Array types:

numericcharacter

logicalcell

structurefunction handle

Java

MATrixLABoratory 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

m rows n columns mn 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) 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 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 Row vector (1 by n matrix)

a = [ 1 , 2 , 3 , 4 ] Column vector (m by 1 matrix)

b = [ 1 ; 2 ; 3 ; 4 ] Transpose of a matrix -

Interchange rows with columns c = a' c is now the same as b

Matrix A and its transpose A'

1 2 3

4 5 6

7 8 9

10 11 12

1 4 7 10

2 5 8 11

3 6 9 12

Referencing elements of an array (matrix) 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 via a specific value range of values created via

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 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

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:

A=magic(4) ; A(2,3:5)=[21,47,45] is ok in Matlab

Functions operating on vectors 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.

1 length(v) for a row vector or

length(v) 1 for a column vector

Representation of arrays in Matlab A = [2, 6, 9; 4, 2, 8; 3, 5, 1] A = 2 6 9 4 2 8 3 5 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

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

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) x = [ -10 : 3 : 10 ]

same as x = [-10 , -7, -4, -1, 2, 5, 8 ] y = [ 10 : -3 : -10]

same as y =[ 10, 7, 4, 1, -2, -5, -8] z = [ 0 : 10 ]

default increment is 1 linspace( x1, x2, n )

n number of points (values) between x1 and x2 inclusive endpoints x1 and x2

xx = linspace( 5, 8, 10 ) same as xx = [5 : 0.3333333333333333 : 10]

logspace( a, b, n) n number of points between 10a and 10b includes

endpoints xlog = logspace( 1, 2, 10 )

logspaceprogram that does the same

a = 1 b = 2n = 10inc = 10^((b – a ) / n )myxlog(1) = 10^afor k = 2 : n myxlog(k) = myxlog(k-1) * inc end

Remarks If the program is converted to a script

do not call it logspace.m 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

should be at the beginning of the script old values from a previous use of

mylogspace will still be visible

Remarks Scalars are arrays of length 1

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 any missing elements will be set to 0 k(4) = 23 the array k has now elements [1, 0, 0, 23]

Remarks A = [1,2,3 ; 4,5,6] A(1,5) = 23 gives new 3 by 5 matrix

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

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 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

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 Common operations

vector addition: u+v vector subtraction: u-v multiplication with a scalar r: ru length of a vector |u| dot product: uv =|u| |v| cos()

with angle between u and v dot product multiplies elements of u and v

pairwise and adds them together uu = |u|2

Use of matrices in geometry defines linear transformation y = A x

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 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

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 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 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) ;

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 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 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

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

625.14.0

1625.06.0

)()( xxL