Numerics on wild oscillating functions

[Theia]

Hello!

I'd like to ask for a help about how to compute accurately functions which has very intense oscillations. My example is to estimate

\(\displaystyle I = \int_0^{\infty} \sin(x^2) dx= \int_0^{\infty}\frac{\sin(t)}{2\sqrt{t}} dt\).I tried trapezoid rule over one oscillation at a time, but result is poor. My next though is to collect positive parts and negative parts together and add them later on in the code...

Any comments?

Thank you!

 

[I like Serena]

Hello!

I'd like to ask for a help about how to compute accurately functions which has very intense oscillations. My example is to estimate

\(\displaystyle I = \int_0^{\infty} \sin(x^2) dx= \int_0^{\infty}\frac{\sin(t)}{2\sqrt{t}} dt\).I tried trapezoid rule over one oscillation at a time, but result is poor. My next though is to collect positive parts and negative parts together and add them later on in the code...

Any comments?

Thank you!

I think integrating to infinity is typically unstable, isn't it?
So we should probably split the integral in two parts to eliminate infinity.
That is:
$$I = \int_0^{\infty} \sin(x^2)\,dx = \int_0^{\sqrt{2\pi}} \sin(x^2)\, dx + \int_0^{1/\sqrt{2\pi}} u^{-2}\sin(u^{-2})\, du$$
Perhaps we can integrate the second integral period by period now?

 

[Theia]

Thank you! I computed first

\(\displaystyle \int_0^{2\pi}\frac{\sin t}{2\sqrt{t}}\),

and then the improper part by using integration by parts to increase \(\displaystyle \sqrt{t}\) to \(\displaystyle \sqrt{t^3}\) to obtain faster convergence.