integration - Error of midpoint, trapezoidal and simpson rule in matlab -

i'm writing script numerically find integral using these 3 methods. right can compute right value integral, need print error , upper analytical bound of it. have no idea how in matlab. formulas need find maximal value of f'' , f'''', dont know ho that. can me?

thank you

recall upper bound of error either trapezoidal rule, simpson's rule or midpoint rule on f(x) must bound between interval of x in [a,b] or else won't able find error. have 2 solutions you, depending on has been given.

solution #1 - closed form solution f(x) given

if have closed form solution of integral, use symbolic toolbox in matlab first define f(x), use diff command differentiate find f'(x). if want second derivative, apply diff command it.

example:

syms x;  f = x^4; df = diff(f); d2f = diff(df);  f =  x^4  df =  4*x^3  d2f =  12*x^2 

if want find maximum within interval, assuming step size t between lower bound a , b, can try doing:

a = 1; b = 3; t = 0.01; maxf = max(abs(double(subs(f,a:t:b))));  maxf =   81 

what above code finds maximum value between [1,3] of x^4 in step size of t = 0.01. can replace fourth line's first parameter f whatever function want. note: error terms require **absolute* value, why abs function there.

solution #2 - closed form solution f(x) not given

if don't have function, have set of (x,y) pairs, can still use diff function on output y values implement discrete approximation derivative. such, derivative approximately equal to:

x_i' = x_{i+1} - x_i 

this takes (i+1) point , subtracts i point. if apply diff on array of n points, return array of n-1 points accommodate looking 1 sample ahead find difference. after, can use array , find maximum of derivative. if want find maximum of second derivative, call diff again on first outpt of diff. example started:

x = 0:0.01:2*pi; y = sin(x); ymax = max(abs(diff(y)));  ymax =      0.0100 

what above finds maximum value of discrete approximation derivative y = sin(x), denoted diff(y).

hopefully 1 of these work you!

aside

i used teach numerical methods when instructor @ university. take @ these slides intro numerical integration using simpson's rule , trapezoidal rule. there slides on how code in matlab (without using built-in functions), slides on how compute error, , how many points need within fixed interval guarantee particular approximation error.

http://www.rnet.ryerson.ca/~rphan/mth510/f2012/lab_10/mth510f2012_lab10.pdf