How to handle the indices of a 9-dimensional matrix - c++

Question

I am a physicist currently writing a C++ program dealing with multidimensional integration; in particular, the functions I am considering can have up to D=9 dimensions.
From a mathematical perspective, I need to handle a NxNxN...xN (D times) matrix, but from a programming point of view, I was instructed to use an array of NxNxN...xN elements instead. From what I know, an array is better for the sake of generality and for all the ensuing calculations involving pointers.
However, now I am stuck with a problem I cannot solve.
I need to perform some calculations where a single index of my matrix is fixed and all the other ones take all their different values.
If it were a 3x3x3 matrix, the code would be something similar to the following:
double test[3][3][3];
for(int i=0;i<3;i++) {
for(int j=0;j<3;j++) {
test[0][i][j]=i*j;
}
}
i.e. I could have an index fixed and cycle through the other ones.
The same process could be extended to the second and the third index as well.
How can I accomplish the same effect with a double test[3*3*3]? Please keep in mind that the three dimensional matrix is just an example; the real matrices I am dealing with are 9-dimensional, and so I need a general way to keep a single index of my matrix fixed and cycle through all the other ones.
TL;DR: I have an array which represents a NxNxN...xN (9 times) matrix.
I need to perform some calculations on the array as if a single index of my matrix were fixed and all the other ones were cycling through all their possible values.
I know there is a simple expression for the case where a 2-D matrix is mapped in a 1-D array; does something similar exist here?

Answer 1

Raster scan is the standard way of ordering elements for two dimensions.
If you have a 2-D array test[3][3], and you access it by test[i][j], the corresponding one-dimensional array would be
double raster[3 * 3];
and you would access it as follows:
raster[i * 3 + j];
This can be generalized to 3 dimensions:
double raster[3 * 3 * 3];
...
raster[a * 9 + b * 3 + c];
Or to 9 dimensions:
double raster[3 * 3 * 3 * 3 * 3 * 3 * 3 * 3 * 3];
...
raster[a * 6561 + b * 2187 + c * 729 + d * 243 + e * 81 + f * 27 + g * 9 + h * 3 + i];
Having any of the a ... i index variables constant, and changing the rest in a loop, will access a 8-D slice in your 9-D array.
You might want to define some struct to hold all these indices, for example:
struct Pos
{
int a, b, c, d, e, f, g, h, i;
};
Then you can convert a position to a 1-D index easily:
int index(Pos p)
{
return p.a * 6561 + p.b * 2187 + p.c * 729 + p.d * 243 + p.e * 81 + p.f * 27 + p.g * 9 + p.h * 3 + p.i;
}

Answer 2

Generally, a flattened array will contain its elements in the following way: the elements of the last dimension will be mapped into repeated groups, the inner-most groups will be the second dimension from the back and so on:
values[x][y][z] => { x0 = { y0_0 = { z0_0_0, z0_0_1, ..., z0_0_N }, y0_1 = { z0_1_0, z0_1_1, ... }, ... y0_N }, x1 = ... }
values[x*y*z] => { z0_0_0, z0_0_1, ..., z0_0_N, z0_1_0, z0_0_1, ... }
I hope this makes sense outside my brain.
So, any element access will need to calculate, how many blocks of elements come before it:
Accessing [2][1][3] means, skip 2 blocks of x, each containing y blocks with z elements, then skip another 1 block of y containing z elements and access the 3rd element from the next block:
values[2 * y * z + 1 * z + 3];
So more generally for N dimensions d1, d2, d3 .. dn, and an n-dimensional index i1, i2, .. iN to be accessed:
[i1 * d2 * ... * dN + i2 * d3 * ... * dN + ... + iN]
Back to your example:
double test[3*3*3];
for(int i = 0; i < 3; i++)
{
for(int j = 0; j < 3; j++)
{
// test[0*3*3 + i*3 + j] = i * j;
test[i*3 + j] = i * j;
}
}

Answer 3

If the matrix has the same size for all dimensions, then you can access them like this:
m[x + y*N + z*N*N + w*N*N*N ...]
In the case that the sizes are different, it is a little bit more complicated:
m[x + y*N1 + z*N1*N2 + w*N1*N2*N3 ...]