MATLAB Arrays - Computational Methods - Lecture Slides, Slides of Calculus for Engineers

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Engr/Math/Physics 25

Chp2 MATLAB

Arrays: Part-

Learning Goals

• Learn to Construct 1-Dimensional

Row and Column Vectors

• Create MULTI-Dimensional

ARRAYS and MATRICES

• Perform Arithmetic Operations on Vectors

and Arrays/Matrices

• Analyze Polynomial Functions

ROW & COLUMN Vectors

• To create a ROW vector, separate the elements by COMMAS

 Use the Transpose operator (‘) to make a COLUMN Vector

p = [ 3 , 7 , 9 ]  Create Col-Vector Directly with SEMICOLONS

>>p = [3,7,9] p = 3 7 9

 The TRANSPOSE Operation Swaps Rows↔Columns

>>p = [3,7,9]' p = 3 7 9

p

>>g = [3;7;9] g = 3 7 9

“Appending” Vectors

• Can create a NEW vector by ''appending'' one vector to another
• e.g.; to create the row vector u whose first three columns contain the values of r = [2,4,20] and whose 4th, 5 th, & 6 th^ columns contain the values of w = [9,-6,3] - Typing u = [r,w] yields vector u = [2,4,20,9,-6,3] >> r = [2,4,20]; w = [9,-6,3]; >> u = [r,w] u = 2 4 20 9 -6 3

Colon (:) Operator cont.

• To create a row vector z consisting of the values from 5 to 8 in steps of 0.1, type z = [5:0.1:8].
• If the increment q is OMITTED, it is taken as the DEFAULT Value of +1.

>> s =[-11:-6] s = -11 -10 -9 -8 -7 - >> t = [-11:1:-6] t = -11 -10 -9 -8 -7 -

linspace Comand

• The linspace command also creates a linearly spaced row vector, but instead you specify the number of values rather than the increment
• The syntax is linspace(x1,x2,n), where x and x2 are the lower and upper limits and n is the number of points - If n is omitted, then it Defaults to 100
• Thus EQUIVALENT statements

>> linspace(5,8,31) = >> [5:0.1:8]

logspace Example

• Consider this command >> y =logspace(-1,2,6) y = 0.1000 0.3981 1.5849 6.3096 25.1189 100.

 Calculate the Power of 10 increment

∆ p = ( b − a ) ( n − 1 )

 In this case

( ( )) ( ) 3 5 0. 6

2 1 6 1 = =

∆ p = − − −

= 10 (^ a +∆ p^ (^ k −^1 ))

yk

 In this Case k Power yk 1 -1 0. 2 -0.4 0. 3 0.2 1. 4 0.8 6. 5 1.4 25. 6 2 100.

 for k = 1→6, then the k th^ y-value

logspace Example (alternative)

• Consider this command >> y =logspace(-1,2,6) y = 0.1000 0.3981 1.5849 6.3096 25.1189 100.

 Take base-10 log of the above

 Calc Spacing Between Adjacent Elements with diff command

>> yLog10 = log10(y) yLog10 = -1.0000 -0.4000 0.2000 0.8000 1.4000 2.

>> yspc = diff(yLog10) yspc = 0.6000 0.6000 0.6000 0.6000 0.

∆ p = ( b − a ) ( n − 1 ) = ( 2 −[− 1 ]) ( 6 − 1 ) = 3 5

Mag, Length, and Abs-Val

• Geometric Length

 By Pythagoras Find vector a magnitude

2 2 2 2

2 2 2

(^2 22) and

a x y z

a w z

w x y

= + +

= +

= + so

a = x^2 + y^2 + z^2

 Thus the box diagonal

>> a =[2,-4,5]; >> length(a) ans = 3 >> % the Magnitude M = norm(a) >> Mag_a = norm(a) Mag_a =

>> abs(a) ans = 2 4 5

3D Vector Length Example

% Bruce Mayer * ENGR % 25Aug09 * Lec % file = VectorLength_0908.m % NOTE: using “norm” command is much easier % % Find the length of a Vector AB with %% tail CoOrds = (0, 0, 0) %% tip CoOrds = (2, 1, -5) % % Do Pythagorus in steps vAB = [2 1 -5] % define vector sqs = vAB.*vAB % sq each CoOrd sum_sqs = sum(sqs) % Add all sqs LAB = sqrt(sum_sqs) % Take SQRT of the sum- of-sqs

Matrices/Arrays

• A MATRIX has multiple ROWS and COLUMNS. For example, the matrix M :

























=

3 12 15

8 4 9

16 3 7

2 4 10

M

 VECTORS are SPECIAL CASES of matrices having ONE ROW or ONE COLUMN.

 M contains

• FOUR rows
• THREE columns

Making Matrices

• For a small matrix you can type it row by row, separating the elements in a given row with spaces or commas and separating the rows with semicolons. For example, typing

 

  



16 3 7

2 4 10 A

 spaces or commas separate elements in different columns , whereas semicolons separate elements in different rows.

 The Command

• >>A = [2,4,10;16,3,7];  Generates Matrix

Directly Make Matrix

• Use COMMAS/SPACES combined with SEMICOLONS to Construct Matrices

>> D = [[1,3,5]; [7 9 11]] D = 1 3 5 7 9 11

 Note the use of

• BOTH Commas and Spaces as COLUMN-Separators
• The SEMICOLON as the ROW-Separator

Make Matrix Example

>> r1 = [1:5] r1 = 1 2 3 4 5 >> r2 = [6:10] r2 = 6 7 8 9 10 >> r3 = [11:15]r3 = 11 12 13 14 15 >> r4 = [16:20]r4 = 16 17 18 19 20

r5 =>> r5 = [21:25]

>> A = [r1;r2;r3;r4;r5]^21 22 23 24 A = 1 2 3 4 5

116 127 138 149 1015 (^1621 1722 1823 1924 )