Solving a linked recurrent relations
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I am trying to solving a linked recurrent relations.
$$
left{begin{matrix}
T_t &= &(a_{t}p_{n}+c_{t}+b_{t}p_r+d_{t})*(1-P_o)\
a_{t+1} &= &T_t*f(T_t)+(1-p_n)*a_t\
b_{t+1} &= &T_t*(1-f(T_t))+(1-p_r)*b_t\
c_{t+1} &= &(a_tp_n+c_t)Po\
d_{t+1} &= &(b_tp_r+d_t)Po\
end{matrix}right.
$$
where $f(T_t)$ is kind of probability function which is not linear function. $p_n$, $p_r$ and $P_o$ are less than 1 and larger than 0.
The initial values are
$$
left{begin{matrix}
a_{1} &= & 100\
b_{1} &= & 0\
c_{1} &= & 0\
d_{1} &= & 0\
end{matrix}right.
$$
Since $f(T_t)$ is in $a_t$ and $b_t$, I thought it can not be solved as matrix form.
How can I determine whether the linked recurrent relations converge or not?
Is there an idea to solve the convergence value?
I put the initial values to the linked recurrent relations in MATLAB, and find out the relations converge.
convergence recurrence-relations
asked Nov 14 at 12:06
hoesang choi
64
add a comment |
up vote
0
down vote
favorite
I am trying to solving a linked recurrent relations.
$$
left{begin{matrix}
T_t &= &(a_{t}p_{n}+c_{t}+b_{t}p_r+d_{t})*(1-P_o)\
a_{t+1} &= &T_t*f(T_t)+(1-p_n)*a_t\
b_{t+1} &= &T_t*(1-f(T_t))+(1-p_r)*b_t\
c_{t+1} &= &(a_tp_n+c_t)Po\
d_{t+1} &= &(b_tp_r+d_t)Po\
end{matrix}right.
$$
where $f(T_t)$ is kind of probability function which is not linear function. $p_n$, $p_r$ and $P_o$ are less than 1 and larger than 0.
The initial values are
$$
left{begin{matrix}
a_{1} &= & 100\
b_{1} &= & 0\
c_{1} &= & 0\
d_{1} &= & 0\
end{matrix}right.
$$
Since $f(T_t)$ is in $a_t$ and $b_t$, I thought it can not be solved as matrix form.
How can I determine whether the linked recurrent relations converge or not?
Is there an idea to solve the convergence value?
I put the initial values to the linked recurrent relations in MATLAB, and find out the relations converge.
convergence recurrence-relations
asked Nov 14 at 12:06
hoesang choi
64
just checking so we have two indices, $n$ and $t$ for variable $a,b,c,d$?
– Siong Thye Goh
Nov 15 at 1:47
There were typo, so I corrected it. Thank you
– hoesang choi
Nov 15 at 1:52
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I am trying to solving a linked recurrent relations.
$$
left{begin{matrix}
T_t &= &(a_{t}p_{n}+c_{t}+b_{t}p_r+d_{t})*(1-P_o)\
a_{t+1} &= &T_t*f(T_t)+(1-p_n)*a_t\
b_{t+1} &= &T_t*(1-f(T_t))+(1-p_r)*b_t\
c_{t+1} &= &(a_tp_n+c_t)Po\
d_{t+1} &= &(b_tp_r+d_t)Po\
end{matrix}right.
$$
where $f(T_t)$ is kind of probability function which is not linear function. $p_n$, $p_r$ and $P_o$ are less than 1 and larger than 0.
The initial values are
$$
left{begin{matrix}
a_{1} &= & 100\
b_{1} &= & 0\
c_{1} &= & 0\
d_{1} &= & 0\
end{matrix}right.
$$
Since $f(T_t)$ is in $a_t$ and $b_t$, I thought it can not be solved as matrix form.
How can I determine whether the linked recurrent relations converge or not?
Is there an idea to solve the convergence value?
I put the initial values to the linked recurrent relations in MATLAB, and find out the relations converge.
convergence recurrence-relations
asked Nov 14 at 12:06
hoesang choi
64
I am trying to solving a linked recurrent relations.
$$
left{begin{matrix}
T_t &= &(a_{t}p_{n}+c_{t}+b_{t}p_r+d_{t})*(1-P_o)\
a_{t+1} &= &T_t*f(T_t)+(1-p_n)*a_t\
b_{t+1} &= &T_t*(1-f(T_t))+(1-p_r)*b_t\
c_{t+1} &= &(a_tp_n+c_t)Po\
d_{t+1} &= &(b_tp_r+d_t)Po\
end{matrix}right.
$$
where $f(T_t)$ is kind of probability function which is not linear function. $p_n$, $p_r$ and $P_o$ are less than 1 and larger than 0.
The initial values are
$$
left{begin{matrix}
a_{1} &= & 100\
b_{1} &= & 0\
c_{1} &= & 0\
d_{1} &= & 0\
end{matrix}right.
$$
Since $f(T_t)$ is in $a_t$ and $b_t$, I thought it can not be solved as matrix form.
How can I determine whether the linked recurrent relations converge or not?
Is there an idea to solve the convergence value?
I put the initial values to the linked recurrent relations in MATLAB, and find out the relations converge.
convergence recurrence-relations
convergence recurrence-relations
asked Nov 14 at 12:06
hoesang choi
64
asked Nov 14 at 12:06
hoesang choi
64
edited Nov 15 at 1:51
asked Nov 14 at 12:06
hoesang choi
64
asked Nov 14 at 12:06
hoesang choi
64
asked Nov 14 at 12:06
hoesang choi
64
64
just checking so we have two indices, $n$ and $t$ for variable $a,b,c,d$?
– Siong Thye Goh
Nov 15 at 1:47
There were typo, so I corrected it. Thank you
– hoesang choi
Nov 15 at 1:52
add a comment |
just checking so we have two indices, $n$ and $t$ for variable $a,b,c,d$?
– Siong Thye Goh
Nov 15 at 1:47
There were typo, so I corrected it. Thank you
– hoesang choi
Nov 15 at 1:52
just checking so we have two indices, $n$ and $t$ for variable $a,b,c,d$?
– Siong Thye Goh
Nov 15 at 1:47
just checking so we have two indices, $n$ and $t$ for variable $a,b,c,d$?
– Siong Thye Goh
Nov 15 at 1:47
There were typo, so I corrected it. Thank you
– hoesang choi
Nov 15 at 1:52
There were typo, so I corrected it. Thank you
– hoesang choi
Nov 15 at 1:52
add a comment |
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just checking so we have two indices, $n$ and $t$ for variable $a,b,c,d$?
– Siong Thye Goh
Nov 15 at 1:47
There were typo, so I corrected it. Thank you
– hoesang choi
Nov 15 at 1:52