17-Dec-2012

1> f = inline('x./log(x)')
f = f(x) = x./log(x)

2> I1 = 2/6 * (f(3) + 4*f(4) + f(5))
I1 =  5.7930

3> I2 = (f(3) + f(5) + 2*f(4) + 4*(f(3.5) + f(4.5)))/6
I2 =  5.7918

4> I1 - I2
ans =  0.0011609

5> quad(f,3,4)
ans =  2.7985

6> quad(f,3,5)
ans =  5.7917

7> help quad
`quad' is a function from the file /usr/lib/octave/3.6.2/oct/i686-pc-cygwin/quad.oct

 -- Loadable Function: Q = quad (F, A, B)
 -- Loadable Function: Q = quad (F, A, B, TOL)
 -- Loadable Function: Q = quad (F, A, B, TOL, SING)
 -- Loadable Function: [Q, IER, NFUN, ERR] = quad (...)
     Numerically evaluate the integral of F from A to B using Fortran
     routines from QUADPACK.  F is a function handle, inline function,
     or a string containing the name of the function to evaluate.  The
     function must have the form `y = f (x)' where Y and X are scalars.

     A and B are the lower and upper limits of integration.  Either or
     both may be infinite.

     The optional argument TOL is a vector that specifies the desired
     accuracy of the result.  The first element of the vector is the
     desired absolute tolerance, and the second element is the desired
     relative tolerance.  To choose a relative test only, set the
     absolute tolerance to zero.  To choose an absolute test only, set
     the relative tolerance to zero.  Both tolerances default to
     `sqrt(eps)' or approximately 1.5e^-8.

     The optional argument SING is a vector of values at which the
     integrand is known to be singular.

     The result of the integration is returned in Q.  IER contains an
     integer error code (0 indicates a successful integration).  NFUN
     indicates the number of function evaluations that were made, and
     ERR contains an estimate of the error in the solution.

     The function `quad_options' can set other optional parameters for
     `quad'.

     Note: because `quad' is written in Fortran it cannot be called
     recursively.  This prevents its use in integrating over more than
     one variable by routines `dblquad' and `triplequad'.

     See also: quad_options, quadv, quadl, quadgk, quadcc, trapz,
     dblquad, triplequad



Additional help for built-in functions and operators is
available in the on-line version of the manual.  Use the command
`doc <topic>' to search the manual index.

Help and information about Octave is also available on the WWW
at http://www.octave.org and via the help@octave.org
mailing list.

8> M = 1000;

9> x = linspace(3, 5-2/M, M);

10> Ips = 2/M * sum(f(x))
Ips =  5.7913

11> xd = linspace(3+2/M, 5, M);

12> Ipd = 2/M * sum(f(xd))
Ipd =  5.7921

13> M = 1000000;

14> M = 1.e6;

15> M
M =  1000000

16> x = linspace(3, 5-2/M, M);

17> xd = linspace(3+2/M, 5, M);

18> Ips = 2/M * sum(f(x))
Ips =  5.7917

19> Ipd = 2/M * sum(f(xd))
Ipd =  5.7917

20> format long

21> Ips = 2/M * sum(f(x))
Ips =  5.79171832753196

22> Ipd = 2/M * sum(f(xd))
Ipd =  5.79171907944595

23> M = 1.e12;

24> x = linspace(3, 5-2/M, M);

24> xd = linspace(3+2/M, 5, M);

24> M = 1.e10;

25> x = linspace(3, 5-2/M, M);

25> M = 1.e9;

26> x = linspace(3, 5-2/M, M);

26> M = 1.e8;

27> x = linspace(3, 5-2/M, M);

28> xd = linspace(3+2/M, 5, M);

29> Ips = 2/M * sum(f(x))

29> M = 1.e7;

30> x = linspace(3, 5-2/M, M);

31> xd = linspace(3+2/M, 5, M);

32> Ips = 2/M * sum(f(x))
Ips =  5.79171866589320

33> Ipd = 2/M * sum(f(xd))
Ipd =  5.79171874108460

34> quad(f,3,5)
ans =  5.79171870348890

35> quadl(f,3,5)
ans =  5.79171870348890

36> 2*cos(0.64)-1.28*sin(0.64)
ans =  0.839781350824723

37> 0.4 * sin(0.64) 
ans =  0.238878176544957

38> quad('sin(x.^2)', 0, 0.8)
warning: quad: passing function body as a string is obsolete; please use anonymous functions
ans =  0.16574

39> 0.4 * sin(0.64)
ans =  0.23888

40> quit;