This Primer is based on a TeX/dvi/ps copy freely distributed via anonymous ftp from:
MATLAB is an interactive, matrix-based system for scientific and engineering calculations. You can solve complex numerical problems without actually writing a program. The name MATLAB is an abbreviation for MATrix LABoratory.
The purpose of this Primer is to help you begin to use MATLAB. They can best be used hands-on. You are encouraged to work at the computer as you read the Primer and freely experiment with examples.
You should liberally use the on-line help facility for more detailed information. After entering MATLAB as described in section 1, the command help (or helpwin ) will display a list of functions for which on-line help is available; the command help functionname will give information about a specific function. The command help eig, for example, will give information about the eigenvalue function eig. You can preview some of the features of MATLAB by entering the command demo.
lookfor etsintään
The scope and power of MATLAB go far beyond these notes. Eventually you will want to consult the MATLAB User's Guide and Reference Guide. The documentation which accompanies the Student Edition of MATLAB is an excellent source. Copies of the complete User's Guide are often available for review at locations such as consulting desks, terminal rooms, computing labs, and the reserve desk of the library. Consult your instructor or your local computing center to learn where the User's Guides are located at your institution.
MATLAB is available for a number of environments. (Kaikkiin saa) There is a relatively inexpensive Student Edition available from Prentice Hall publishers. The information in these notes applies generally to all of these environments.
MATLAB is licensed by The MathWorks, Inc., Cochituate Place, 24 Prime Park Way, Natick, MA 01760, (508)653-1415, Fax: (508)653-2997, Email: info@mathworks.com. Copyright (C) 1989, 1992 by Kermit Sigmon
On most systems, after logging in one can enter MATLAB with the
system command matlab and exit MATLAB with the command
exit or quit.
TKK:n UNIX:ssa
use matlab
matlab
and exit it with the command:
>> quit
On systems permitting multiple processes, such as a Unix system, you will find it convenient, for reasons discussed in section 14, to
keep both MATLAB and your local editor active. Nyt se on uskottava!
MATLAB works with essentially only one kind of object-a rectangular numerical matrix with possibly complex entries; all variables represent matrices. In some situations, 1-by-1 matrices are interpreted as scalars and matrices with only one row or one column are interpreted as vectors.
Versiossa 5 on myös sisäkkäisiä ja moniulotteisia rakenteita.
Matrices can be introduced into MATLAB in several different ways:
A = [1 2 3; 4 5 6; 7 8 9]
and
A = [
1 2 3
4 5 6
7 8 9 ]
creates the obvious 3-by-3 matrix and assigns it to a variable
A. Try it. The elements within a row of a matrix may
be separated by commas as well as a blank.
When listing a number in exponential form (e.g. 2.34e-9), blank spaces must be avoided. Listing entries of a large matrix is best done in an M-file, where errors can be easily edited away (see sections 12 and 14).
magic(m,n) will create an integral n x n matrix which is a magic square (rows and columns have common sum); hilb(m,n) will create the m x n Hilbert matrix, the king of ill-conditioned matrices ( m and n denote, of course, positive integers). Matrices can also be generated with a for-loop (see section 6 below).
A(1,:) ilmaisee A-matriisin ensimmäisen rivin (yksinäinen kaksoispiste jälkimmäisenä indeksinä tarkoittaa, että sarakeindeksi käy läpi kaikki arvonsa . Vastaavasti A(:,2) tarkoitta A:n toista saraketta.
Indeksointia käsitellään tarkemmin ... Kts. myös [SKK] 2.9, ss. 23 - 25.
The following matrix operations are available in MATLAB:
+ addition
- subtraction
* multiplication
^ power
' transpose
\ left division
/ right division
These matrix operations apply, of course, to scalars (1-by-1
matrices) as well. If the sizes of the matrices are incompatible
for the matrix operation, an error message will result, except in
the case of scalar-matrix operations (for addition, subtraction,
and division as well as for multiplication) in which case each
entry of the matrix is operated on by the scalar.
The "matrix division" operations deserve special comment. If A is an invertible square matrix and b is a compatible column, resp. row, vector, then
x=A \ b
is the solution of A*x=b and, resp.,
x=b/A
is the solution of x*A=b . In left division, if A
is square, then it is factored using Gaussian elimination
and these factors are used to solve A*x=b. If A is not
square, it is factored using Householder orthogonalization with
column pivoting and the factors are used to solve the under- or
over- determined system in the least squares sense. Right
division is defined in terms of left division by
b/A=(A' \ b')'
Array operations. The matrix operations of addition and
subtraction already operate entry-wise but the other matrix
operations given above do not-they are matrix operations.
It is important to observe that these other operations,
* , ^ , \ , and /,
can be made to operate entry-wise by preceding them by a period.
For example, either
[1,2,3,4].*[1,2,3,4]
or
[1,2,3,4].^2
will
yield [1,4,9,16]. Try it. This is particularly useful when
using Matlab graphics.
Tee itsellesi selväksi pisteittäiset, vastinalkioittain toimivat operaatiot. *** Ota opas.tex-tiedoston yhteenveto tähän ****
MATLAB is an expression language; the expressions you type are interpreted and evaluated. MATLAB statements are usually of the form
variable = expression,
or simply
expression
Expressions are usually composed from operators, functions, and
variable names. Evaluation of the expression produces a matrix,
which is then displayed on the screen and assigned to the
variable for future use. If the variable name and = sign are
omitted, a variable ans (for answer) is automatically
created to which the result is assigned.
A statement is normally terminated with the carriage return. However, a statement can be continued to the next line with three or more periods followed by a carriage return. On the other hand, several statements can be placed on a single line if separated by commas or semicolons.
If the last character of a statement is a semicolon, the printing is suppressed, but the assignment is carried out. This is essential in suppressing unwanted printing of intermediate results.
| Matlab | Maple |
|---|---|
|
>> muuttuja=lauseke Loppumerkki ei pakollinen (;) estää tulostuksen |
> muuttuja:=lauseke; Loppumerkki (; tai :) pakollinen, kaksoispiste estää tulostuksen. |
Matemaattisten lausekkeiden syntaksi on hyvin pitkälle sama, paitsi Matlabissa käytetään lausekkeissa hyvin usein piste-alkuisia kerto- ja jako-operaattoreita. Maplessa maksetaan "peruspisteettömyydestä" hankalampana matriisikertosyntaksina evalm( A&*B);
MATLAB is case-sensitive in the names of commands, functions, and variables. For example, solveUT is not the same as solveut. Samoin Maple.
The command who will list the variables currently in the workspace. Komento whos antaa tarkempaa tietoa. A variable can be cleared from the workspace with the command clear variablename. The command clear alone will clear all nonpermanent variables.
The permanent variable eps (epsilon) gives the machine precision-about 10-16 on most machines. It is useful in determining tolerences for convergence of iterative processes.
A runaway display or computation can be stopped on most machines without leaving MATLAB with CTRL-C (CTRL-BREAK on a PC).
Saving a session. When one logs out or exits MATLAB all variables are lost. However, invoking the command save before exiting causes all variables to be written to a non-human-readable diskfile named matlab.mat. When one later reenters MATLAB, the command load will restore the workspace to its former state.
Huom! Yllä oleva save ei tarkoita, että Matlab- työ tallettuisi. Siis muuttujat kyllä, mutta komennot eivät. Siten save -komento on tavallisesti varsin hyödytön. Työn talletus tehdään pitämällä editori-ikkuna auki ("köyhän miehen worksheet", kuten L1.html:ssä puhuttiin, mutta ei ollenkaan huono.)
Convenient matrix building functions are
eye(10,10) identity matrix
zeros(10,10) matrix of zeros
ones(10,10) matrix of ones
diag(A) returns diagonal M.E. as vector
triu(A) upper triangular part of a matrix
tril(A) lower triangular part of a matrix
rand(10,10) randomly generated matrix
hilb(5,5) Hilbert matrix
magic(5,5) magic square
toeplitz see help toeplitz
For example, zeros(m,n) produces an m-by-n matrix of
zeros. size(A) gives the size of A (big surprise !), ai
niin, suomeksi, olkoon ...
Thus if A
is a matrix, then zeros(size(A)) produces a matrix of zeros of
the same size as A .
If x is a vector, diag(x) is the diagonal matrix with x down the diagonal; if A is a square matrix, then diag(A) is a vector consisting of the diagonal of A . What is diag(diag(A))?
Try it, joo kokeile!
Matrices can be built from blocks. For example, if A is a 3-by-3 matrix, then
B = [A, zeros(3,2); zeros(2,3), eye(2)]
will build a certain 5-by-5 matrix. Try it.
1:10 antaa vektorin [1,2,3,4,5,6,7,8,9,10] ja yleisemmin a:h:b vektorin [a,a+h,a+2h,...,b] . Tarkista help : , jos haluat tietää, mikä on viimeinen alkio, ellei mene tasan.)
Usein on tarpeen jakaa annettu väli tasan n:ään osaan. Se voitaisiin tehdä tyyliin
h=(b-a)/n x=a:h:bTähän on myös valmis funktio linspace .
x=linspace(a,b,n);
Pieni ero on siinä, että linspace ottaa loppupisteen aina mukaan, jolloin viimeinen osaväli saattaa olla lyhyempi.
Funktiolla linspace on myös muoto linspace(a,b), jolloin n:n oletusarvona käytetään lukua 100. Tämä on tavallisessa käyrien piirtämisessä usein sopiva arvo.
Jos haluamme muodostaa matriisin, jonka 1. sarake on väli [-1,1] jaettuna tasavälisesti 10 osaan ja toisena kunkin jakopisteen neliö, ja laitetaan nyt vielä kolmanneksi jakopisteiden kuutiot, saamme sen aikaan (riemastuttavan helposti):
x=linspace(-1,1,10);
x1x2x3=[x' (x.^2)' (x.^3)']; % taulukointi *** find-hakusana ***
x1x2x3 % Katsotaan arvo, pitemmän x-vektorin tapauksessa kannattaa katsoa
% osissa, tyyliin x1x2x3(1:5,:)
Huomaa, että vaakavektorista saadaan pystyvektori transponoimalla, eli
operaatiolla x'
Tässä ei kukaan malta enää olla sanomatta plot(x1x2x3);
Kokeile myös
plot(x1x2x3(:,1),x1x2x3(:,3)) % tässä tapauksessa lyhemmin suoraan: plot(x,x.^3)
In their basic forms, these MATLAB flow control statements operate like those in most computer languages.
For. For example, for a given n, the statement
x = []; for i = 1:n, x=[x,i^2 ], end
or
x = [];
for i = 1:n
x = [x,i^2 ]
end
will produce a certain n-vector and the statement
x = []; for i = n:-1:1, x=[x,i^2 ], end
will produce the same vector in reverse order. Try them.
Huom ! Tällaisia vektorin alkiotason silmukoita ei ole syytä suosia (ne ovat karkeaa pilaa, Matlabin väärinkäyttöä). Tietysti tuollainen hoidettaisiin edellä esitetyillä keinoilla. (Vektorin kääntämiseen takaperin on fliplr (left-right) ja ylösalaisin flipud (updown) .) No teepä !
Note that a matrix may be empty (such as x = []). The statements
for i = 1:m
for j = 1:n
H(i, j) = 1/(i+j-1);
end
end
H
will produce and print to the screen the m-by-n hilbert matrix.
The semicolon on the
inner statement suppresses printing of unwanted intermediate results while the
last H displays the final result.
Jos Matlabissa ei sattuisi olemaan komentoa hilb tai emme sitä muistaisi, niin yllä oleva muodostamistapa olisi ihan asiallinen (eikä sitä voitaisi tuomita alkiotason silmukoista huolimatta pilailuksi).
While. The general form of a while loop is
while relation
statements
end
The statements will be repeatedly executed as long as the
relation remains true. For example, for a given number
a, the following will compute and display the smallest
nonnegative integer n such that 2n >=a:
n = 0;
while 2^n < a
n = n + 1;
end
n
If. The general form of a simple if statement is
if relation
statements
end
The statements will be executed only if the relation is true.
Multiple branching is also possible, as is illustrated by
if n < 0
parity = 0;
elseif rem(n,2) == 0
parity = 2;
else
parity = 1;
end
In two-way branching the elseif portion would, of course, be omitted.
Relations. The relational operators in MATLAB are
< less than
> greater than
<= less than or equal
>= greater than or equal
== equal
~= not equal.
Note that "=" is used in an assignment statement while "==" is
used in a relation. Relations may be connected or quantified by
the logical operators
& and
| or
~ not.
When applied to scalars, a relation is actually the scalar 1 or 0
depending on whether the relation is true or false. Try 3
< 5, 3 > 5, 3 == 5, and 3 == 3. When applied
to matrices of the same size, a relation is a matrix of 0's and
1's giving the value of the relation between corresponding
entries. Try a = rand(5), b = triu(a), a == b.
A relation between matrices is interpreted by while and if to be true if each entry of the relation matrix is nonzero. Hence, if you wish to execute statement when matrices A and B are equal you could type
if A == B
statement
end
but if you wish to execute statement when A and
B are not equal, you would type
if any(any(A ~ B)) ** Tarkista, pitäiskö olla: A ~= B **
statement
end
or, more simply,
if A == B else
statement
end
Note that the seemingly obvious
if A ~= B, statement, end
will not give what is intended since statement would
execute only if each of the corresponding entries
ofA and B differ. The functions any
and all can be creatively used to reduce matrix
relations to vectors or scalars. Two any's are required
above since any is a vector operator (see section 8).
The for statement permits any matrix to be used instead of 1:n. See the User's Guide for details of how this feature expands the power of the for statement.
Certain MATLAB functions operate essentially on scalars, but operate element-wise when applied to a matrix. Näin toimivat mm. kaikki matemaattiset perusfunktiot. Omat funktiot on useimmiten syytä kirjoittaa näin toimiviksi.
Jos f on skalaarifunktio, niin
f(skalaari) -> skaalaari
| f(a1 1) ... f(a1 n) |
| f(a2 1) ... f(a2 n) |
f(A) -> | ... |
| f(am 1) ... f(am n) |
The most common
such functions are
sin asin exp abs round
cos acos log (natural log) sqrt floor
tan atan rem (remainder) sign ceil
Other MATLAB functions operate essentially on a vector (row or column), but act on an m-by-n matrix (m>=2) in a column-by-column fashion to produce a row vector containing the results of their application to each column. Row-by-row action can be obtained by using the transpose; for example, mean(A')'.
f(vekt) -> skal
f(A) operoi sarakkeittain:
| | | |
| f(a. 1) ... f(a. n) |
f(A) -> | | | |
| | | |
A few of these
functions are
max sum median any floor
min prod mean all ceil round
sort std
For example, the maximum entry in a matrix A is given by
max(max(A)) rather than max(A). Try it.
Matriisin jonoutus pitkäksi sarakevektoriksi saadaan aikaan kirjoittamalla
A(:)
Niinpä matriisin A suurin alkio saadaan myös näin: max(A(:)) .
Kokeile tätäkin.
Jonoutetusta pitkästä vektorista päästään takaisin matriisiin komennolla reshape , joka on varsin kätevä moneen hommaan, katso helpwin reshape .
Tehtävä: (a) Kirjoita lauseke, joka laskee matriisin A alkioiden
itseisarvojen sarakesummien maksimin.
(b) Sama rivisummille.
Edellinen on matriisin ns. 1-normi, jälkimmäinen taas "ääretön-normi",
näistä enemmän tuonnempana.
Much of MATLAB's power comes from its matrix functions. Matriisifunktion argumenttina on matriisi (joissakin tapauksessa kelpaa vain neliömatriisi). Tuloksena saattaa olla oikeastaan mitä vain, kuten
eig eigenvalues and eigenvectors
chol cholesky factorization
svd singular value decomposition
inv inverse
lu LU factorization
qr QR factorization
hess hessenberg form
schur schur decomposition
rref reduced row echelon form
expm matrix exponential
sqrtm matrix square root
poly characteristic polynomial
det determinant
size size
norm 1-norm, 2-norm, F-norm, infinity-norm
cond condition number in the 2-norm
rank rank
MATLAB functions may have single or multiple output arguments.
For example,
y = eig(A) or simply eig(A)
produces a column vector containing the eigenvalues of A
while
[U,D] = eig(A)
produces a matrix U whose columns are the eigenvectors
of A and a diagonal matrix D with the
eigenvalues of A on its diagonal. Try it.
Tehtävä: Muodosta symmetrinen 10 x 10 satunnaismatriisi (käyttäen rand -funktiota), nimeltään A. Huomaa, että symmetrisen matriisin saat esim. kertomalla mielivaltaisen matriisin ja sen transpoosin keskenään. Muodosta matriisit V ja D, missä V:n sarakkeet ovat A:n ominaisvektorit ja D:n diagonaalialkiot ovat A:n ominaisarvot. Laske A*V ja V*D ja totea ne samoiksi. Muistele ominaisarvojen ja - vektoreiden määritelmää ja perustele, miksi näin on.
The command line in MATLAB can be easily edited. The cursor can be positioned with the left/right arrows and the Backspace (or Delete) key used to delete the character to the left of the cursor. Other editing features are also available. On a PC try the Home, End, and Delete keys; UNIX:ssa toimivat Emacs-tyyliset (tcshell-tyyliset) CTR-A (alkuun, "Anfang"), CTR-E (loppuun, "Ende"), CTR-K (rivin lopun poisto "kill").
A convenient feature is use of the up/down arrows to scroll through the stack of previous commands. One can, therefore, recall a previous command line, edit it, and execute the revised command line. For small routines, this is convenient. Muista kuitenkin aina pitää editori-ikkuna työssä mukana (pienestä alusta kehittyy usein isompaa). (see sections 12 and 14). For example, flopcounts (see section 15) for computing the inverse of matrices of various sizes could be compared by repeatedly recalling, editing, and executing
a = rand(8,8); flops(0), inv(a); flops
If one wanted to compare plots of the functions y = sin
mx and y = sin nx on the interval [0,2*pi] for
various m and n, one might do the same for the
command line:
m=2; n=3; x=0:.01:2*pi; y=sin(m*x); z=cos(n*x); plot(x,y,x,z)
Vectors and submatrices are often used in MATLAB to achieve fairly complex data manipulation effects. Colon notation" (which is used both to generate vectors and reference submatrices) and subscripting by vectors are keys to efficient manipulation of these objects. Creative use of these features permits one to minimize the use of loops (which slows MATLAB) and to make code simple and readable. Special effort should be made to become familiar with them.
The expression 1:5 (met earlier in for statements) is actually the row vector [1 2 3 4 5]. The numbers need not be integers nor the increment one. For example,
0.2:0.2:1.2
gives [0.2, 0.4, 0.6, 0.8, 1.0, 1.2], and
5:-1:1 gives [5 4 3 2 1].
The following statements will, for example, generate a table of
sines. Try it.
x = [0.0:0.1:2.0]' ;
y = sin(x);
[x y]
Note that since sin operates entry-wise, it produces a
vector y from the vector x.
The colon notation can be used to access submatrices of a matrix. For example,
A(1:4,3) is the column vector consisting of the first four entries of the third column of A.A colon by itself denotes an entire row or column:
A(:,3) is the third column of A , and A(1:4,:) is the first four rows.Arbitrary integral vectors can be used as subscripts:
A(:,[2 4]) contains as columns, columns 2 and 4 of A .Such subscripting can be used on both sides of an assignment statement:
A(:,[2 4 5]) = B(:,1:3) replaces columns 2,4,5 of A with the first three columns of B. Note that the entire altered matrix A is printed and assigned. Try it.Columns 2 and 4 of A can be multiplied on the right by the 2-by-2 matrix [1 2;3 4]:
A(:,[2,4]) = A(:,[2,4])*[1 2;3 4]Once again, the entire altered matrix is printed and assigned.
If x is an n-vector, what is the effect of the statement x = x(n:-1:1)? Try it.
To appreciate the usefulness of these features, compare these MATLAB statements with a Pascal, FORTRAN, or C routine to effect the same.
MATLAB can execute a sequence of statements stored on diskfiles. Such files are called M-files" because they must have the file type of .m" as the last part of their filename. Much of your work with MATLAB will be in creating and refining M-files.
There are two types of M-files: script files and function files.
Script files. A script file consists of a sequence of normal MATLAB statements. If the file has the filename, say, rotate.m, then the MATLAB command rotate will cause the statements in the file to be executed. Variables in a script file are global and will change the value of variables of the same name in the environment of the current MATLAB session.
Script files are often used to enter data into a large matrix; in such a file, entry errors can be easily edited out. If, for example, one enters in a diskfile data.m
A = [
1 2 3 4
5 6 7 8
];
then the MATLAB statement data will cause the assignment given in
data.m to be carried out.
An M-file can reference other M-files, including referencing itself recursively.
Function files. Function files provide extensibility to MATLAB. You can create new functions specific to your problem which will then have the same status as other MATLAB functions. Variables in a function file are by default local. However, version 4.0 (nyt on jo 5.3) permits a variable to be declared global.
We first illustrate with a simple example of a function file.
function a = randint(m,n)
%RANDINT Randomly generated integral matrix.
% randint(m,n) returns an m-by-n such matrix with entries
% between 0 and 9.
a = floor(10*rand(m,n));
A more general version of this function is the following:
function a = randint(m,n,a,b)
%RANDINT Randomly generated integral matrix.
% randint(m,n) returns an m-by-n such matrix with entries
% between 0 and 9.
% rand(m,n,a,b) return entries between integers a and b .
if nargin < 3, a = 0; b = 9; end
a = floor((b-a+1)*rand(m,n)) + a;
This should be placed in a diskfile with filename
randint.m (corresponding to the function name). The
first line declares the function name, input arguments, and
output arguments; without this line the file would be a script
file. Then a MATLAB statement z = randint(4,5), for
example, will cause the numbers 4 and 5 to be passed to the
variables a and b in the function file with the
output result being passed out to the variable z. Since
variables in a function file are local, their names are
independent of those in the current MATLAB environment.
Note that use of nargin ("number of input arguments") permits one to set a default value of an omitted input variable-such as a and b in the example.
Tehtävä Kirjoita tämä tiedostoon randint.m (Voit leikata/liimata tästä suoraan, onhan sinulla emacs auki!) (Tietysti UNIX:ssa näinkin: Hiiren vasen, maalaus, cat > randint.m, hiiren keskimmäinen, CTR-D). Kokeile help randint ja testaa.
A function may also have multiple output arguments. For example:
function [mean, stdev] = stat(x)
% STAT Mean and standard deviation
% For a vector x, stat(x) returns the
% mean and standard deviation of x.
% For a matrix x, stat(x) returns two row vectors containing,
% respectively, the mean and standard deviation of each column.
[m n] = size(x);
if m == 1
m = n; % handle case of a row vector
end
mean = sum(x)/m;
stdev = sqrt(sum(x.^ 2)/m - mean.^2);
Once this is placed in a diskfile stat.m, a MATLAB
command [xm, xd] = stat(x), for example, will assign the
mean and standard deviation of the entries in the vector
x to m and xd, respectively. Single
assignments can also be made with a function having multiple
output arguments. For example, xm = stat(x) (no
brackets needed around xm) will assign the mean of
x to xm.
The % symbol indicates that the rest of the line is a comment; MATLAB will ignore the rest of the line. However, the first few comment lines, which document the M-file, are available to the on-line help facility and will be displayed if, for example, help stat is entered. Such documentation should always be included in a function file.
This function illustrates some of the MATLAB features that can be used to produce efficient code. Note, for example, that x.^2 is the matrix of squares of the entries of x, that sum is a vector function (section 8), that sqrt is a scalar function (section 7), and that the division in sum(x)/m is a matrix-scalar operation.
The following function, which gives the greatest common divisor of two integers via the Euclidean algorithm, illustrates the use of an error message (see the next section).
function a = gcd(a,b)
% GCD Greatest common divisor
% gcd(a,b) is the greatest common divisor of
% the integers a and b, not both zero.
a = round(abs(a)); b = round(abs(b));
if a == 0 & b == 0
error('The gcd is not defined when both numbers are zero')
else
while b ~= 0
r = rem(a,b);
a = b; b = r;
end
end
Some more advanced features are illustrated by the following
function. As noted earlier, some of the input arguments of a
function-such as tol in the example, may be made
optional through use of nargin ("number of input
arguments"). The variable nargout can be similarly
used. Note that the fact that a relation is a number (1 when
true; 0 when false) is used and that, when while or
if evaluates a relation, "nonzero" means "true" and
0 means "false". Finally, the MATLAB function
feval permits one to have as an input variable a string
naming another function.
function [b, steps] = bisect(fun, x, tol)
%BISECT Zero of a function of one variable via the bisection method.
% bisect(fun,x) returns a zero of the function. fun is a string
% containing the name of a real-valued function of a single
% real variable; ordinarily functions are defined in M-files.
% x is a starting guess. The value returned is near a point
% where fun changes sign. For example,
% bisect('sin',3) is pi. Note the quotes around sin.
%
% An optional third input argument sets a tolerence for the
% relative accuracy of the result. The default is eps.
% An optional second output argument gives a matrix containing a
% trace of the steps; the rows are of form [c f(c)].
% Initialization
if nargin < 3, tol = eps; end
trace = (nargout == 2);
if x ~= 0, dx = x/20; else, dx = 1/20; end
a = x - dx; fa = feval(fun,a);
b = x + dx; fb = feval(fun,b);
% Find change of sign.
while (fa > 0) == (fb > 0)
dx = 2.0*dx;
a = x - dx; fa = feval(fun,a);
if (fa > 0) ~= (fb > 0), break, end
b = x + dx; fb = feval(fun,b);
end
if trace, steps = [a fa; b fb]; end
% Main loop
while abs(b - a) > 2.0*tol*max(abs(b),1.0)
c = a + 0.5*(b - a); fc = feval(fun,c);
if trace, steps = [steps; [c fc]]; end
if (fb > 0) == (fc > 0)
b = c; fb = fc;
else
a = c; fa = fc;
end
end
Some of MATLAB's functions are built-in while others are
distributed as M-files. The actual listing of any
M-file-MATLAB's or your own-can be viewed with the MATLAB command
type functionname. Try entering type
eig, type vander, and type rank.
Opettavaista katsoa, miten "isot pojat/tytöt" Mathworks:ssa ovat funktioita kirjoitelleet. Sama pätee Mapleen (Maplesoft).
Tehtäviä Kirjoita funktio hypot , joka laskee argumentteina annettuja kateettimatriiseja A ja B vastaavan hypotenuusamatriisin Pythagoraan lauseen avulla.
Tiedosto hypot.m
function C=hypot(A,B) % Selitys ... C=... ;
Text strings are entered into MATLAB surrounded by single quotes. For example,
s = 'This is a test'
assigns the given text string to the variable s.
Text strings can be displayed with the function disp. For example:
disp('this message is hereby displayed')
Error messages are best displayed with the function error
error('Sorry, the matrix must be symmetric')
since when placed in an M-File, it causes execution to exit the M-file.
In an M-file the user can be prompted to interactively enter input data with the function input. When, for example, the statement
iter = input('Enter the number of iterations: ')
is encountered, the prompt message is displayed and execution
pauses while the user keys in the input data. Upon pressing the
return key, the data is assigned to the variable iter
and execution resumes.
As noted in section 1, on systems permitting multiple processes, such as one running Unix, it may be preferable to keep both MATLAB and your local editor active, keeping one process suspended while working in the other. (CTR-Z bg , takaisin Matlabiin: fg )
If these processes can be run in multiple windows, as on a workstation, you will want to keep MATLAB active in one window and your editor active in another. YES, INDEED !
Version 4.0 has many debbugging tools (even more in 5.3). See help dbtype and references given there.
M-files must be accessible to MATLAB. On most mainframe or workstation network installations, personal M-files which are stored in a subdirectory of one's home directory named matlab will be accessible to MATLAB from any directory in which one is working. See the discussion of MATLABPATH in the User's Guide for further information.
Two measures of the efficiency of an algorithm are the number of floating point operations (flops) performed and the elapsed time.
The MATLAB function flops keeps a running total of the flops performed. The command flops(0) (not flops = 0!) will reset flops to 0. Hence, entering flops(0) immediately before executing an algorithm and flops immediately after gives the flop count for the algorithm.
The MATLAB function clock gives the current time accurate to a hundreth of a second (see help clock). Given two such times t1 and t2, etime(t2,t1) gives the elapsed time from t1 to t2. One can, for example, measure the time required to solve a given linear system Ax=b using Gaussian elimination as follows:
t = clock; x = A \ b; time = etime(clock,t)
You may wish to compare this time-and flop count-with that for
solving the system using x = inv(A)*b;. Try it.
Version 4.0 has the more convenient tic and
toc.
It should be noted that, on timesharing machines, etime may not be a reliable measure of the efficiency of an algorithm since the rate of execution depends on how busy the computer is at the time.
While all computations in MATLAB are performed in double precision, the format of the displayed output can be controlled by the following commands.
format short fixed point with 4 decimal places (the default)
format long fixed point with 14 decimal places
format short e scientific notation with 4 decimal places
format long e scientific notation with 15 decimal places
Once invoked, the chosen format remains in effect until changed.
The command format compact will suppress most blank lines allowing more information to be placed on the screen or page. It is independent of the other format commands.
Hardcopy is most easily obtained with the diary command. The command
diary filename
causes what appears subsequently on the screen (except graphics)
to be written to the named diskfile (if the filename is omitted
it will be written to a default file named diary) until
one gives the command diary off; the command diary
on will cause writing to the file to resume, etc. When
finished, you can edit the file as desired and print it out on
the local system.
MATLAB can produce both planar plots and 3-D mesh surface plots. To preview some of these capabilities in version 3.5, enter the command plotdemo.
Planar plots. The plot command creates linear x-y plots; if x and y are vectors of the same length, the command plot(x,y) opens a graphics window and draws an x-y plot of the elements of x versus the elements of y. You can, for example, draw the graph of the sine function over the interval -4 to 4 with the following commands:
x = -4:.01:4; y = sin(x); plot(x,y)
Try it. The vector x is a partition of the domain with
meshsize 0.01 while y is a vector giving the values of
sine at the nodes of this partition (recall that sin
operates entrywise).
Kuten edellä oli puhe, usein linspace voi olla kätevämpikin (saadaan aina 100:aan osaan jako, jos käytetään oletusarvoa). Siis edellä olisi yhtä hyvin voitu kirjoittaa:
x=linspace(-4,4);
plot-funktion toiminnasta saa havainnollisen kuvan ottamalla harvan
diskretoinnin. Kokeile vaikka näin:
x=linspace(-4,4,10); y=sin(x); [x' y'], plot(x,y)
Varmista, että ymmärrät kaiken, kiinnitä myös huomiota puolipisteisiin
ja pilkkuun, tässä on kaikki harkittua.
Huomaa, että Matlabin nuolinäppäineditointityylillä on kätevää operoida,
kun kirjoitetaan pitkiä rivejä. Tässäkin tarvitaan vain yksi nuolipainallus,
jonka jälkeen voidaan esim. diskretointimäärää käydä muuttamassa ja lähettää
rivi uudelleen suoritukseen.
Tehtävä Piirrä funktion
As a second example, you can draw the graph of y=e-x^2 over the interval -1.5 to 1.5 as follows:
x = -1.5:.01:1.5; y = exp(-x.^2); plot(x,y)
Note that one must precede the power-to sign by a period to
ensure that it operates entrywise (see section
3).
Parametrimuodossa annetut käyrät
Plots of parametrically defined curves can also be made. Try, for example,
t=0:.001:2*pi; x=cos(3*t); y=sin(2*t); plot(x,y)
Ennen tätä kannattaa kokeilla kaikkein perustavanlaatuisinta,
eli yksikköympyrää (piirrämme oikeastaan 100-kulmion):
t=linspace(0,2*pi);x=cos(t);y=sin(t);plot(x,y);
axis('equal');axis('square')
Vertaa tätä kuvaan
t=linspace(0,2*pi);x=cos(t);y=sin(t);plot(t,x,'r',t,y,'b');Ymmärräthän nyt varmasti! Vrt. Maplen parametripiirto, harj0.mws , [HAM] s. 93
The command grid will place grid lines on the current graph. Kannattaa käyttää !
Jos ympyrä on perusparametrikäyrä tasossa, niin ruuviviiva on sitä avarudessa.
t=linspace(0,4*pi);x=cox(t);y=sin(t);z=t; plot3(x,y,z); % Ruuviviiva
Napakoordinaatistossa annettu käyrä
fii=linspace(fii1,fii2);x=r(fii).*cos(fii);y=r(fii).*sin(fii);
plot(x,y);
axis('equal');axis('square');grid % Usein näissä halutaan sama
% skaala akseleille
Matlabissa on myös funktio polar , jossa napakoordinaattimuunnos
on sisäänrakennettu. Sitä käyttäen voidaan piirtää suoraan näin:
fii=linspace(fii1,fii2);polar(fii,r(fii));
grid % Lisäbonuksena saadaan myös hilaviivat napakoordinaattiviivoina
Lisää napakoordinaattiesim. [BB] s. 53 -> ja sec 10.3 ja exe 10.1, 10.3
The graphs can be given titles, axes labeled, and text placed
within the graph with the following commands which take a string
as an argument.
title graph title
xlabel x-axis label
ylabel y-axis label
gtext interactively-positioned text
text position text at specified coordinates
For example, the command
title('Best Least Squares Fit')
gives a graph a title. The command gtext('The Spot')
allows a mouse or the arrow keys to position a crosshair on the
graph, at which the text will be placed when any key is pressed.
Tämäkin voidaan nykyisin hoitaa kuvaikkunan valikosta.
By default, the axes are auto-scaled. This can be overridden by the command axis. If c=[xmin,xmax,ymin,ymax] is a 4-element vector, then axis(c) sets the axis scaling to the precribed limits. By itself, axis freezes the current scaling for subsequent graphs; entering axis again returns to auto-scaling. The command axis('square') ensures that the same scale is used on both axes. In version 4.0, axis has been significantly changed; see help axis.
Two ways to make multiple plots on a single graph are illustrated by
x=0:.01:2*pi;y1=sin(x);y2=sin(2*x);
y3=sin(4*x);plot(x,y1,x,y2,x,y3)
and by forming a matrix Y containing the functional
values as columns
x=0:.01:2*pi; Y=[sin(x)', sin(2*x)', sin(4*x)']; plot(x,Y)
Another way is with hold. The command hold
freezes the current graphics screen so that subsequent plots are
superimposed on it. Entering hold again releases the
"hold." The commands hold on and hold off are
also available in version 4.0.
One can override the default linetypes and pointtypes. For example,
x=0:.01:2*pi; y1=sin(x); y2=sin(2*x); y3=sin(4*x);
plot(x,y1,'--',x,y2,':',x,y3,'+')
plot(x,y1,'--b',x,y2,'g',x,y3,'+y')
renders a dashed line and dotted line for the first two graphs
while for the third the symbol is placed at each node. The line-
and mark-types are
Linetypes: solid (-), dashed (-). dotted (:), dashdot (-.)See help plot for line and mark colors.Marktypes: point (.), plus (+), star (*), circle (o), x-mark (x)
Viimeisellä rivillä lisättiin vielä värimääreet (lisää help plot, help color).
'b' - blue, 'g' - green, 'y' - yellow
The command subplot can be used to partition the screen so that up to four plots can be viewed simultaneously. See help subplot.
Kuva voidaan myös tulostaa tiedostoon monissa eri muodoissa, kuten eps, gif, jpeg, ym. Voidaan käyttää print-komennon valitsimia tai tulostaa suoraan grafiikkaikkunan valikosta. Versiossa 5.3 on mahdollisuus myös editoida kuvaa piirtämällä siihen viivoja, lisäämällä tekstiä ym. Valikko selittää itse itsensä.
Piirtoikkunat, figure, clf, shg
Grafiikkakomento (kuten plot) avaa ikkunan numero 1. Jos halutaan ohjata grafiikka uuteen ikkunaan, komennetaan figure tai figure(n). Aiemmin avattuihin ikkunoihin päästään käsiksi komenttamalla vaikkapa figure(1) , jolloin voidaan jatkaa piirtämistä ensimmäiseen ikkunaan, arvatenkin käytteän hyväksi hold on komentoa, tai voidaan editoida kuvaa, tulostaa se, ym.
3-D mesh plots. Three dimensional mesh surface plots are drawn with the function mesh. The command mesh(z) creates a three-dimensional perspective plot of the elements of the matrix z. The mesh surface is defined by the z-coordinates of points above a rectangular grid in the x-y plane. Try mesh(eye(10)).
To draw the graph of a function z=f(x,y) over a rectangle, one first defines vectors x and y which give partitions of the sides of the rectangle. With the function meshgrid one then creates a matrix X, each row of which equals x and whose column length is the length of y, and similarly a matrix Y, each column of which equals y, as follows:
[X,Y] = meshgrid(x,y);
One then computes a matrix Z, obtained by evaluating
f entrywise over the matrices X and Y,
to which mesh can be applied.
You can, for example, draw the graph of z=e-x^2-y^2 over the square [-2,2] x [-2,2] as follows (try it):
x = -2:.1:2;
y = x;
[X,Y] = meshgrid(x,y);
Z = exp(-X.^2 - Y.^2);
mesh(x,y,Z) % yhtä hyvin mesh(X,Y,Z)
You are referred to the User's Guide for further details
regarding mesh.
Lisää [BB] s. 54 ->
Kts. myös surf, surfc, contour, waterfall, etc.
In version 4.0, the 3-D graphics capabilities of MATLAB have been considerably expanded. Consult the on-line help for plot3, mesh, and surf.
Katselusuuntaa voi säädellä view-komennolla ja versiossa 5.3 voi kuvaa pyöritellä myös grafiikkaikkunan valikosta hiiren avulla. (Ei ihan niin hienosti kuin Maplessa mutta toisaalta grafiikka on Matlabissa paljon nopeampaa.)
Jatketaan vielä yllä olevaa esimerkkiä piirtämällä vierekkäin pintakuva ja korkeuskäyrät, lisätään tekstejä:
subplot(1,2,1),surfc(x,y,Z)
title('Pintapiirros')
subplot(1,2,2),contour(x,y,Z)
title('Korkeuskäyrät')
x=linspace(a,b);plot(x,f(x))
Hieman vaivattomammin voimme tehdä näin
fplot('f',[a,b])
Tämä vastaa läheisemmin sitä tapaa, jota esim. Maplen plot noudattaa.
(Käyttäjän ei tarvitse huolehtia diskretoinnista.)
Kokeile nyt aluksi vaikka fplot('log',[0.5,2])
Tehtävä Piirrä fplot:n avulla funktion
There are many MATLAB features which cannot be included in these introductory notes. Listed below are some of the MATLAB functions and operators available, grouped by subject area (Source: MATLAB User's Guide, version 3.5). Use the on-line help facility or consult the User's Guide for more detailed information on the functions.
There are many functions beyond these. There exist, in particular, several "toolboxes" of functions for specific areas; included among such are signal processing, control systems, robust-control, system identification, optimization, splines, chemometrics, mu-analysis and synthesis, state-space identification, and neural networks. (The toolboxes, which are optional, may not be installed on your system.) These can be explored via the command help.
General
help help facility
demo run demonstrations
who list variables in memory
what list M-files on disk
size row and column dimensions
length vector length
clear clear workspace
computer type of computer
^C local abort
exit exit MATLAB
quit same as exit
Matrix/Array Operators
Matrix Operators Array Operators
---------------------------------------------------
+ addition + addition
- subtraction - subtraction
* multiplication .* multiplication
/ right division ./ right division
\ left division .\ left division
^ power .^ power
' conjugate transpose .' transpose
Relational and Logical Operators
< less than
<= less than or equal
> greater than
>= greater than or equal
== equal
~= not equal
& and
| or
~ not
Special Characters
= assignment statement
[ used to form vectors and matrices
] see [
( arithmetic expression precedence
) see (
. decimal point
... continue statement to next line
, separate subscripts and function arguments
; end rows, suppress printing
% comments
: subscripting, vector generation
! execute operating system command
Special Values
ans answer when expression not assigned
eps floating point precision
pi pi
i, j sqrt(-1)
inf infinity
NaN Not-a-Number
clock wall clock
date date
flops floating point operation count
nargin number of function input arguments
nargout number of function output arguments
tic, toc
Disk Files
chdir change current directory
delete delete file
diary diary of the session
dir directory of files on disk
load load variables from file
save save variables to file
type list function or file
what show M-files on disk
fprintf write to a file
pack compact memory via save
Special Matrices
compan companion
diag diagonal
eye identity
gallery esoteric
hadamard Hadamard
hankel Hankel
hilb Hilbert
invhilb inverse Hilbert
linspace linearly spaced vectors
logspace logarithmically spaced vectors
magic magic square
meshgrid domain for mesh points
ones constant
pascal Pascal
rand random elements
toeplitz Toeplitz
vander Vandermonde
zeros zero
Matrix Manipulation
rot90 rotation
fliplr flip matrix left-to-right
flipud flip matrix up-to-down
diag diagonal matrices
tril lower triangular part
triu upper triangular part
reshape reshape
.' transpose
: convert matrix to single column; A(:)
Relational and Logical Functions
any logical conditions
all logical conditions
find find array indices of logical values
isnan detect NaNs
finite detect infinities
isempty detect empty matrices
isstr detect string variables
strcmp compare string variables
Control Flow
if conditionally execute statements
elseif used with if
else used with if
end terminate bif, for, while
for repeat statements a number of times
while do while
break break out of for and while loops
return return from functions
pause pause until key pressed
Programming and M-files
input get numbers from keyboard
keyboard call keyboard as M-file
error display error message
function define function
eval interpret text in variables
feval evaluate function given by string
echo enable command echoing
exist check if variables exist
casesen set case sensitivity
global define global variables
startup startup M-file
getenv get environment string
menu select item from menu
etime elapsed time
tic,toc
Text and Strings
abs convert string to ASCII values
eval evaluate text macro
num2str convert number to string
int2str convert integer to string
setstr set flag indicating matrix is a string
sprintf convert number to string
isstr detect string variables
strcomp compare string variables
hex2num convert hex string to number
Command Window
clc clear command screen
home move cursor home
format set output display format
disp display matrix or text
fprintf print formatted number
echo enable command echoing
Graph Paper
plot linear X-Y plot
plot3
loglog loglog X-Y plot
semilogx semi-log X-Y plot
semilogy semi-log X-Y plot
polar polar plot
mesh 3-dimensional mesh surface
surf
contour contour plot
meshgrid domain for mesh plots
bar bar charts
stairs stairstep graph
errorbar add error bars
Graph Annotation
title plot title
xlabel x-axis label
ylabel y-axis label
grid draw grid lines
text arbitrarily position text
gtext mouse-positioned text
ginput graphics input
zoom
Graph Window Control
axis manual axis scaling
hold hold plot on screen
shg show graph window
clf clear figure
subplot split graph window
Graph Window Hardcopy
print send graph to printer or file, several formats
Elementary Math Functions
abs absolute value or complex magnitude
angle phase angle
sqrt square root
real real part
imag imaginary part
conj complex conjugate
round round to nearest integer
fix round toward zero
floor round toward -infinity
ceil round toward infinity
sign signum function
rem remainder
exp exponential base e
log natural logarithm
log10 log base 10
Trigonometric Functions
sin sine
cos cosine
tan tangent
asin arcsine
acos arccosine
atan arctangent
atan2 four quadrant arctangent
sinh hyperbolic sine
cosh hyperbolic cosine
tanh hyperbolic tangent
asinh hyperbolic arcsine
acosh hyperbolic arccosine
atanh hyperbolic arctangent
Special Functions
bessel bessel function
gamma gamma function
rat rational approximation
erf error function
inverf inverse error function
ellipk complete elliptic integral of first kind
ellipj Jacobian elliptic integral
Decompositions and Factorizations
balance balanced form
backsub backsubstitution
cdf2rdf convert complex-diagonal to real-diagonal
chol Cholesky factorization
eig eigenvalues and eigenvectors
hess Hessenberg form
inv inverse
lu factors from Gaussian elimination
nnls nonnegative least squares
null null space
orth orthogonalization
pinv pseudoinverse
qr orthogonal-triangular decomposition
qz QZ algorithm
rref reduced row echelon form
schur Schur decomposition
svd singular value decomposition
Matrix Conditioning
cond condition number in 2-norm
norm 1-norm,2-norm,F-norm, infinity-norm
rank rank
rcond condition estimate (reciprocal)
Elementary Matrix Functions
expm matrix exponential
logm matrix logarithm
sqrtm matrix square root
funm arbitrary matrix function
poly characteristic polynomial
det determinant
trace trace
kron Kronecker tensor product
Polynomials
poly characteristic polynomial
roots polynomial roots---companion matrix method
roots1 polynomial roots---Laguerre's method
polyval polynomial evaluation
polyvalm matrix polynomial evaluation
conv multiplication
deconv division
residue partial-fraction expansion
polyfit polynomial curve fitting
Column-wise Data Analysis
max maximum value
min minimum value
mean mean value
median median value
std standard deviation
sort sorting
sum sum of elements
prod product of elements
cumsum cumulative sum of elements
cumprod cumulative product of elements
diff approximate derivatives
hist histograms
corrcoef correlation coefficients
cov covariance matrix
cplxpair reorder into complex pairs
Signal Processing
abs complex magnitude
angle phase angle
conv convolution
corrcoef correlation coefficients
cov covariance
deconv deconvolution
fft radix-2 fast Fourier transform
fft2 two-dimensional FFT
ifft inverse fast Fourier transform
ifft2 inverse 2-D FFT
fftshift FFT rearrangement
Numerical Integration
quad numerical function integration
quad8 numerical function integration
Differential Equation Solution
ode23 2nd/3rd order Runge-Kutta method
ode45 4th/5th order Runge-Kutta-Fehlberg method
Nonlinear Equations and Optimization
fmin minimum of a function of one variable
fmins minimum of a multivariable function
fsolve solution of a system of nonlinear equations
(zeros of a multivariable function)
fzero zero of a function of one variable
Interpolation
polyfit
spline cubic spline
table1 1-D table look-up
table2 2-D table look-up
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