MATLAB Central Newsreader - symbolic computation

symbolic computation

devi — Fri, 08 May 2009 19:00:04 -0400

Hello,

I have an equation
y = b*sech(b*acosh(1/x))*tanh(b*acosh(1/x))/(sqrt((1/x)-1)*sqrt(1+(1/x))*(x^2));

where b varies from 2 to 101,as b = M+1 ; M varies from 1 : 100
x = 1.001:0.001:1.01

The problem is I am simulating a chain network with M routers.For M =1 and 2 that is b = 2 and 3 the analytical and simulated results are almost concordant with 4 and 9 respectively which are the limits.But when I increase the b values they tend to hit numerical stability issues and lose accuracy.

I need help to compute the limits analytically using symbolic computation or maple

Thanks
devi

Re: symbolic computation

Peter — Fri, 08 May 2009 22:41:01 -0400

On May 8, 12:00 pm, "devi " <rama...@gmail.com> wrote:
> Hello,
>
> I have an equation
> y = b*sech(b*acosh(1/x))*tanh(b*acosh(1/x))/(sqrt((1/x)-1)*sqrt(1+(1/x)=
)*(x^2));
>
> where b varies from 2 to 101,as b = M+1 ; M varies from 1 : 100
> x = 1.001:0.001:1.01
>
> The problem is I am simulating a chain network with M routers.For M =1 =
and 2 that is b = 2 and 3 the analytical and simulated results are almost=
concordant with 4 and 9 respectively which are the limits.But when I incre=
ase the b values they tend to hit numerical stability issues and lose accur=
acy.
>
> I need help to compute the limits analytically using symbolic computation=
or maple
>
> Thanks
> devi

The following power series in the variable dx = x - 1 should provide
sufficient accuracy (about 6 to 8 significant digits) for your desired
range 0.001 <= dx <= 0.01

    b^2 + (5*b*(-b + b^3)*dx)/3. +
       (b*(64*b - 125*b^3 + 61*b^5)*dx^2)/
        30. + (b*(-312*b + 889*b^3 - 854*b^5 +
            277*b^7)*dx^3)/126. +
       (b*(62136*b - 229000*b^3 + 324093*b^5 -
            207750*b^7 + 50521*b^9)*dx^4)/
        22680

--Peter

Re: symbolic computation

devi — Sat, 09 May 2009 01:15:02 -0400

Well y is already a differentiated equation.To explain in a better way,
For example:The mean of y when x varies from 1.001:0.001:1.01 at b = 2 is 4
So this mean value is compared with simulated values.This value is varying to a large extent and is not concordant with analytical values as b increases from 2:100.

I am basically finding the average number of hops in a chain network by simulation which has to agree with analytical results that is average number of hops by simulation ishould agree with the mean from analytical calculations of the equation.

Thanks
Devi

Peter <petersamsimon2@hotmail.com> wrote in message <80cd7c70-2c48-4035-bb89-a154702eb83d@d38g2000prn.googlegroups.com>...
> On May 8, 12:00?pm, "devi " <rama...@gmail.com> wrote:
> > Hello,
> >
> > I have an equation
> > y = b*sech(b*acosh(1/x))*tanh(b*acosh(1/x))/(sqrt((1/x)-1)*sqrt(1+(1/x)=
> )*(x^2));
> >
> > where b varies from 2 to 101,as b = M+1 ; M varies from 1 : 100
> > x = 1.001:0.001:1.01
> >
> > The problem is I am simulating a chain network with M routers.For M =1 =
> and 2 that is b = 2 and 3 the analytical and simulated results are almost=
> concordant with 4 and 9 respectively which are the limits.But when I incre=
> ase the b values they tend to hit numerical stability issues and lose accur=
> acy.
> >
> > I need help to compute the limits analytically using symbolic computation=
> or maple
> >
> > Thanks
> > devi
>
> The following power series in the variable dx = x - 1 should provide
> sufficient accuracy (about 6 to 8 significant digits) for your desired
> range 0.001 <= dx <= 0.01
>
> b^2 + (5*b*(-b + b^3)*dx)/3. +
> (b*(64*b - 125*b^3 + 61*b^5)*dx^2)/
> 30. + (b*(-312*b + 889*b^3 - 854*b^5 +
> 277*b^7)*dx^3)/126. +
> (b*(62136*b - 229000*b^3 + 324093*b^5 -
> 207750*b^7 + 50521*b^9)*dx^4)/
> 22680
>
> --Peter