- #1
Telemachus
- 835
- 30
Hi. I want to normalize a discretized function ##p_{k,k'}##, to satisfy simultaneously two conditions. The normalized function ##p^*_{k,k'}## has to satisfy simultaneously:
1) ##\sum_{k=1}^{M} p^*_{k,k'} w_k=1##, for all ##k'=1,2,...,M##;
2) ##\sum_{k=1}^{M} p^*_{k,k'} w_k \hat \Omega_k \cdot \hat \Omega_{k'}=g##, for all ##k'=1,2,...,M##
The ##w_k## are weights, the ##\Omega## are versors, and I have to modify ##p_{k,k'}##, which is a given function in ##\Omega_k## in order to satisfy this both conditions.
For example, if I only had to satisfy condition 1), I would have the trivial solution:
##p^*_{k,k'}=p_{k,k'}/\sum_{k=1}^{M} p_{k',k} w_k##.
However, I'm not sure on how to proceed now that I have to satisfy both conditions simultaneously.
I was thinking that I could pose this problem this way. I could try to find the vector ##p^*_{k,k'}## considering the system of equations in matrix form. So, for example, for the first condition I would search the ##p^*_{k,k'}## that satisfy:
##(\sum_{k=1}^{M} p_{k,k'} w_k) p^*_{k,k'}=p_{k,k'}## for each ##k'##
I am trying to write these conditions in matrix form. Something like (this is clearly wrong):
##\begin{bmatrix}
p_{1,1}w_1 & p_{2,1}w_2 & p_{3,1}w_3 & \dots & p_{M,1}w_M \\
p_{1,2}w_1 & p_{2,2}w_2 & p_{3,2}w_3 & \dots & p_{M,2}w_M \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
p_{1,M}w_1 & p_{2,M}w_2 & p_{3,M}w_3 & \dots & p_{M,M} w_M\\
\vdots & \vdots & \vdots & \ddots & \vdots
\end{bmatrix}
\begin{bmatrix}
p^*_{1,1} \\
p^*_{1,2} \\
\vdots\\
p^*_{1,M} \\
\vdots\
\end{bmatrix}=\begin{bmatrix}
p_{1,1} \\
p_{1,2} \\
\vdots\\
p_{1,M} \\
\vdots\
\end{bmatrix}
##
This matrix should extend, and also include the systems for condition 2). However, I am having some trouble on figuring out the matrix coefficients and how to properly set the systems of equations.
1) ##\sum_{k=1}^{M} p^*_{k,k'} w_k=1##, for all ##k'=1,2,...,M##;
2) ##\sum_{k=1}^{M} p^*_{k,k'} w_k \hat \Omega_k \cdot \hat \Omega_{k'}=g##, for all ##k'=1,2,...,M##
The ##w_k## are weights, the ##\Omega## are versors, and I have to modify ##p_{k,k'}##, which is a given function in ##\Omega_k## in order to satisfy this both conditions.
For example, if I only had to satisfy condition 1), I would have the trivial solution:
##p^*_{k,k'}=p_{k,k'}/\sum_{k=1}^{M} p_{k',k} w_k##.
However, I'm not sure on how to proceed now that I have to satisfy both conditions simultaneously.
I was thinking that I could pose this problem this way. I could try to find the vector ##p^*_{k,k'}## considering the system of equations in matrix form. So, for example, for the first condition I would search the ##p^*_{k,k'}## that satisfy:
##(\sum_{k=1}^{M} p_{k,k'} w_k) p^*_{k,k'}=p_{k,k'}## for each ##k'##
I am trying to write these conditions in matrix form. Something like (this is clearly wrong):
##\begin{bmatrix}
p_{1,1}w_1 & p_{2,1}w_2 & p_{3,1}w_3 & \dots & p_{M,1}w_M \\
p_{1,2}w_1 & p_{2,2}w_2 & p_{3,2}w_3 & \dots & p_{M,2}w_M \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
p_{1,M}w_1 & p_{2,M}w_2 & p_{3,M}w_3 & \dots & p_{M,M} w_M\\
\vdots & \vdots & \vdots & \ddots & \vdots
\end{bmatrix}
\begin{bmatrix}
p^*_{1,1} \\
p^*_{1,2} \\
\vdots\\
p^*_{1,M} \\
\vdots\
\end{bmatrix}=\begin{bmatrix}
p_{1,1} \\
p_{1,2} \\
\vdots\\
p_{1,M} \\
\vdots\
\end{bmatrix}
##
This matrix should extend, and also include the systems for condition 2). However, I am having some trouble on figuring out the matrix coefficients and how to properly set the systems of equations.
Last edited: