MatLab for biologists
Lecture 1
May 11, 2010
1
1
Start MatLab , environment, history
To start MatLab type ‘matlab’, or click on its icon.
1.1
MatLab environment
The following elements are the typical components of a MatLab environment:
1. Command window – interprets and executes commands
2. Command history – list of earlier commands
3. Workspace and variable editor – list and edit current variables
4. Current Directory – file/folder browser
5. Editor – create/edit/execute scripts and functions.
Figure 1: Typical MatLab environment.
1.2
History
MatLab stands for ”MAtrix LABoratory”1 and is a numerical computing
environment. Developed by The MathWorks, MatLab allows matrix manipulations, plotting of functions and data, implementation of algorithms,
creation of user interfaces.
MatLab was created in the late 1970s by Cleve Moler at the University
of New Mexico. He designed it to give his students access to LINPACK
and EISPACK without having to learn Fortran. It soon spread to other
universities and found a strong audience within the applied mathematics
community. Jack Little, an engineer rewrote MatLab in C and founded The
MathWorks in 1984 to continue its development.
2
MatLab as a calculator
MatLab can be used as a calculator with the basic arithmetic operators + * / and brackets ( ). An extra symbol, ^ can be used to compute powers;
e.g. 3^3 = 27.
>>4+1/2*3
ans=
5.5000
From the result we can see that the evaluation of the arithmetic operations
is not directly from left to right, but some of them has higher priority. In
MatLab the order of the evaluation of the most important operators is the
following:
1. expressions in brackets,
2. powers,
3. * /, working left to right,
4. + -, working left to right.
Therefore, the expression is evaluated as 4+3 1/1 2*2 3.
What is the result of the following expressions?
• 1+2^3*4/5
• 5+(4*3)^2-1
• (1+2)^3*4/4
1
Source: Wikipedia
Exercise
• 3*4/(4-2^2)
We can separate operations/expressions using ‘,’ or ‘;’ signs. Using ‘,’
the result will be printed after the calculation, while expressions with ‘;’
will be evaluated but not written to the display, this plays important role in
more difficult computations, where we can determine which results we want
to visualize e.g.
>>3+3, 3*10+23; 9*3.5, 23;
ans=
6
ans=
31.5000
The last ‘,’ and ‘;’ can be left, the result will than be displayed.
3
Remark
Data types
Different data types are available in MatLab . These are:
• logical
• char
• NUMERIC
– INTEGER
∗
∗
∗
∗
int8, uint8
int16, uint16
int32, uint32
int64, uint64
– REAL
∗ single
∗ double
• cell
The e (exponent) notation is used for very large or very small numbers
e.g.
-1.3412e+03 = -1.3412 ×103 = -1341.2
1.3412e-01 = 1.3412 ×101 = 0.13412
The who command shows the stored variables and the whos prints the
type of them.
Remark
Remark
4
Variables
Type the following to the command window!
>>16 * 7
ans=
112
>>ans / 4
ans=
28
Here we used the output (ans) (answer) of the first calculation as a variable in the second one. To store our data and make our work much flexible
we can store the numbers in variables. These variables can have different
data types.
Compute the size of a cell population after a given hour if the rate of
growth/hour and the current size is known.
>>CellNumber = 1500; GrowthRate = 5; Hours = 10;
>>NewPopulation = CellNumber*(1+GrowthRate/100)^Hours
NewPopulation =
2.4433e+003
Exercise
The variables can be used in subsequent calculations.
Assignment statements: values are assigned to variables, each variable
must be assigned a value before it may be used on the right of an assignment
statement.
4.1
Naming variables
Variable names must begin with a letter, which may be followed by any
combination of letters, digits, and underscores. MatLab distinguishes between upper and lowercase characters. Variable names can be of any length,
MatLab uses only the first 63 characters of the name and ignores the rest.
One can use isvarname to make sure the name is valid. The answer is 1
if the name is valid, 0 otherwise.
Are the following names valid variable names in MatLab ?
21st century
var.name
pi
hello^world
is this valid
Exercise
4.2
Special (reserved) names in MatLab
Several names are reserved for special values. These are: ans, eps, realmax,
realmin, pi, i, j, inf, NaN, computer, version.
With help command we can see the corresponding description of MatLab Important
Remark
functions. Try: help NaN.
5
MatLab functions
MatLab has many built-in functions which can be used in the following general form:
function [out1 out2 ... outn ] = function name(in1 in2 ... inn ).
The list of the built-in functions is huge, here we give a short and far not
complete overview on them. To see the detailed description use the help
function.
For detailed description of MatLab functions see the following:
Remark
1. help command
2. MatLab graphical help pages
3. Online help pages: http://www.mathworks.com
4. MatLab Central – open (file) exchange for the user community
5.1
Elementary mathematical functions
sin, cos, tan, asin, ... trigonometric functions (radian)
sqrt, mod, rem
elementary mathematical functions
logarithmic functions
log, log10, exp
Type help sind, cosd.
5.2
System functions
These functions are: clear, tic, toc, format, who whos ...
What is the date function for? What is the command to display the
current time?
5.3
Vector and matrix functions
This category manipulates vectors and matrices, detailed description see
later. Some examples: inv, permute, det, min, max, mean, std,...
Exercise
5.4
Special functions
Special functions are built-in into MatLab or in toolboxes e.g. to manipulate
images, fitting curves, solving PDE-s.
5.5
Exercises
1. We know that the longest side of a right-angle triangle is 8 cm and one
angle is 33◦ . How long are the other two sides?
2. I have 20.000 CHF and I go for shopping, in the first hour I spend the
half of my money, in the second hour the half of the remaining. How
much money I have after 9 hours of shopping? After how many hours
is my money less than 1 CHF?
6
Vectors
We distinguish between row (list of numbers) and column vectors.
Row vectors are lists of numbers separated by either commas or spaces.
The number of entries is known as the length of the vector and the entries
are often referred to as elements or components of the vector. The entries
must be enclosed in square brackets.
>>v1 = [5 9 log(3)], v2 = [sqrt(3) 1 0]
v1 =
5.0000 9.0000 1.0986
v2 =
1.7321 1.0000 0
We can do certain arithmetic operations with vectors of the same length
e.g.
>>v1 + v2
ans =
6.7321 10.0000 1.0986
>>v3 = v1 + 3*v2
v3 =
10.1962 12.0000 1.0986
>>length(v3)
ans =
3
Try to make arithmetic operation with vectors not in the same size. Try
to multiply two vectors.
We can build row vectors from existing one.
>>v4 = [v1 + 3, v2]
v4 =
8.0000 12.0000 4.0986 1.7321 1.0000 0
Exercise
>>sort(v4)
ans =
0 1.0000 1.7321 4.0986 8.0000 12.0000
Remark
We can see or modify the elements of the vectors.
>>v3(2) = 23
v3 =
10.1962 23.0000 1.0986
Important
>>v3(3)
ans =
1.0986
6.1
Creating row vectors with ‘:’
In MatLab ‘:’ is a shortcut producing row vectors.
>>1:5
ans =
1 2 3 4 5
>>3:6
ans =
3 4 5 6
The general form is start : inc : stop where the vector start from start
value, increases with inc and stops before left stop. inc can be negative value.
>>0.32:0.1:0.6
ans =
0.3200 0.4200 0.5200
>>6:-.5:4
ans =
6.0000 5.5000 5.0000 4.5000 4.0000
6.2
Pointing parts of vectors
To get the 2nd to 5th elements of v4 we can use the following:
>>v4(2:5)
ans =
12.0000 4.0986 1.7321 1.0000
To list every second element
>>v4(1:2:6)
ans =
8.0000 4.0986 1.0000
How to list the elements in a decreasing order?
6.3
Exercise
Column vectors
There are two ways to create column vectors. Fist we can make a similar
list as in the previous case but separate the elements with ‘;’, or we can
transpose row vectors with the ‘’’ command e.g.
>>v5 = [2; 3; 8]
v5 =
2
3
8
>>v2’
ans =
1.7321
1.0000
0
Also row vectors can be created from columns by transposing them using
‘’’.
Remark
6.4
Exercises
The plot command plots vectors to a graphical window. Try:
>>plot(v4);
>>scale= 0:0.01:2*pi; plot(sin(scale));
Compute the average cell number in a well from radius 0.5-2.5 mm in 100
µm steps if we know that the cell density in ideal case is 50 cells/mm2 but
depending on the radius it is less with well radius3 * 40. Plot the values.
7
Matrices
Matrix is a rectangular array of numbers. MatLab is a matrix based programming language. The row and column vectors are special cases of matrices.
An m × n matrix is a set of numbers organized into a rectangular array. e.g.
a matrix with m = 2 rows and n = 3 columns
2 3 −2
7 9 8
can be defined in MatLab as:
>>A = [ 2 3 -2
7 9 8]
A =
2 3 -2
7 9 8
An alternative way to define them is using ‘;’ rather than new lines, as:
>>B = [ 4 5; 1 -7; 4 8]
B =
4 5
1 -7
4 8
>>C = [ 7:13; 14:-2:2; 1:7]
C =
7 8 9 10 11 12 13
14 12 10 8 6 4 2
1 2 3 4 5 6 7
7.1
Matrix functions
Most of the functions work on matrices, and we should try to write our
programs being compatible with this concept.
We can query the size of the matrices using the size command. e.g.
>>size(A), size(C)
ans =
2 3
ans =
3 7
Transposing matrices works similar way as we mentioned in case of vectors. Using ‘’’, the rows becomes columns and vice versa. e.g.
>>A’
ans =
2 7
3 9
-2 8
There are built-in functions in MatLab which creates special matrices.
Using zeros(n, m) and ones(n, m) functions we can create n by m matrices
of 0s and 1s respectively.
Try what is the difference between zeros(n, 1) and zeros(n). Check
in the help.
Create a 10 × 10 diagonal matrix and fill the elements with the square of
the numbers from 1 to 10. Use the eye and diag functions.
7.2
Exercise
Exercise
Building and accessing parts of matrices
We can build large matrices from smaller ones. To concatenate matrices
horizontally, similar to the vectors we put the matrices into ‘[]’, tabulating
them with spaces or ‘,’; vertical concatenation can be done separating the
elements with enter or ‘;’. The matrices have to be in the same dimension
along the concatenation. e.g.
>>D = [A’ C]
D =
2 7 7 8 9 10 11 12 13
3 9 14 12 10 8 6 4 2
-2 8 1 2 3 4 5 6 7
>>E = [D; C A’]
E =
2 7 7 8 9 10 11 12 13
3 9 14 12 10 8 6 4 2
-2 8 1 2 3 4 5 6 7
7 8 9 10 11 12 13 2 7
14 12 10 8 6 4 2 3 9
1 2 3 4 5 6 7 -2 8
To access an element of a matrix we need to give its row and column
index. In MatLab the element in the ith row and jth column on matrix A
can be accessed as: A(i, j). e.g.
>>E(2, 3), E(5, 6)
ans =
14
ans =
Important
4
>>E(7, 2)
??? Index exceeds matrix dimensions.
To modify the elements:
>>A(2, 3) = 3
A =
2 3 -2
7 9 3
>>A(1, 1) = A(2, 3) + 5
A =
8 3 -2
7 9 3
It is possible to access not only individual elements of matrices but arbitrary sub-matrices using vectors or intervals instead of single indices.
For example the list of every second elements of the 3rd line of E.
>>E(3, 1:2:9)
ans =
-2 1 3 5 7
The stand alone ‘:’ operator substitutes a vector going from the first
index to the last. e.g.
>>E(:, 4)
ans =
8
12
2
10
8
4
The end statement during indexing refers the last row/column of the matrix.
Create the following 10 × 10 matrix and try the spy command!
Remark
Exercise
0
1
0
1
0
1
0
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
7.3
0
1
0
1
0
1
0
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
1
0
1
0
1
0
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
1
0
1
0
1
0
1
0
1
Arithmetic operations
To calculate the dot product of matrices use ‘.*’. To element-wise add or
subtract matrices we can use ‘+’/‘-’. The matrices have to be in the same
size! e.g.
>>A + A, A .* A
To multiply matrices with vectors or other matrices we use ‘*’. e.g.
>>v1=[1; 3; 6];
>>A*v1
ans =
5
52
>>A’*A
ans =
113 87 5
87 90 21
5 21 13
>>A * E(2:4, 5:7)
ans =
67 52 37
130 128 126
8
Script files and functions
To run more commands after each other or organize computations what we
use frequently in MatLab we have alternative ways.
8.1
Script files
Sometimes we want to collect sequence of commands, for this MatLab offers
the script files (.m). We can edit and run them on the MatLab Editor. It is
also possible to run them, typing the name of the script file.
Create a script which computes the body mass index (weight [kg] * height2
[m]).
8.2
MatLab functions
In many cases there are sequence of operations we want to run more than
once, possibly with different parameters. For this we can write our own MatLab functions. The standard definition of a function is:
function [out1 out2 ... outn ] = function name(in1 in2 ... inn ),
where outi and ini are the output and input variables respectively. The functions are stored in .m files, and in most of the cases the filename is the same
as the function name. The input arguments can be used by referring their
names, and the output variables should be created in the function.
An example function which computes the body mass index:
function bmi = BodyMassIndex(h, w);
bmi = w / h^2;
Write a function, which has one input number n, and no output. The
function creates an n × n matrix, with a chessboard shape (1-white, 0-black)
and plots in a graphical window! Hint: imagesc.
9
Exercise
Exercise
Input, output I.
In the previous section we learned how to run a sequence of commands. Sometimes it is convenient or essential to read or write variables either from/to
standard input or files.
The input command displays a question and reads a variable from the
command line. To display values or text we can use the disp command. The
result of disp command and leaving the ‘;’ from the end of the commands
is similar but disp does not visualizes the variable names.
Remark
To save and open our variables MatLab offers the save and load commands. See the help of them!
9.1
Exercises
• Save the workspace to the lesson 2.mat file!
• Save A, C, and E variables to matrices.mat file. Clear the workspace
and load the matrices.
• Write a script file, which asks the user’s height and weight and computes the body mass index, than displays the result and writes all the
variables into a file.
10
Used material
• An Introduction to MatLab – David F. Griffiths – University of Dundee
• Using MatLab – The MathWorks, Inc.
c ‘primer’ – Ernesto Di Iorio – ETH Zurich
• A MatLab