vectors, arrays and functions

<9/6/2007> Physics 241 -- New Mexico Tech

Vectors, Arrays and FunctionsRichard Sonnenfeld

(with some material from W. Palm)

nn VectorsVectorsPhysics definitionPhysics definitionUse in Use in MatlabMatlab

nn Concept of an ArrayConcept of an Arraynn User Defined FunctionsUser Defined Functionsnn Conditional statementsConditional statements


Matlab can easily represent vectors as defined in the classical physics sense.

The vector r can be specified by three components: x, y, and z, and can be written as:

r = [x, y, z].

However, MATLAB can use vectors having more than three elements.

r


To create a row vector, separate the elements by commas. For example,

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

3 7 9

You can create a column vector by using the transpose notation (').Or by separating the elements by semicolons. For example,

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

379

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

379


You can create vectors by ''appending'' one vector to another.

For example, if

>> r = [2,4,20] and >> w = [9,-6,3], Then>> u = [r,w]. The result is the vector u = [2,4,20,9,-6,3].You can also create a subset of a vector.Assume we wanted to use the 2nd, 3rd, and 4th elements of u. How could this be done?


The colon operator (:) easily generates a large vector of regularly spaced elements.

Typing

>>x = [m:q:n]

creates a vector x of values with a spacing q. The first value is m. The last value is n if m - n is an integer multiple of q. If not, the last value is less than n.


For example>> x = [0:2:8]x =

[0,2,4,6,8]

>> x = [0:2:7]x =

[0,2,4,6]

To create 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, then q=1. y = [-3:2]y = [-3,-2,-1,0,1,2].


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 x1 and x2are the lower and upper limits and n is the number of points.

For example, linspace(5,8,31) is equivalent to [5:0.1:8].

If n is omitted, the spacing is 1.

Example:t=linspace(0,5);x=3*t-4.9*t.^2;plot(t,x)


Matlab flexibly and cleverly handles row and column vectors and matrices of different sizes.

Most operations are defined on vectors or matrices of the same size. For example, adding a row vector to acolumn vector causes an error.

>> a=[0:10]; b=a’+7; c=a+b

??? Error using ==> +Matrix dimensions must agree.

Get used to looking for these – they are one of the most common causes of strange behavior.

Potential confusionPotential confusion


“Array” (noun) In programming languages, a name given to sequences of variables that have a relationship to eachother.

Examples:Successive points in a scientific data set. Adjacent values of a mathematical function.

Successive array elements are often allocated physically adjacent locations in computer memory.

In Matlab, arrays are referred to as “Vectors” or “Matrices”.

Programming concept called “Array”Programming concept called “Array”


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 x1 and x2are the lower and upper limits and n is the number of points.

For example, linspace(5,8,31) is equivalent to [5:0.1:8].

If n is omitted, the spacing is 1.

Example:t=linspace(0,5);x=3*t-4.9*t.^2;plot(t,x)

Vectors and relative motionVectors and relative motion


Magnitude, Length, and Absolute Value of a Vector

In physics, length, magnitude and absolute value of a vector all mean the same thing. In Matlab, they do not.

Magnitude of a vector x having elements x1, x2, …, xn is a scalar, given by √(x1

2 + x22 + … + xn

2), and is the same as the physics definition of magnitude.

length command gives the number of elements in the vector.

The absolute value of a vector x is a vector whose elements are the absolute values of the elements of x.


For example:>> x = [2,-4,5]

• its length is 3 (length(x))

• its magnitude is √[22 + (–4)2 + 52] = 6.7082 (sqrt(x’*x))

• its absolute value is [2,4,5] (abs(x)).


Matrices

A matrix has multiple rows and columns. For example, the matrix

has four rows and three columns.

Vectors are special cases of matrices having one row or one column.

M =2 4 10

16 3 7 8 4 9 3 12 15


Creating Matrices from Vectors

Suppose a = [1,3,5] and b = [7,9,11] (row vectors). Note the difference between the results given by [a b] and [a;b] in the following session:

>>c = [a b];c =

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

alternatively>>D = [a;b]D =

1 3 57 9 11


Array Addressing

The colon operator selects individual elements, rows, columns, or ''subarrays'' of arrays.

Examples:n v(:) represents all the elements of the vector v.n v(2:5) represents the second through fifth

elements; that is v(2), v(3), v(4), v(5). n A(:,3) denotes all the elements in the third column

of the matrix A.n A(:,2:5) denotes all the elements in the second

through fifth columns of A.n A(3,:) denotes all the elements in the third row


You can use array indices to extract a smaller array from another array. For example, if you first create the array B

B =

C =16 3 7

8 4 9

2 4 10 1316 3 7 18 8 4 9 253 12 15 17

then type C = B(2:3,1:3), you can produce the following array:


size(A) Returns a row vector [m n]containing the sizes of them x n array A.

sort(A) Sorts each column of the array A in ascending order and returns an array the same size as A.

sum(A) Sums the elements in each column of the array A andreturns a row vector containing the sums.

max(A) ?

Additional Array Functions


The Workspace Browser


The Array Editor


Element-by-element operations: Table 2.3–1

Symbol

+

-

+

-

.*

./

.\

.^

Examples

[6,3]+2=[8,5]

[8,3]-5=[3,-2]

[6,5]+[4,8]=[10,13]

[6,5]-[4,8]=[2,-3]

[3,5].*[4,8]=[12,40]

[2,5]./[4,8]=[2/4,5/8]

[2,5].\[4,8]=[2\4,5\8]

[3,5].^2=[3^2,5^2]

2.^[3,5]=[2^3,2^5]

[3,5].^[2,4]=[3^2,5^4]

Operation

Scalar-array addition

Scalar-array subtraction

Array addition

Array subtraction

Array multiplication

Array right division

Array left division

Array exponentiation

Form

A + b

A – b

A + B

A – B

A.*B

A./B

A.\B

A.^B


User defined functionsRichard Sonnenfeld


Operations on Arrays

MATLAB will treat a variable as an array automatically. For example, to compute the square roots of 5, 7, and 15, type

>>x = [5,7,15];>>y = sqrt(x)y =

2.2361 2.6358 3.8730


User-Defined Functions

The first line in a function file must begin with a function definition line that has a list of inputs and outputs. This line distinguishes a function M-file from a script M-file. Its syntax is as follows:

function [output variables] = name(input variables)

Note that the output variables are enclosed in square brackets, while the input variables must be enclosed with parentheses. The function name (here, name) should be the same as the file name in which it is saved (with the .m extension).


User-Defined Functions: Example

function z = fun(x,y)u = 3*x;z = u + 6*y.^2;

Note the use of a semicolon at the end of the lines. This prevents the values of u and z from being displayed.

Note also the use of the array exponentiation operator (.^). This enables the function to accept y as an array.


User-Defined Functions: Example (continued)

Call this function with its output argument:

>>z = fun(3,7)z =

303

The function uses x = 3 and y = 7 to compute z.



Call this function without its output argument and try to access its value. You will see an error message.

>>fun(3,7)ans =

303>>z??? Undefined function or variable ’z’.



Assign the output argument to another variable:

>>q = fun(3,7)q =

303

You can suppress the output by putting a semicolon after the function call.

For example, if you type q = fun(3,7); the value of qwill be computed but not displayed (because of the semicolon).


The variables x and y are local to the function fun, so unless you pass their values by naming them x and y, their values will not be available in the workspaceoutside the function. The variable u is also local to the function. For example,

>>x = 3;y = 7;>>q = fun(x,y);>>xx =3

>>yy =7

>>u??? Undefined function or variable ’u’.


Only the order of the arguments is important, not the names of the arguments:

>>x = 7;y = 3;>>z = fun(y,x) z =

303

The second line is equivalent to z = fun(3,7).


You can use arrays as input arguments:

>>r = fun([2:4],[7:9])r =

300 393 498


A function may have more than one output. These are enclosed in square brackets.

For example, the function circle computes the area A and circumference C of a circle, given its radius as an input argument.

function [A, C] = circle(r)A = pi*r.^2;C = 2*pi*r;


The function is called as follows, if the radius is 4.

>>[A, C] = circle(4)A =50.2655

C =25.1327


A function may have no input arguments and no output list.

For example, the function show_date computes and stores the date in the variable today, and displays the value of today.

function show_datetoday = date


1. One input, one output:

function [area_square] = square(side)

2. Brackets are optional for one input, one output:

function area_square = square(side)

3. Three inputs, one output:

function [volume_box] = box(height,width,length)

4. One input, two outputs:

function [area_circle,circumf] = circle(radius)

5. No output: function sqplot(side)

Examples of Function Definition Lines

vectors, arrays and functions

Documents