Although Algorithms 1 and 2 only divide the spectrum along the pure imaginary axis and the unit circle, respectively, we can use Möbius and other simple transformations of the input matrix to divide along other more general curves. As a result, we can compute the eigenvalues (and corresponding invariant subspace) inside any region defined as the intersection of regions defined by these curves. This is a major attraction of this kind of algorithm.
Let us show how to use Möbius transformations to divide the spectrum along arbitrary lines and circles. Transform the eigenproblem to
Then if we apply Algorithm 1 to we can split the spectrum with respect to a region
If we apply Algorithm 2 to , we can split along the curve
For example, by computing the matrix sign function of , then Algorithm 1 will split the spectrum of along a circle centered at with radius . If is real, and we choose to be real, then all arithmetic will be real.
If and , then Algorithm 2 will split the spectrum of along a circle centered at with radius . If is real, and we choose to be real, then all arithmetic in the algorithm will be real.
Other more general regions can be obtained by taking as a polynomial function of . For example, by computing the matrix sign function of , we can divide the spectrum within a ``bowtie'' shaped region centered at . Figure 1 illustrates the regions which the algorithms can deal with assuming that is real and the algorithms use only real arithmetic.
Figure 1: Different Geometric Regions for the Spectral Decomposition