Transcript Array types: - University of Cincinnati

Array types:
numeric
character
logical
cell
structure
function handle
Java
MATrixLABoratory
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The name says it all: Matlab is designed
for work with matrices
A matrix is an arrangement of numbers
in a rectangular pattern
An m by n matrix has
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m rows
n columns
mn elements of (floating point) numbers
Example: Magic square
16
2
3
13
5
11
10
8
9
7
6
12
4
14
15
1
A = magic(4)
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Each row adds to 34
Each colum adds to 34
Each diagonal adds to 34
A(1,1) element in upper left corner
A(i,j) element in i-th row and j-th column
Note: Numbering starts with 1 (and not with 0, as
some other programming languages do)
Starting with 1 is used frequently in mathematics
Starting with 0 is more natural for implementation in
a computer
Referencing elements of an
array
A(1,1) A(1,2) A(1,3) A(1,4)
A(2,1) A(2,2) A(2,3) A(2,4)
A(3,1) A(3,2) A(3,3) A(3,4)
A(4,1) A(4,2) A(4,3) A(4,4)
Creating matrices
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B=[16,2,3,13;5,11,10,8;9,7,6,12;4,14,15,1]
Values are given as integers, but stored as
floating point numbers
Elements of a row are separated by a space
or a comma
Rows are separated by semicolon
Number of elements have to be the same in
each row
Special Matrices
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Row vector (1 by n matrix)
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Column vector (m by 1 matrix)
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a=[1,2,3,4]
b=[1;2;3;4]
Transpose of a matrix - Interchange
rows with columns
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c = a'
c is now the same as b
Matrix A and its transpose A'
1
2
3
1
4
7
10
4
5
6
2
5
8
11
7
8
9
3
6
9
12
10
11
12
Referencing elements of an
array (matrix)
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A= magic(4) 4 by 4 matrix
A(2,3) specific element in row 2
A(2, 2:4) row vector from row 2 of A with
elements 2 to 4
A(1:3,3) column vector from column 3 of A
with elements 1 to 3
A(1:2:3,4) column vector from column 4 of A
with elements 1 and 3
A(:,3) entire column 3 of A
A(1:2, 2:3) submatrix of A
Indexing
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via a specific value
range of values created via
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start:inc:stop
default for inc is 1
if value for stop is not known can use
keyword 'end' instead, which will refer to
largest value for that index
: refers to all defined values for that index
Array operations
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Matlab allows use of index notation on the right and
also on the left hand side of an assignment
statement
Most other programming languages are much more
restrictive
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in C++ A[2][3] references an element
but A[2] gives the address of row 2, which can not be
changed
Since Matlab extends arrays automatically need to
watch out for referencing undefined elements:
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A=magic(4) ; A(2,3:5)=[21,47,45]
is ok in Matlab
Functions operating on vectors
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v vector (row or column)
max(v) returns largest element in v
min(v) returns smallest element in v
find(v) returns indices of nonzero elements in v
sort(v) returns vector sorted in ascending order
sum(v) returns sum of the elements in v
length(v)
size(v) gives dimensions of the array v, i.e.
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or
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1 length(v)
for a row vector
length(v) 1
for a column vector
Representation of arrays in
Matlab
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A = [2,
A=
2
4
3
6, 9; 4, 2, 8; 3, 5, 1]
6
2
5
9
8
1
is actually stored in memory as the sequence
2, 4, 3, 6, 2, 5, 9, 8, 1
that is in column major form, as done by Fortran
Most other programming languages use row major
form, i.e. order in which elements are entered
Functions applied to matrices
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A
m , n matrix
max(A), min(A), sort(A), sum(A) return row vectors with
operation performed on each column
find(v) returns linear indices of nonzero elements in A
linear index of element A(i,j) is (j-1)m + i
[u,v,w] = find(A) gives a more useful result
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u vector for row indices
v vector for column indices
w vector with non-zero elements
length(A) returns maximum of m and n
size(A) returns dimension of A, that is m n
Other methods for creating a
vector (one dimensional array)
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x = [ -10 : 3 : 10 ]
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y = [ 10 : -3 : -10]
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same as xx = [5 : 0.3333333333333333 : 10]
logspace( a, b, n)
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n number of points (values) between x1 and x2 inclusive endpoints
x1 and x2
xx = linspace( 5, 8, 10 )
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default increment is 1
linspace( x1, x2, n )
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same as y =[ 10, 7, 4, 1, -2, -5, -8]
z = [ 0 : 10 ]
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same as x = [-10 , -7, -4, -1, 2, 5, 8 ]
n number of points between 10a and 10b includes endpoints
xlog = logspace( 1, 2, 10 )
logspace
program that does the same
a=1
b=2
n = 10
inc = 10^((b – a ) / n )
myxlog(1) = 10^a
for k = 2 : n
myxlog(k) = myxlog(k-1) * inc
end
Remarks
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If the program is converted to a script do not
call it logspace.m
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otherwise original command is no longer available.
removing file logspace.m from directory causes
also problems since Matlab maintains history of
functions used
information kept in tool_path_cache
clear myxlog
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should be at the beginning of the script
old values from a previous use of mylogspace will
still be visible
Remarks
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Scalars are arrays of length 1
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k=1
k(1) is defined but not k(2)
Arrays (vectors and matrices) can be
extended by assigning an array element
outside the current bounds
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any missing elements will be set to 0
k(4) = 23
the array k has now elements [1, 0, 0, 23]
Remarks
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A = [1,2,3 ; 4,5,6]
A(1,5) = 23
gives new 3 by 5 matrix
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1 2 3 0 23
4 5 6 0 0
Most programming languages do not allow the extension of an
existing array. For these languages
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Size of the array has to declared at the beginning
Exceeding the bounds of an array causes a run time error (not a
compile time error)
Some programming languages i.e. Java will detect this type of error
and terminate the run with an error message
Other programming languages i.e. C and C++ will overwrite an
adjacent storage location with unpredictable results
Multi dimensional arrays
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A(3,2,4) element of a three dimensional array
Note how Matlab fills in missing elements and display
new array
Information about workspace variables is updated
automatically
Creating multi dimensional arrays
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by assigning individual elements
or with notation cat(n, A, B, C,…)
where A, B, C,…all are arrays of the same size
n the dimension where the matrices are catenated
example cat(1,A,B) cat(2,A,B) cat(3,A,B)
Use of vectors in geometry
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Common operations
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vector addition:
u+v
vector subtraction: u-v
multiplication with a scalar r: ru
length of a vector |u|
dot product: uv =|u| |v| cos()
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with  angle between u and v
dot product multiplies elements of u and v pairwise and
adds them together
uu = |u|2
Use of matrices in geometry
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defines linear transformation y = A x
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x column vector of length n
y column vector of length m
A matrix of size m by n
for the multiplication A  x take each row of
A and form the dot product with x
Note the requirement that the number of
elements in a row of A has to be the same as
the number of elements in the column of x
Operations on arrays in Matlab
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Matlab provides many different operations for
arrays (vectors, matrices, multi dimensional
arrays)
Some of the operations have a geometric
interpretation, others do not
If there are certain requirements for
performing on operation it comes from
conforming to the geometric interpretation of
the operation
See course on 'Matrix Methods'
Element by element operations
for arrays of the same size
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Scalar b added to array A: A + b
Scalar b subtracted from array A: A – b
Addition of two arrays: A + B
Subtraction of two arrays: A – B
Multiplication of two arrays: A . B
Right division of two arrays: A ./ B
Left division of two arrays: A .\ B
Array exponentiation: A .^ B
Remarks
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The last 4 operations have no geometric
interpretation
The first two operations are mathematically
incorrect, b should be vector with all of its
elements set to b
Matlab allows standard functions to operate
on each element of an array and produces an
array of the same size
For example sqrt(A) or cos(A), again the
resulting matrices do not conform to the
mathematical definition of sqrt(A) or cos(A)
Array operations in Matlab
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Geometric interpretation not needed, when
evaluating functions at many points
t=[ 0 : 0.003 : 0.5 ] ;
array t(1)=0 to t(167)=0.498
y = exp(-8 * t ) .* sin(9.7 * t +pi/2) ;
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exp(-8*t) evaluates to an array of length 167
sin(9.7*t+pi/2) evaluates to an array
in order to multiply individual terms of the two
arrays need to use .*
General Comments for Plotting
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Use … in order to continue a statement on
next line
Number of points for plotting should be
enough to give a smooth graph
but not so many so that it would take too
long to evaluate functions and exceed
available memory
labels for each axis should give dimension
Current and power dissipation
in resistors
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current: I
volt: V
resistance: R
Ohm's law I = V/R
Power dissipation Watts: I*V=I2/R
in Matlab given arrays for V and R
need to use .^ and ./
watts = V.^2./R
In order to make statement clearer use parentheses
watts = (V.^2) ./ R
Solving equations by graphical
means
x 0.625 1 x 1.625
0.4
0.6
L( )
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(
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Given L = 70 find approximate value for x
[x,y]=ginput(n)
graphical input from mouse and cursor
option to plot, allows fixing of n coordinate
lines with the click of the mouse