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Chapter 3 Arrays (2)

1. Array Referencing2. Common Operations

1. Slicing2. Diminution3. Augmentation

3. List of Commonly Used Built-in Functions1. Applied to vectors2. Applied to matrices

4. Traversing an Array1

ARRAY REFERENCINGAccessing the elements within the array

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Array Referencing

Once you have an array, we sometimes need to refer to the elements contained within it – as smaller portions of the array or even individually.

Because our programs cannot know the values contained within the variable when we write it, we must refer to the positions of the elements as a more general means of accessing them.

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? …

? …

3RD

Array Referencing, cont.

We refer to the positions using “dimensions” – based upon how many dimensions make up a variable:

A scalar has no dimensions – we simply refer to the variable itself.

A vector has one dimension – either a number of rows or a number of columns. We use a single number to reference the values in a vector.

A matrix has two or more dimensions. We use a number for EACH dimension: a row number, AND a column number (Typically - matrices with more dimensions would use another number for each additional dimension)

Array Referencing - Scalars

x = 7;

y = x + 5;

We referenced the scalar x as the variable itself.

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Array Referencing - Vectors

Vectors use a single value. Each value is called an “index”:

x = [5; -1; 4]; %original vectorsum = 0; %start sum at zerosum = sum + x(1); %add first elementsum = sum + x(2); %add second

elementsum = sum + x(3); %add third element

Vectors have one dimension, so we use a single index in parentheses to specify which element we are using. Indexing starts at 1, and can go as high as how-many-elements-there-are.

Yes, it seems quite repetitive… couldn’t we use a loop or something to make it easier? Hang in there…

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Array Referencing - Matrices

Matrices are similar. To access the 6 in this matrix:

M = [1, 2, 3; 4, 5, 6; 7, 8, 9]

we would say: M(2,3)

It can directly be used in equations:

x = 7 * M(2,3); %Result? _____

The row and column positions specified in the parentheses are referred to as “indices” (plural of “index”). We say that 2 is the “row index” and 3 is the “column index”.

Row number always first!

Column number always second!

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Array Referencing, cont.

But, we don’t have to use constants for the indices. We can also use variables:

Instead of saying M(2,3), we can say:

r=2;

c=3;

x = y + M(r, c);

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Referencing - (Nested FOR Loops)

“So what?”, you say.

Well, with the power of loops it should become obvious. We can add up all of the values in a matrix, but only if we refer to each one:

sum = 0;for r = 1:3

for c = 1:3sum = sum + M(r, c);

endend

The row index is gotten from a variable!

The column index is gotten from a variable!9

Referencing – Nested loops

sum = 0;for r = 1:3for c = 1:3

sum = sum + M(r, c);end

end

Notice that the inner loop is changing its value while the outer loop waits. Only when the inner loop is complete does the outer loop increment.

Why did we choose “r” and “c” for the loop variables?

In this version, we move across all columns in a row before we proceed to the next row. Did we have to do this?

If we choose to move down through all rows of a column before moving onto the next column, how would these loops change?

Referencing – (The range operator…)

If you want to refer to “all” of a column or row, you can use the range operator by itself:

V = M(:, 3); % from M, copy all rows in columns 3 to V

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Note: The same could be done with columns!

V = M(2, :); % Copy all columns of row 2 to V

ARRAY SLICINGUsed for:

Accessing more than one element of an array

Eliminating bad elements

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Array Slicing

In general, when we “pull out” part of an array, we call that a “slice”. The range operator is frequently used when getting a slice.

% Pull out all elements in rows 1 and 2

% that are in columns 1 through 4

M1 = M(1:2, 1:4);

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Real-life #1: Eliminating bad data In wind tunnels, the data is

obtained throughout the tunnel. However, data is usually

flawed around the walls, or far away form the object itself.

Given an array of pressure/temperature/or density obtained, only the ones far from the wall are kept for analysis!

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ARRAY DIMINUTION

Making arrays smaller

Deleting an element

Deleting a row

Deleting a column

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Pronounce:“Dim’ – min – yoo’ – shun”

Array Diminution

To eliminate the whole content, just re-define it as an empty-vector:

scores = []; %delete all scores

To eliminate a single value from a vector, you can either take a slice:

HighScores = [757, 65, -13, -89];HighScores = HighScores(1:3); %deletes last

%score

Or you can use the empty-vector:HighScores(4) = []; %removes 4th score

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Example Diminution After analyzing data, you may wish to get rid of some data: in this

case, assign the empty brackets []

For example, get rid of the number 8 in b below:

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This action changes the original vector and cannot be undone.

Array Diminution, cont.

To eliminate an entire row/column:1. Use the range operator, combined with

2. the empty-vector

M = [1, 2, 3; 4, 5, 6];

M(:, 1) = [] … Read it as:

QUESTION: Can we eliminate a single value from a matrix? M(2,2) = [] ?

No – because that would mean some rows or columns would have more values than others. 18

%”M , all-rows, 1stcolumn , delete!”

Real life#2 – similar example

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Clearly, bad results on the walls…

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Real life#2 – similar example

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Real life#2 – similar example

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Real life#2 – similar example

Suppose you want to delete the top now, since that is also a wall in the wind tunnel. What would be the command line?____________________________________

AUGMENTING AN ARRAY

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Array Augmentation, reviewAugmentation = “Adding to” = making an array bigger. For example:V = [1, 2, 3];

To augment more columns, it’s much like doing a running total or running product: to the current variable, perform an action:

V = [V, 4, 5, 6];

To augment with another row vector variable:V1 = [3, 4, 5];V2 = [6, 7, 8];V1 = [V1; V2];

To augment with a column vector variable:V1 = [6; 8; 9];V2 = [10; 20; 30];V1 = [V1, V2];

Makes a matrix!

Why did we use a comma? ________________

Result: _ [ _________________ ] _?

Result:

__ __ __. __ __ __.

Result:

__ __ .

__ __ __ __

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Array Augmentation, reviewWorks for matrices, too:

M1 = [1, 2, 3; 4, 5, 6]; %original matrixM1 = [M1; 7, 8, 9]; %add a row to M1M1 = [M1, [11, 2, 33; 44, 33, 22; 1, 0, 2]]

M1 =

1 2 3 11 2 33 4 5 6 44 33 22 7 8 9 1 0 2

Be sure that you augment with the correct number of rows / columns!

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Extending an array

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Array b does not have 4 columns… mmm… what will it do?

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Commonly Used Built-In Functions

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Common Array Functions

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1. To create arrays 2. To analyze arrays 2. To loop through arrays

zeros() max() length()

ones() min() numel()

rand() mean() size()

sum()

prod()

sort()

1. TO CREATE ARRAYS

No hardcoding

Arrays useful and easy to create

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Creating Special Arrays

zeros() to create arrays with zeros e = zeros(nb of rows, nb of columns)

ones() to create arrays with ones f = ones(nbRows, nbCols)

rand() to create arrays with numbers between 0 and 1. g = rand(nbRows, nbCols)

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Creating Special Arrays, cont.

zeros() to create arrays with zeros e = zeros(one argument only)

ones() to create arrays with ones f = ones(one argument only)

rand() to create arrays with numbers between 0 and 1. g = rand(one argument only)

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>> zeros(3)

ans =

0 0 0 0 0 0 0 0 0

>> ones(3)

ans =

1 1 1 1 1 1 1 1 1

>> rand(3)

ans =

0.8147 0.9134 0.2785 0.9058 0.6324 0.5469 0.1270 0.0975 0.9575

2. VECTOR ANALYSISmax(), min(), mean(), sum(), prod(), sort()

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Example of using max() 2 different ways

Note: max_of_b and where are arbitrary names. You could (but not should) pick blabla1 and blabla2 instead… HOWEVER: you must have two variables in THAT order: 1st will represent the value, 2nd will represent the position of the maximum

This is the first time we’ve seen a function return multiple values. As you can see, the only change we made was that we decided that we wanted to save the second value instead of ignoring it

Only collected a single return value

Collecting two return values – but same function call!

3. MATRIX ANALYSISmax(),min(),mean(),sum(),prod(),sort() - huh? Again? Yes!

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Library functions sum(), prod(), mean(), max(), min() have a different result than when applied

to vectors vs. matrices

By default, the operations are done on each column, thus giving the sum/product/average/max or min of each column.

You can choose to apply the operation to any dimension, but you have to use the two-argument version: sum(x, 2)

1 = sum the elements in all rows (i.e. down the column) 2 = sum the elements in all columns (i.e. across the row) 3 = sum the elements in each row and column (i.e across “pages”) – 3Dimensional

The result of these functions on an array is a vector.

If you want the sum of a whole array, use sum(sum(arrayName))

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Using the sum()

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The same rules apply for mean(), prod()

NOT formin(), max(),

Sorting arrays

There are two fundamental tasks when working with computer data: sorting and searching.

You can build a PhD thesis out of efficiency of sorting and searching!

Fortunately, we don’t have to – we’ll just use some MATLAB functions

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Sorting Arrays, cont.

Like sum(), max(), min(), etc., the sort() function works along the columns by default. But you can change that:

>> a

a =

1 2 3 4 5 6

>> sort(a, 2, 'descend')

ans =

3 2 1 6 5 4

2 = sort all the columns in a row2

This is the first time we’ve seen a non-numeric argument.

Like numeric arguments, which non-numeric values are valid is determined by the function. Refer to the Help (F1 in MATLAB) for more info about each function.

The sort() function has two possible values for the 3rd argument: ‘ascend’ and ‘descend’. By default (if you don’t specify a third argument) the function sorts in ascending order.

Dimension of an array size() is a built-in function that returns a new vector with 2 values

inside: the number of rows (first), the number of columns (second)

length() is another built in function that returns the maximum value between the number of rows and columns of an array

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TRAVERSING AN ARRAY

My friend, the FOR loop

Analyze an array in other ways than max,min,sort…

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Linear Searching

“Linear searching” is exactly how we typically work a search – we look at each element and see if it matches. We stop when we reach the end of the array.

For example:

Our program has generated a matrix. We would like to know if any value in the matrix exceeds the value of 10.

Linear Searching, cont.

s = size(M);found = [];for r = 1:s(1)for c = 1:s(2)

if M(r, c) > 10found(1) = r;found(2) = c;

endend

end

size() gives us # of rows and # of columns

found will contain the position of the last value found that is greater than 10

Binary Searching

Binary searching is a fancy name for how we use the phone book – an extremely fast way to search.

Since names are sorted in the phone book, we can skip massive portions. For example, if the last name starts with “T”, we can skip about 75% of the phonebook!

Binary Searching, cont.

Just like with the phonebook, we can search arrays using binary searching. And like the phonebook, this method requires that the information already is sorted.

Although it can be modified for matrices, it is most useful when searching vectors.

Binary Searching, cont.

To use binary searching:

Sort the vector

Repeat until found or run out of values to check

Compute the ½ way index for the current range

Is it what you want?

If not, is it too high or too low?

If too high, move to middle of lower range

If too low, move to middle of upper range

Binary Searching, cont.clearclc

% Give us some valuesv = floor(25*rand(1, 10))

% Search for the value 10value = 10;

% Sort the datav = sort(v)

% Setup the varshi = 10;lo = 1;done = false;found = false;

while (~done && ~found) range = hi - lo + 1; index = floor(range / 2) + lo; if (v(index) == value) found = true; elseif (v(index) > value) hi = index-1; else lo = index+1; end if hi<lo done = true; end end

Repeat until found or run out of values to check

Compute the ½ way index for the current range

Is it what you want?

If too high, move to middle of lower range

If too low, move to middle of upper range

Have we used up all the values?