Permutation of indices in fortran

[Telemachus]

Hi there. I am working with a numerical quadrature in some scheme to solve a set of equations. At this point I am working in two dimensions. The thing is that I have some function $\psi_m(x,y,\Omega_m)$ with $\Omega_m=(\Omega_{x,i},\Omega_{y,j})$ with $\displaystyle m=1,2,...,N,N+1,...,M=\frac{N(N+2)}{2}$ where I am using a quadrature for the angular variables $\Omega$, and the index N denotes the number of discrete directions cosines in each direction, N for x, and N for y.

Now, I have to give a determined weight to each set of the $\Omega_m$ (the weights comes from the angular quadrature of an integral), such that, for example $\Omega_1=(\Omega_{x,1},\Omega_{y,1})$ has assigned the weight $w_1$, then for $\Omega_2=(\Omega_{x,2},\Omega_{y,2})$ we have $w_2$,..., and for $\Omega_N=(\Omega_{x,N},\Omega_{y,N})$ we have $w_N$. But the story continues, when I have $\Omega_{N+1}=(\Omega_{x,1},\Omega_{y,2})$, this will have the weight $w_{N+1}$, $\Omega_{N+2}=(\Omega_{x,1},\Omega_{y,3})$ this will have the weight t $w_{N+2}$, and so on. But then, when I get, for example the direction $\Omega_{2N+1}=(\Omega_{x,2},\Omega_{y,1})$, this weight must be equal to the one with the permuted indices, id est: $w(\Omega_{x,2},\Omega_{y,1})=w(\Omega_{x,1},\Omega_{y,2})=w_{N+1}$ and, in general $w(\Omega_{x,i},\Omega_{y,j})=w(\Omega_{x,j},\Omega_{y,i})$.

This whole thing comes into a program I have written, which is automatized to solve the equation for an arbitrary set of directions M. I don't really know how to do this assignment of weights, specially the part of permuting the indices.

I thought perhaps somebody here could give me some idea.

Thanks in advance.

 

[andrewkirk]

I'm afraid I can't make sense of the problem as it's described. In the first para it says $\Omega_m=(\Omega_{x,m},\Omega_{y,m})$ for $1\leq m\leq N(N+1)/2$. But then in the second para it says $\Omega_{N+1}=(\Omega_{x,1},\Omega_{y,2})$, which contradicts that, since $1\neq 2$.

What is needed is a clear, comprehensive specification of the function $\Omega$.

 

[Telemachus]

Yes, I know, my notation was just stupid. I should use different letters $\Omega_m=(\Omega_{x,i},\Omega_{x,j})$ with $i,j=1,2,3...,N$ and $m=1,2,...,M=\frac{N(N+2)}{2}$

 

[andrewkirk]

It looks like $i$ and $j$ are supposed to be functions of $m$. What are those functions?
So far you have specified that:

• for $1\leq m\leq N$: $i=j=m$
• for $N+1\leq m\leq 2N-1$: $i=1$ and $j=m-N+1$

Are you sure that's the relation you want? It looks very unintuitive to me. Also, what is the relation for $m\geq 2N$? It has to change there, otherwise $j$ will exceed $N$.

 

[Telemachus]

I am willing to use a Gauss-Legendre quadrature set in two dimensions, by discretizing the directions, I obtain a set of discrete directions cosines in both directions. You are right, the correct expression would be a function of the direction cosines $\hat \Omega_m=\hat \Omega_m(\hat \Omega_{x,m}, \hat \Omega_{y,m})$ that's how it is written in the book). I might have misunderstood something in the discretization scheme. I have an integral of this type:

$\int_0^{2\pi} \Psi (x,y,\Omega_x,\Omega_y) d\phi$ which I am willing to evaluate numerically by discretization of the angular variables.

The equation I am solving is: $\displaystyle \Omega_x \frac{\partial \Psi}{\partial x} + \Omega_y \frac{\partial \Psi}{\partial y}+\sigma_t \Psi=\sigma_s \int_0^{2\pi} \Psi (x,y,\Omega_x,\Omega_y) d\phi+q(x,y,\Omega_x,\Omega_y)$