Mixing
Determine $kappa$ / does there exist a closed form solution?
$begingroup$
How can I determine the value for $kappa$ for e.g. $n in {200, 400, 600}$. Is there a closed form solution?
Let $a_j = j^{0.51}$ and let $kappa$ be the solution to the equation
$$
sigma^2_{epsilon} n^{-1} sum_{j=1}^{infty} a_j lambda_j = kappa,
$$
where $lambda_j = max(1-kappa a_j,0)$ and $sigma^2_{epsilon}$ is chosen to be 1.
For $n=100$ the solution for $kappa$ is given as 0.199 according to the paper.
Are there any computer programs which can solve such equations without the knowledge of a closed form solution?
real-analysis closed-form
asked Jan 6 at 11:58
user483161user483161
446
$endgroup$
add a comment |
$begingroup$
How can I determine the value for $kappa$ for e.g. $n in {200, 400, 600}$. Is there a closed form solution?
Let $a_j = j^{0.51}$ and let $kappa$ be the solution to the equation
$$
sigma^2_{epsilon} n^{-1} sum_{j=1}^{infty} a_j lambda_j = kappa,
$$
where $lambda_j = max(1-kappa a_j,0)$ and $sigma^2_{epsilon}$ is chosen to be 1.
For $n=100$ the solution for $kappa$ is given as 0.199 according to the paper.
Are there any computer programs which can solve such equations without the knowledge of a closed form solution?
real-analysis closed-form
asked Jan 6 at 11:58
user483161user483161
446
$endgroup$
$begingroup$
I don't think you'll get a closed form analytic solution for this. Numerical methods seem to work fine, however. If you don't need many digits, a simple binary search works without difficulty. You could speed it up by using numerical derivatives and Newton's method...it really just depends on how much accuracy you want (and how important speed is).
$endgroup$
– lulu
Jan 6 at 12:21
add a comment |
$begingroup$
How can I determine the value for $kappa$ for e.g. $n in {200, 400, 600}$. Is there a closed form solution?
Let $a_j = j^{0.51}$ and let $kappa$ be the solution to the equation
$$
sigma^2_{epsilon} n^{-1} sum_{j=1}^{infty} a_j lambda_j = kappa,
$$
where $lambda_j = max(1-kappa a_j,0)$ and $sigma^2_{epsilon}$ is chosen to be 1.
For $n=100$ the solution for $kappa$ is given as 0.199 according to the paper.
Are there any computer programs which can solve such equations without the knowledge of a closed form solution?
real-analysis closed-form
asked Jan 6 at 11:58
user483161user483161
446
$endgroup$
How can I determine the value for $kappa$ for e.g. $n in {200, 400, 600}$. Is there a closed form solution?
Let $a_j = j^{0.51}$ and let $kappa$ be the solution to the equation
$$
sigma^2_{epsilon} n^{-1} sum_{j=1}^{infty} a_j lambda_j = kappa,
$$
where $lambda_j = max(1-kappa a_j,0)$ and $sigma^2_{epsilon}$ is chosen to be 1.
For $n=100$ the solution for $kappa$ is given as 0.199 according to the paper.
Are there any computer programs which can solve such equations without the knowledge of a closed form solution?
real-analysis closed-form
real-analysis closed-form
asked Jan 6 at 11:58
user483161user483161
446
asked Jan 6 at 11:58
user483161user483161
446
asked Jan 6 at 11:58
user483161user483161
446
asked Jan 6 at 11:58
user483161user483161
446
asked Jan 6 at 11:58
user483161user483161
446
446
$begingroup$
I don't think you'll get a closed form analytic solution for this. Numerical methods seem to work fine, however. If you don't need many digits, a simple binary search works without difficulty. You could speed it up by using numerical derivatives and Newton's method...it really just depends on how much accuracy you want (and how important speed is).
$endgroup$
– lulu
Jan 6 at 12:21
add a comment |
$begingroup$
I don't think you'll get a closed form analytic solution for this. Numerical methods seem to work fine, however. If you don't need many digits, a simple binary search works without difficulty. You could speed it up by using numerical derivatives and Newton's method...it really just depends on how much accuracy you want (and how important speed is).
$endgroup$
– lulu
Jan 6 at 12:21
$begingroup$
I don't think you'll get a closed form analytic solution for this. Numerical methods seem to work fine, however. If you don't need many digits, a simple binary search works without difficulty. You could speed it up by using numerical derivatives and Newton's method...it really just depends on how much accuracy you want (and how important speed is).
$endgroup$
– lulu
Jan 6 at 12:21
$begingroup$
I don't think you'll get a closed form analytic solution for this. Numerical methods seem to work fine, however. If you don't need many digits, a simple binary search works without difficulty. You could speed it up by using numerical derivatives and Newton's method...it really just depends on how much accuracy you want (and how important speed is).
$endgroup$
– lulu
Jan 6 at 12:21
add a comment |
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$begingroup$
I don't think you'll get a closed form analytic solution for this. Numerical methods seem to work fine, however. If you don't need many digits, a simple binary search works without difficulty. You could speed it up by using numerical derivatives and Newton's method...it really just depends on how much accuracy you want (and how important speed is).
$endgroup$
– lulu
Jan 6 at 12:21