
a
a
a
a
A 2
1
2
22
12
1
21
11
...
...
...
...
nx
x
x
X...
2
1
mb
b
b
B...
2
1
In general a system of m equations in n unknowns can be written as:
In matrix form:
66
BAX2
1
2
1
22
222
212
11
121
111
...
mnmn
nn
nn
mm b
b
b
xa
xa
xa
xa
xa
xa
xa
xa
xa
mn
n
n
mm a
a
a
a
a
a
a
a
a
2
1
2
22
12
1
21
11
...
nx
x
x
...2
1
mb
b
b
...2
1
In general a system of m equations in n unknowns can be written as:
The solution to the linear system:X=A\B (matrix left division)
67
5105
321
123
3
2
1
x
x
x
0
13
5
Example:
BAX
xxx
xxx
xxx
0
13
5
5105
32
23
321
321
321
A X B
X=A\B, the MATLAB solution>> A=[3 2 1; 1 2 3; -5 -10 -5]A =
3 2 1 1 2 3 -5 -10 -5
>> B=[5;13;0]B =
5 13 0
>> X=A\BX = 2.5000 -4.5000 6.5000
68
Verify the answer:
>> B= A*X <E>
B = 5 13 0
Matrix Inverse
• A is the coefficient matrix• X is the solution vector• m = n, A is square matrixi.e., number of rows equal the number of columns
• det A is non-zero,
69
IFThen
A-1 exist
Inverse
100
010
001
I
• 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
70
• If A is order 3, the identity matrix is also order 3:
ExampleA system of 2 equations and 2 unknowns:2x1- x2 = 2
x1+ x2 = 5
71
>> A=[2 -1; 1 1]A = 2 -1 1 1>> B=[2;5]B = 2 5>> X=inv(A)*BX = 2.3333 2.6667
72
Logical Arrays and MasksSection 4.3 Textbook
73
Two possible values
Logical Data Type
74
True—(1)
False—(0)
Logical Arrays
Example:
n=10;ii=[1:1:n+1];c= mod(ii,2)==0c= 0 1 0 1 0 1 0 1 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
Memory:
ii(1)=1
ii(2)=2
ii(3)=3
…
ii(11)=11
Memory:
c(1)=0
c(2)=1
c(3)=0
…
c(11)=0
Application-Midpoint Rule
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);
76Only f’s with coefficients equal to 1 survive
)dx
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
78
Mascaras sirven para “enmascarar” los elementos que no queremos que entren en efecto
mask, example:>> a=[4, 5, 6]a = 4 5 6
>> b= a > 5b = 0 0 1
>> a(b)=sqrt(a(b))a = 4.0000 5.0000 2.4495
79
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., 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))
b is alogicalarray
mask, example:>> a=[1 2 3; 4 5 6; 7 8 9]a = 1 2 3 4 5 6 7 8 9>> b=a>5b = 0 0 0 0 0 1 1 1 1
>> a(b)=sqrt(a(b))
a =
1.0000 2.0000 3.0000 4.0000 5.0000 2.4495 2.6458 2.8284 3.0000
80
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)
To understand these instructions after defining a,and b=a>5, run sequentially sqrt(a(b))and a(b)=sqrt(a(b))
b is alogicalarray
mask, example:>> a=[1 2 3; 4 5 6; 7 8 9]a = 1 2 3 4 5 6 7 8 9>> b=a>5b = 0 0 0 0 0 1 1 1 1
>> sqrt(a(b))
• ans =
• 2.6458• 2.8284• 2.4495• 3.0000
81
To understand these instructions after defining a,and b=a>5, run sequentially sqrt(a(b))and a(b)=sqrt(a(b))
b is alogicalarray
With loops and if statement
for ii=1:1:3for jj=1:1:3
if a(ii,jj)>5a(ii,jj)= sqrt(a(ii,jj));
endend
end
82
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.
masks: simpson rule example
SOLUTION-1c=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-2c=ones(1,n+1);ii=[1:1:n+1];b=mod(ii,2)==0c(b)=4*c(b);c(~b)=2*c(~b); c(1)=1;c(n+1)=1;(7 lines)
83
Produce: c=[1 4 2 4 2 4 2 … 4 1]
mask: simpson rule example
SOLUTION-3c=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-4c=ones(1,n+1);b=mod(1:1:n+1,2)==0c(b)=4*c(b);c(~b)=2*c(~b); c(1)=1;c(n+1)=1;(6 lines)
84
Produce: c=[1 4 2 4 2 4 2 … 4 1]
mask: simpson rule example
SOLUTION-5c=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-6c=ones(1,n+1);b=mod(1:1:n+1,2)==0c(b)=4;c(~b)=2; c(1)=1;c(n+1)=1;(6 lines)
85
Produce: c=[1 4 2 4 2 4 2 … 4 1]
mask: simpson rule example
SOLUTION-7:
c=2*ones(1,n+1);b=mod(1:1:n+1,2)==0c(b)=4;c(1)=1;c(n+1)=1;(5 lines)
86
Produce: c=[1 4 2 4 2 4 2 … 4 1]
SOLUTION-8:
c(3:2:n-1)=2;c(2:2:n)=4c(1)=1;c(n+1)=1;(4 lines)
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 3ii=[1:1:n+1];b=mod(ii,3)==1; % also b=mod(1:1:n+1,3)==1c(b)=(2/3)*c(b); % also c(b)=2/3c(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 10,000-
elements 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
Solutionii=[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 two-dimensional 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
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] for ii=1:1:3 for jj=1:1:4 if a(ii,jj)>5 a(ii,jj)=sqrt(a(ii,jj)) else a(ii,jj)=a(ii,jj)^2 end endend 92
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.
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