makefullmatrix {ResistorArray}

Conductance matrix for a lattice of unit resistors

Description

Conductance matrix for a lattice of unit resistors

Usage

makefullmatrix(R, C)
makefullmatrix_strict(R, C,toroidal)

Arguments

R

Number of rows of nodes

C

Number of columns of nodes

toroidal

Boolean, with TRUE meaning to return a toroidally connected lattice, and FALSE meaning to return a lattice with edges

Details

The array produced by makefullmatrix_strict(R,C,TRUE) is toroidally connected.

Function makefullmatrix() is not entirely straightforward. The array produced is sort of toroidally connected. I regard this function as the canonical one because it is more elegant (see example image). Consider, for concreteness, the case with four rows and seven columns of nodes giving 28 nodes altogether. Number these columnwise so the top row is 1,5,9,13,17,21,25. Then number n corresponds to the row n and column n of the returned matrix.

Now, ‘interior’ nodes are as expected: node 6, for example, is connected to 2,5,10,7. And the wrapping is as expected in the horizontal: 1-25, 2-26, 3-27, and 4-28, are all connected.

However, the vertical wrapping is not as might be expected. One might expect node 9, say, to be connected to 5,10 13,12; but in fact node 9 is connected to nodes 5,8,10,13. So there is a Hamiltonian path comprising entirely of vertical connections (function makefullmatrix_strict(R,C,TRUE) returns the “expected” adjacency graph).

For the arrays returned by functions documented here, one can determine pairwise resistances using function array.resistance().

Value

Returns matrix of size RC\times RC. Note that this matrix is singular.

Author(s)

Robin K. S. Hankin

See Also

array.resistance

Examples

makefullmatrix(3,3)
image(makefullmatrix(4,7))              # A beautiful natural structure
image(makefullmatrix_strict(4,7,TRUE))  # A dog's breakfast