Computation of 3-d matrices of maximal trace over rotations

These programs compute 3x3 matrices of maximal trace over rotation matrices from input matrices. A dxd matrix M is of maximal trace over rotation matrices if given any dxd rotation matrix U, the trace of UM does not exceed that of M. Given a dxd matrix M, the problem of finding among all dxd rotation matrices U one such that UM is of maximal trace over rotation matrices is intricately related to the so-called constrained orthogonal Procrustes problem which is the least-squares problem that calls for a rotation matrix that optimally aligns two corresponding sets of points in d-dimensional Euclidean space.

The package consists of Fortran and Matlab programs, Matlab mex file of Fortran program, compiled mex file, and sample data file of 3x3 matrices. The software executes a procedure based on the Cayley transform and Newton's method that for each input matrix M computes a rotation matrix U such that UM is symmetric. If the procedure is successful (Newton's method didn't fail) a rotation matrix R (without the SVD) is computed such that RUM is of maximal trace over rotation matrices. If Newton's method fails, then using our version of the SVD method, the program computes a rotation matrix R such that RM is of maximal trace over rotation matrices.

Note the following: (a) The option exists in the Fortran code to do everything using our version of the SVD (singular value decomposition) method only. (b) In the Fortran code the integer variable ITEX is set to the maximum number of allowed iterations per input matrix of Newton's method. (c) The Matlab program can be used for the same purposes. (d) The option also exists in the Matlab code to do everything using the Matlab version of the SVD method only. Otherwise the Matlab mex file of the Fortran program is used by the Matlab code. This has the effect of executing the Fortran code as described above from the Matlab code.