non linear recurrence equations
in [Math]
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From: Panita Gomez on 10 Jul 2010 12:17 I'm serching to solve this non linear recurrence equations of this form with no hope :(. Anyone has an idea how to solve this kind of equations? a+ b X_{i-1} +c X_{i} + d X_{i+1} + e X_{i-1} X_{i+1}=0 without the last term, I can use the guassian elimination method to solve it. But when the last term appear I don't know how to deal with the multiplication between two reccurence anymore. help appreciated. Thank you Panita.
From: Ray Vickson on 10 Jul 2010 13:54 On Jul 10, 9:17 am, Panita Gomez <gatitablan...(a)gmail.com> wrote: > I'm serching to solve this non linear recurrence equations of this > form with no hope :(. > Anyone has an idea how to solve this kind of equations? > > a+ b X_{i-1} +c X_{i} + d X_{i+1} + e X_{i-1} X_{i+1}=0 > > without the last term, I can use the guassian elimination method to > solve it. But when the last term appear I don't know how to deal with > the multiplication between two reccurence anymore. help appreciated. > Thank you > Panita. If you know X_0 and X_1 you can get all the X_i: re-write the recursion as X_{i+1} = -[a + b*X_{i-1}+c*X_i]/[d + e*X_{i-1}]. You can get X_2, then X_3, ... . R.G. Vickson
From: Panita Gomez on 11 Jul 2010 07:39 On 10 juil, 19:54, Ray Vickson <RGVick...(a)shaw.ca> wrote: > On Jul 10, 9:17 am, Panita Gomez <gatitablan...(a)gmail.com> wrote: > > > I'm serching to solve this non linear recurrence equations of this > > form with no hope :(. > > Anyone has an idea how to solve this kind of equations? > > > a+ b X_{i-1} +c X_{i} + d X_{i+1} + e X_{i-1} X_{i+1}=0 > > > without the last term, I can use the guassian elimination method to > > solve it. But when the last term appear I don't know how to deal with > > the multiplication between two reccurence anymore. help appreciated. > > Thank you > > Panita. > > If you know X_0 and X_1 you can get all the X_i: re-write the > recursion as > X_{i+1} = -[a + b*X_{i-1}+c*X_i]/[d + e*X_{i-1}]. You can get X_2, > then X_3, ... . > > R.G. Vickson Dear Ray, Thank you very much for your help. That cheer me up that it is possible to solve the equations. However, I forgot to precise at the beginning that I do not know the two first terms. I only the first term and the last term a[0]=0 and a[n]=0.5. In the case that we know the first and the last recurrence, is it still possible to solve the rest of the terms? Thank you very much
From: Panita Gomez on 11 Jul 2010 08:10 On 11 juil, 13:39, Panita Gomez <gatitablan...(a)gmail.com> wrote: > On 10 juil, 19:54, Ray Vickson <RGVick...(a)shaw.ca> wrote: > > > > > > > On Jul 10, 9:17 am, Panita Gomez <gatitablan...(a)gmail.com> wrote: > > > > I'm serching to solve this non linear recurrence equations of this > > > form with no hope :(. > > > Anyone has an idea how to solve this kind of equations? > > > > a+ b X_{i-1} +c X_{i} + d X_{i+1} + e X_{i-1} X_{i+1}=0 > > > > without the last term, I can use the guassian elimination method to > > > solve it. But when the last term appear I don't know how to deal with > > > the multiplication between two reccurence anymore. help appreciated. > > > Thank you > > > Panita. > > > If you know X_0 and X_1 you can get all the X_i: re-write the > > recursion as > > X_{i+1} = -[a + b*X_{i-1}+c*X_i]/[d + e*X_{i-1}]. You can get X_2, > > then X_3, ... . > > > R.G. Vickson > > Dear Ray, > Thank you very much for your help. That cheer me up that it is > possible to solve the equations. However, I forgot to precise at the > beginning that I do not know the two first terms. I only the first > term and the last term > a[0]=0 and a[n]=0.5. > In the case that we know the first and the last recurrence, is it > still possible to solve the rest of the terms? Thank you very much Actually, I calculated it wrong. I know only the first and the last recurrence a[0]=0 and a[n+1]=0 the other terms are unknown :(
From: Robert Israel on 11 Jul 2010 11:10
Panita Gomez <gatitablanca3(a)gmail.com> writes: > On 11 juil, 13:39, Panita Gomez <gatitablan...(a)gmail.com> wrote: > > On 10 juil, 19:54, Ray Vickson <RGVick...(a)shaw.ca> wrote: > > > > > > > > > > > > > On Jul 10, 9:17=A0am, Panita Gomez <gatitablan...(a)gmail.com> wrote: > > > > > > I'm serching to solve this non linear recurrence equations of this > > > > form with no hope :(. > > > > Anyone has an idea how to solve this kind of equations? > > > > > > a+ b X_{i-1} +c X_{i} + d X_{i+1} + e X_{i-1} X_{i+1}=3D0 > > > > > > without the last term, I can use the guassian elimination method to > > > > solve it. But when the last term appear I don't know how to deal with > > > > the multiplication between two reccurence anymore. help appreciated. > > > > Thank you > > > > Panita. > > > > > If you know X_0 and X_1 you can get all the X_i: re-write the > > > recursion as > > > X_{i+1} =3D -[a + b*X_{i-1}+c*X_i]/[d + e*X_{i-1}]. You can get X_2, > > > then X_3, ... . > > > > > R.G. Vickson > > > > Dear Ray, > > Thank you very much for your help. That cheer me up that it is > > possible to solve the equations. However, I forgot to precise at the > > beginning that I do not know the two first terms. I only the first > > term and the last term > > a[0]=3D0 and a[n]=3D0.5. > > In the =A0case that we know the first and the last recurrence, is it > > still possible to solve the rest of the terms? Thank you very much > > Actually, I calculated it wrong. I know only the first and the last > recurrence > a[0]=3D0 and a[n+1]=3D0 > the other terms are unknown :( Then it's very unlikely that you could get a closed-form solution. You could try expressing the a[j] in terms of the unknown a[1], and solving the resulting equation a[n+1] = 0 for a[n], but this will require solving a (probably irreducible) high-degree polynomial. -- Robert Israel israel(a)math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada |