DISTANCE_TO_POSITION is a MATLAB program which estimates the positions of cities given a city-to-city distance table.
In 2D, the problem is singular. In particular, the position of one city is completely arbitrary, and one component of a second city is completely arbitrary (and a third city's position can be "flipped" about the line connecting cities one and two). To remove some of this singularity, the program assigns city #1 the position (0,0) and city #2 is given a 0 Y coordinate.
In N-dimensional space, a similar set of constraints must be placed on the first N cities, or the least squares solver is likely to fail. (That is, we specify all N components of the first city to be 0, N-1 components of the second one, and so on, up to the N-th city which has a single 0 component).
The computations carried out by this program assume that the cities lie on a plane. If the distance data is for international cities, the effect of spherical geometry may make the planar approximation very bad.
Once the nonlinear least squares problem is set up, MATLAB's LSQNONLIN function is called to seek a solution.
distance_to_position ( 'distance.txt' )reads the distance information in 'distance.txt', estimates the positions of the cities, and writes out a position table in distance.coord.txt.
The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.
CITIES, a dataset directory which contains sets of information about cities and the distances between them;
CITIES is a FORTRAN90 library which handles various problems associated with a set of "cities" on a map.
DISTANCE_TO_POSITION is available in a FORTRAN90 version and a MATLAB version.
DISTANCE_TO_POSITION_SPHERE is a MATLAB program which estimates the positions of cities on a sphere (such as the earth) based on a city-to-city distance table.
LAU_NP is a FORTRAN90 library which implements heuristic algorithms for various NP-hard combinatorial problems.
NMS is a FORTRAN90 library which includes a wide variety of numerical software.
PARTIAL_DIGEST is a FORTRAN90 library which solves the partial digest problem.
You can go up one level to the MATLAB source codes.