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Re: Numerical Differentiation and Integration of Array Data
From: |
c. |
Subject: |
Re: Numerical Differentiation and Integration of Array Data |
Date: |
Sun, 4 Dec 2011 13:43:28 +0100 |
On 4 Dec 2011, at 12:15, syberraith wrote:
> Actually I kind of understand what your saying. I've been playing with this:
>
> n = 1000000
> p = 2 * pi;
> h = p / (n-1);
> g = h / 2;
>
> function y = f ( x )
> y = sin( x );
> endfunction
>
> t = linspace(0 - g, p + g, n + 1);
> a = (f(t+g) - f(t-g))./h;
> a = h.*(a(1:end-1) + a(2:end))./2;
>
> u = linspace(0, p, n);
> s = sum(a.*h)
> s1 = a(end) - a(1)
> #plot(u, a,"-1")
>
> The more data points I used the closer s got to s1.
The comparison you are making here is wrong: s and s1 do not represent the same
quantity,
they jus happen to have similar values because f = sin(x) and a is an
approximation
of cos (x) which are functions with the same periodicity.
the correct comparison is between s and
s1 = f(p) - f(0)
If you do the comparison correctly you get the maximum possible accuracy
already with about 50 samples.
---------
n = 100
p = 2 * pi;
h = p / (n-1);
g = h / 2;
function y = f ( x )
y = sin( x );
endfunction
t = linspace(0, p, n);
a = (diff (f (t))) / h;
u = linspace(0, p, n);
s = sum(a * h)
s1 = f(p) - f(0)
figure (1)
plot (t(1:end-1)+g, a, t, cos (t))
title ("compare a and cos(t)")
figure (2)
plot (t, f(t), t, sin (t))
title ("compare f(t) and sin(t)")
---------
n = 50
s = -2.4980e-16
s1 = -2.4493e-16
>> s-s1
ans = -4.8708e-18
This said, I think you should sit back and carefully rethink your model,
you are spending a lot of time on a numerical method to compute an
approximation "s"
of the quantity "s1" which you already know analytically from the beginning.
c.
- Re: Numerical Differentiation and Integration of Array Data, (continued)
- Re: Numerical Differentiation and Integration of Array Data, c., 2011/12/03
- Message not available
- Re: Numerical Differentiation and Integration of Array Data, Carlo de Falco, 2011/12/03
- Re: Numerical Differentiation and Integration of Array Data, syberraith, 2011/12/03
- Re: Numerical Differentiation and Integration of Array Data, c., 2011/12/04
- Re: Numerical Differentiation and Integration of Array Data, Carlo de Falco, 2011/12/04
- Re: Numerical Differentiation and Integration of Array Data, syberraith, 2011/12/04
- Re: Numerical Differentiation and Integration of Array Data,
c. <=
- Re: Numerical Differentiation and Integration of Array Data, syberraith, 2011/12/04
- Re: Numerical Differentiation and Integration of Array Data, syberraith, 2011/12/04
- Re: Numerical Differentiation and Integration of Array Data, syberraith, 2011/12/04
- Re: Numerical Differentiation and Integration of Array Data, Francesco Potortì, 2011/12/05
- Re: Numerical Differentiation and Integration of Array Data, syberraith, 2011/12/05
- Re: Numerical Differentiation and Integration of Array Data, syberraith, 2011/12/04
- Re: Numerical Differentiation and Integration of Array Data, c., 2011/12/04
- Re: Numerical Differentiation and Integration of Array Data, syberraith, 2011/12/05
- Re: Numerical Differentiation and Integration of Array Data, syberraith, 2011/12/05
- Re: Numerical Differentiation and Integration of Array Data, Francesco Potortì, 2011/12/05