function [mean,stdev] = stat(x)
            % Keskeisiä tilastollisia tunnuslukuja
            % x - vektori
            % esim: [ke,std]=stat(1:10)
            n = length(x);
            mean = sum(x) / n;
            stdev = sqrt(sum((x - mean).^2)/n);

        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));

        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;

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.

Kehitetään yllä oleva stat-funktio tomimaan vektorifunktioiden tavoin, jos syötteenä on matriisi, siis saraketyyliin, kuten sum, max jne.

        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);

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

        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

Opettavaista katsoa, miten "isot pojat/tytöt" Mathworks:ssa ovat funktioita kirjoitelleet. Sama pätee Mapleen (Maplesoft).

13. Text strings, error messages, input

        s = 'This is a test'

Text strings can be displayed with the function disp. For example:

        disp('this message is hereby displayed')

        error('Sorry, the matrix must be symmetric')

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:  ')

14. Managing M-files, Matlab environment

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.

Matlabin tuntemia UNIX/Dos-komentoja

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.

15. Comparing efficiency of algorithms: flops and (etime) tic/toc

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.

tic ja toc, esim:

   A=rand(100,100);b=ones(100,1); tic; x = A \ b;  toc

16. Output format

        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

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.

17. Hardcopy

        diary  filename 

18. Graphics

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)

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);

       x=linspace(-4,4,10); y=sin(x); [x' y'], plot(x,y)

Tehtävä Piirrä funktion x³-2x +1 kuvaaja välillä [-2,2] punaisella värillä.

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)

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)

   t=linspace(0,2*pi);x=cos(t);y=sin(t);plot(x,y);
   axis('equal');axis('square')

   t=linspace(0,2*pi);x=cos(t);y=sin(t);plot(t,x,'r',t,y,'b');

The command grid will place grid lines on the current graph. Kannattaa käyttää !

Käyrä avaruudessa

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

    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

    fii=linspace(fii1,fii2);polar(fii,r(fii));
    grid  % Lisäbonuksena saadaan myös hilaviivat napakoordinaattiviivoina

        title         graph title
        xlabel        x-axis label
        ylabel        y-axis label
        gtext         interactively-positioned text
        text          position text at specified coordinates

        title('Best Least Squares Fit')

By default, the axes are auto-scaled. This can be overridden by the command axis. If c=[x_min,x_max,y_min,y_max] 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)

        x=0:.01:2*pi; Y=[sin(x)', sin(2*x)', sin(4*x)']; plot(x,Y)

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')

Linetypes: solid (-), dashed (-). dotted (:), dashdot (-.)

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

Graphics hardcopy, kuvan editointi

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);

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)

Lisää [BB] s. 54 ->

Kts. myös surf, surfc, contour, waterfall, etc.

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')

Funktion piirto, fplot

    x=linspace(a,b);plot(x,f(x))

   fplot('f',[a,b])

Kokeile nyt aluksi vaikka fplot('log',[0.5,2])

Tehtävä Piirrä fplot:n avulla funktion f(x)=e^-x2 sin(x)/x kuvaaja erilaisilla väleillä.

20. Uudet tietorakenteet

A={'Otsikko';[1 2;3 4];magic(10)}
A
celldisp(A)
cellplot(A)
A{1,1},A{2,1},A{3,1}
%% ei toimi näin: %[ots,ykakone,taika10]=deal([A{1,1},A{2,1},A{3,1}])
A={'Otsikko';{1 2;3 4};magic(10)}

A
celldisp(A)
cellplot(A)

clear
AA={'A';magic(3)}
nimi=AA{1}
A=AA{2}

clear
M=magic(4)
A.nimi='M'
A.data=M
A2.nimi='A2'
A2.data=A.data^2

for i=1:5
nimet{i}=['nimi',num2str(i)]
  end;
 cellplot(nimet);shg

A(:,:,1)=[1 2 3;-2 3 -1;5 -4 -1];A(:,:,2)=vander(1:3);

A(:,:,1) =
     1     2     3
    -2     3    -1
     5    -4    -1
A(:,:,2) =
     1     1     1
     4     2     1
     9     3     1
» max(A)
ans(:,:,1) =
     5     3     3
ans(:,:,2) =
     9     3     1
» min(A)
ans(:,:,1) =
    -2    -4    -1
ans(:,:,2) =
     1     1     1

» whos
  Name      Size         Bytes  Class

  A         3x3x2          144  double array   % 3 riviä, 3 saraketta, 2 tasoa
  ans       1x3x2           48  double array

min(min(A))

ans(:,:,1) =
    -4
ans(:,:,2) =
     1
» min(min(min(A)))
ans =
    -4
» min(A(:))
ans =
    -4

A
sum(A)
A(:,:,1) =
     1     2     3
    -2     3    -1
     5    -4    -1
A(:,:,2) =
     1     1     1
     4     2     1
     9     3     1
ans(:,:,1) =
     4     1     1     % 1. tason sarakesummat
ans(:,:,2) =
    14     6     3     % 2. tason sarakesummat

cumsum(A)
ans(:,:,1) =
     1     2     3   % 1. tason sarakkeittain otetut kumulatiiviset summat
    -1     5     2
     4     1     1
ans(:,:,2) =
     1     1     1   % 2. tason sarakkeittain otetut kumulatiiviset summat
     5     3     2
    14     6     3

format compact
x=[1 1 3 4 5 1 9 8]
[m,y]=hist(x,10)  % Oletusarvo 10, toimii siten muodossa hist(x). 
bar(y,m)

house=ceil(10*rand(3,3,3,4))
A=cell(3,4)
for i=1:3
   for j=1:4
     A{i,j}=ceil(10*rand(3,3));
   end;
end;

celldisp(A)
cellplot(A)

>> help colspace

 --- help for sym/colspace.m ---

 COLSPACE Basis for column space.
    The columns of B = COLSPACE(A) form a basis for the column space of A.
    SIZE(B,2) is the rank of A.
 
    Example:
 
      colspace(sym(magic(4))) is
 
        [  0,  1,  0]
        [  0,  0,  1]
        [  1,  0,  0]
        [ -3,  1,  3]
 
    See also NULL.

20. Reference

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