euler
t

t =

  Columns 1 through 3

                   0   0.300000000000000   0.600000000000000

  Columns 4 through 6

   0.900000000000000   1.200000000000000   1.500000000000000

  Columns 7 through 9

   1.800000000000000   2.100000000000000   2.400000000000000

  Columns 10 through 11

   2.700000000000000   3.000000000000000

euler
{Error: <a href="matlab: opentoline('/Users/pauldj/works/courses/LSC/LSC-14/matlab/ode1.m',5,4)">File: ode1.m Line: 5 Column: 4
</a>This statement is not inside any function.
 (It follows the END that terminates the definition of the function
 "ode1".)

Error in <a href="matlab:helpUtils.errorDocCallback('euler', '/Users/pauldj/works/courses/LSC/LSC-14/matlab/euler.m', 9)" style="font-weight:bold">euler</a> (<a href="matlab: opentoline('/Users/pauldj/works/courses/LSC/LSC-14/matlab/euler.m',9,0)">line 9</a>)
    u(i+1) = u(i) + h * ode1( u(i), t(i) );
} 
euler
plot(t,u)
plot(t,u)
euler
plot(t,u)



help ode45
 <strong>ode45</strong>  Solve non-stiff differential equations, medium order method.
    [TOUT,YOUT] = <strong>ode45</strong>(ODEFUN,TSPAN,Y0) with TSPAN = [T0 TFINAL] integrates 
    the system of differential equations y' = f(t,y) from time T0 to TFINAL 
    with initial conditions Y0. ODEFUN is a function handle. For a scalar T
    and a vector Y, ODEFUN(T,Y) must return a column vector corresponding 
    to f(t,y). Each row in the solution array YOUT corresponds to a time 
    returned in the column vector TOUT.  To obtain solutions at specific 
    times T0,T1,...,TFINAL (all increasing or all decreasing), use TSPAN = 
    [T0 T1 ... TFINAL].     
    
    [TOUT,YOUT] = <strong>ode45</strong>(ODEFUN,TSPAN,Y0,OPTIONS) solves as above with default
    integration properties replaced by values in OPTIONS, an argument created
    with the ODESET function. See ODESET for details. Commonly used options 
    are scalar relative error tolerance 'RelTol' (1e-3 by default) and vector
    of absolute error tolerances 'AbsTol' (all components 1e-6 by default).
    If certain components of the solution must be non-negative, use
    ODESET to set the 'NonNegative' property to the indices of these
    components.
    
    <strong>ode45</strong> can solve problems M(t,y)*y' = f(t,y) with mass matrix M that is
    nonsingular. Use ODESET to set the 'Mass' property to a function handle 
    MASS if MASS(T,Y) returns the value of the mass matrix. If the mass matrix 
    is constant, the matrix can be used as the value of the 'Mass' option. If
    the mass matrix does not depend on the state variable Y and the function
    MASS is to be called with one input argument T, set 'MStateDependence' to
    'none'. ODE15S and ODE23T can solve problems with singular mass matrices.  
 
    [TOUT,YOUT,TE,YE,IE] = <strong>ode45</strong>(ODEFUN,TSPAN,Y0,OPTIONS) with the 'Events'
    property in OPTIONS set to a function handle EVENTS, solves as above 
    while also finding where functions of (T,Y), called event functions, 
    are zero. For each function you specify whether the integration is 
    to terminate at a zero and whether the direction of the zero crossing 
    matters. These are the three column vectors returned by EVENTS: 
    [VALUE,ISTERMINAL,DIRECTION] = EVENTS(T,Y). For the I-th event function: 
    VALUE(I) is the value of the function, ISTERMINAL(I)=1 if the integration 
    is to terminate at a zero of this event function and 0 otherwise. 
    DIRECTION(I)=0 if all zeros are to be computed (the default), +1 if only 
    zeros where the event function is increasing, and -1 if only zeros where 
    the event function is decreasing. Output TE is a column vector of times 
    at which events occur. Rows of YE are the corresponding solutions, and 
    indices in vector IE specify which event occurred.    
 
    SOL = <strong>ode45</strong>(ODEFUN,[T0 TFINAL],Y0...) returns a structure that can be
    used with DEVAL to evaluate the solution or its first derivative at 
    any point between T0 and TFINAL. The steps chosen by <strong>ode45</strong> are returned 
    in a row vector SOL.x.  For each I, the column SOL.y(:,I) contains 
    the solution at SOL.x(I). If events were detected, SOL.xe is a row vector 
    of points at which events occurred. Columns of SOL.ye are the corresponding 
    solutions, and indices in vector SOL.ie specify which event occurred. 
 
    Example    
          [t,y]=ode45(@vdp1,[0 20],[2 0]);   
          plot(t,y(:,1));
      solves the system y' = vdp1(t,y), using the default relative error
      tolerance 1e-3 and the default absolute tolerance of 1e-6 for each
      component, and plots the first component of the solution. 
    
    Class support for inputs TSPAN, Y0, and the result of ODEFUN(T,Y):
      float: double, single
 
    See also <a href="matlab:help ode23">ode23</a>, <a href="matlab:help ode113">ode113</a>, <a href="matlab:help ode15s">ode15s</a>, <a href="matlab:help ode23s">ode23s</a>, <a href="matlab:help ode23t">ode23t</a>, <a href="matlab:help ode23tb">ode23tb</a>, <a href="matlab:help ode15i">ode15i</a>,
             <a href="matlab:help odeset">odeset</a>, <a href="matlab:help odeplot">odeplot</a>, <a href="matlab:help odephas2">odephas2</a>, <a href="matlab:help odephas3">odephas3</a>, <a href="matlab:help odeprint">odeprint</a>, <a href="matlab:help deval">deval</a>,
             <a href="matlab:help odeexamples">odeexamples</a>, <a href="matlab:help rigidode">rigidode</a>, <a href="matlab:help ballode">ballode</a>, <a href="matlab:help orbitode">orbitode</a>, <a href="matlab:help function_handle">function_handle</a>.

    Reference page in Help browser
       <a href="matlab:doc ode45">doc ode45</a>

ode45( @ode1, [0 3], 1 )
ode45( @ode1, [0 3], 1 )
hold on
plot(t,u,'+r')
feuler( @ode1, 1, 0, 3, 10 )

ans =

  Columns 1 through 3

   1.000000000000000   0.700000000000000  -0.618443691672424

  Columns 4 through 6

  -0.549083037934104   0.211793004434057   0.274492599267961

  Columns 7 through 9

  -0.121799308067884  -0.187443606786559   0.030376076322544

  Columns 10 through 11

   0.094278448793071  -0.015033202622791

[fu, ft ] = feuler( @ode1, 1, 0, 3, 1000 );
plot( ft, fu, 'b')
close all
plot( ft, fu, 'b')
hold
Current plot held
hold on
ode45( @ode1, [0 3], 1 )
[fu, ft ] = feuler( @ode2, exp(1), 1, 2, 10 )

fu =

  Columns 1 through 3

   2.718281828459046   3.533766376996759   4.539773443530052

  Columns 4 through 6

   5.774499187659123   7.282994814895872   9.118241839096839

  Columns 7 through 9

  11.342387458469149  14.028162191422943  17.260505010152450

  Columns 10 through 11

  21.138424678639183  25.777129906895595


ft =

  Columns 1 through 3

   1.000000000000000   1.100000000000000   1.200000000000000

  Columns 4 through 6

   1.300000000000000   1.400000000000000   1.500000000000000

  Columns 7 through 9

   1.600000000000000   1.700000000000000   1.800000000000000

  Columns 10 through 11

   1.900000000000000   2.000000000000000

[fu, ft ] = feuler( @ode2, exp(1), 1, 2, 1000 );
plot( ft, fu, 'r')
hold on
ode45( @ode2, [1 2], exp(1) )
[fu, ft ] = feuler( @ode2, exp(1), 1, 2, 100 );
[fu, ft ] = feuler( @ode2, exp(1), 1, 2, 100 );
ode45( @ode2, [1 2], exp(1) )
hold on
plot( ft, fu, 'r')
close all
ode45( @ode2, [1 4], exp(1) )
[fu, ft ] = feuler( @ode2, exp(1), 1, 4, 100 );
hold on
plot( ft, fu, 'r')
[fu, ft ] = feuler( @ode2, exp(1), 1, 4, 1000 );
plot( ft, fu, 'g')
target = -0.037822634055742

target =

  -0.037822634055742

[fu, ft ] = feuler( @ode1, 1, 0, 3, 10 );
abs( target - fu(end) )/abs(target)

ans =

   0.602534223273937

[fu, ft ] = feuler( @ode1, 1, 0, 3, 100 );
abs( target - fu(end) )/abs(target)

ans =

   0.001946729465011

[fu, ft ] = feuler( @ode1, 1, 0, 3, 1000 );
abs( target - fu(end) )/abs(target)

ans =

     7.272437623209290e-04

[fu, ft ] = feuler( @ode1, 1, 0, 3, 10000 );
abs( target - fu(end) )/abs(target)

ans =

     7.799831062937499e-05

[fu, ft ] = feuler( @ode1, 1, 0, 3, 100000 );
abs( target - fu(end) )/abs(target)

ans =

     7.852521557373846e-06

[fu, ft ] = feuler( @ode1, 1, 0, 3, 10000000 );
abs( target - fu(end) )/abs(target)

ans =

     7.858365598684429e-08

diary off