> diary 9_11_00

% first move to the directory where the files are saved.
> pwd
ans =
C:\MATLAB\bin

> cd ../..
> pwd
 ans =
 C:\
> cd _math320_00

% check the files are there:
> ls

.                  f_int.m            simp.m             trap_filter.m      
..                 left_hand.m        simp_filter.m      
f.m                left_hand_filter.m trap.m             

% look at "left_hand.m" to see what the command is to use it
> help left_hand

  this has the integration for the left-hand rule, n can be an even or
  odd number.
 
  [area,h,error] = left_hand(a,b,n)

% use Left Hand Rule to approximate int_0^1 cos(x) using 2, 4, 8, 16
% mesh-points
> [a_lhr1,h1,e_lhr1] = left_hand(0,1,2);
> [a_lhr2,h2,e_lhr2] = left_hand(0,1,4);
> [a_lhr3,h3,e_lhr3] = left_hand(0,1,8);
> [a_lhr4,h4,e_lhr4] = left_hand(0,1,16);

% look at the areas.  They're decreasing, hopefully to some value...
> a_lhr1
 a_lhr1 =
     1

> a_lhr2
a_lhr2 =
    0.9103

> a_lhr3
a_lhr3 =
    0.8729

> a_lhr4
a_lhr4 =
    0.8565

% look at the errors.  they appear to be decreasing by a factor of 2.
% this is a good sign.
> e_lhr1
e_lhr1 =
   -0.1585

> e_lhr2
e_lhr2 =
   -0.0688

> e_lhr3
e_lhr3 =
   -0.0314

> e_lhr4
e_lhr4 =
   -0.0150

% check whether they really are decreasing by a factor of 2...
> e_lhr1/e_lhr2
ans =
    2.3039

> e_lhr2/e_lhr3
ans =
    2.1911

> e_lhr3/e_lhr4
ans =
    2.0920

% what is the error if we have 10000 meshpoints?
> [a_lhr,h,e_lhr] = left_hand(0,1,10000);
> e_lhr
e_lhr =
 -2.2986e-005

% now repeat the excercise with the trapezoidal rule
> [a_trap1,h1,e_trap1] = trap(0,1,2);
> [a_trap1,h1,e_trap2] = trap(0,1,4);
> [a_trap1,h1,e_trap3] = trap(0,1,8);
> [a_trap1,h1,e_trap4] = trap(0,1,16);

% first, recall the errors with the left-hand rule
> e_lhr1, e_lhr2, e_lhr3, e_lhr4
e_lhr1 =
   -0.1585

e_lhr2 =
   -0.0688

e_lhr3 =
   -0.0314

e_lhr4 =
   -0.0150

% the errors are definitely smaller than with the left-hand rule
% also, they appear to be decreasing more quickly.
> e_trap1, e_trap2, e_trap3, e_trap4
e_trap1 =
    0.0713

e_trap2 =
    0.0078

e_trap3 =
    0.0014

e_trap4 =
  3.1168e-004

% look at the ratios of the errors to see the rate they're decreasing
% it's faster than 4.
> e_trap1/e_trap2
ans =
    9.1367

> e_trap2/e_trap3
ans =
    5.4527

> e_trap3/e_trap4
ans =
    4.5931

% what's the error with 10000 mesh-points?  It's 10^(-10).  With the
% left-hand rule, it's 10^(-5).  We're much happier with the
% trapezoidal rule.
> [a_trap,h,e_trap] = trap(0,1,10000);
> e_trap
e_trap =
  7.0137e-010

> diary off