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 May 8th, 2018, 04:17 PM #1 Member   Joined: Nov 2016 From: Kansas Posts: 73 Thanks: 1 Absolute stability f(t,u) = -iu where i$^{2}$ = -1 find theta for which the function has a range of absolute stability
 May 8th, 2018, 06:44 PM #2 Global Moderator   Joined: May 2007 Posts: 6,581 Thanks: 610 There is no theta in the expression. What do you mean by absolute stability? Thanks from topsquark
 May 9th, 2018, 04:41 AM #3 Member   Joined: Nov 2016 From: Kansas Posts: 73 Thanks: 1 Absolute stability: the numerical method/function is absolutely stable if all its roots lie in a unit circle. u(t$_{n+1}$) = u(t$_{n}$) + h[(1-theta) f(t$_{n}$,u(t$_{n}$) ) + (theta)f(t$_{n+1}$,u(t$_{n+1}$) )]
 May 9th, 2018, 01:42 PM #4 Global Moderator   Joined: May 2007 Posts: 6,581 Thanks: 610 It would be helpful if you could explain your notation. What is $t_n$? What is $u(t_n)$?
May 9th, 2018, 04:16 PM   #5
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Quote:
 Originally Posted by ZMD Absolute stability: the numerical method/function is absolutely stable if all its roots lie in a unit circle. u(t$_{n+1}$) = u(t$_{n}$) + h[(1-theta) f(t$_{n}$,u(t$_{n}$) ) + (theta)f(t$_{n+1}$,u(t$_{n+1}$) )]
...And is h[ ] a function or a constant?

If f(t, u) = -i u then why include the t in the notation?

You have
$\displaystyle u( t_{n+1} ) = u( t_{n} ) + h [ ( 1 - \theta) f( t_{n} , u( t_{n} ) ) + \theta f( t_{n+1},u( t_{n+1}) ) ]$

$\displaystyle u( t_{n+1} ) = u( t_{n} ) + h [ ( 1 - \theta) ( -i u( t_{n} ) + \theta ( -i u( t_{n+1}) ) ]$

$\displaystyle u( t_{n+1} ) = u( t_{n} ) + h [ -i u( t_{n} ) + i \theta \left \{ u( t_{n}) - ~ u( t_{n+1}) \right \} ]$

That's as far as I can help until you define the h[ ] thing.

-Dan

Last edited by topsquark; May 9th, 2018 at 04:32 PM.

 May 9th, 2018, 06:45 PM #6 Member   Joined: Nov 2016 From: Kansas Posts: 73 Thanks: 1 h is a constant. u(t$_{n}$) is technically u$_{n}$ General definition of this notation is under theta method (in numerical analysis) while calculating error Thanks from topsquark
May 10th, 2018, 12:41 PM   #7
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Quote:
 Originally Posted by ZMD h is a constant. u(t$_{n}$) is technically u$_{n}$ General definition of this notation is under theta method (in numerical analysis) while calculating error
Asking you to elaborate seems to be like pulling teeth. What is the theta method?

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