Name
Title | ||
|---|---|---|
A Backward Stable Algorithm for Computing the CS Decomposition via the Polar Decomposition |
Abstract | ||
|---|---|---|
We introduce a backward stable algorithm for computing the CS decomposition of a partitioned 2n x n matrix with orthonormal columns, or a rank-deficient partial isometry. The algorithm computes two n x n polar decompositions (which can be carried out in parallel) followed by an eigendecomposition of a judiciously crafted n x n Hermitian matrix. We prove that the algorithm is backward stable whenever the aforementioned decompositions are computed in a backward stable way. Our algorithm can also be adapted to compute the complete CS decomposition of a square orthogonal or unitary matrix. Since the polar decomposition and the symmetric eigendecomposition are highly amenable to parallelization, the algorithm inherits this feature. We illustrate this fact by invoking recently developed algorithms for the polar decomposition and symmetric eigendecomposition that leverage Zolotarev's best rational approximations of the sign function. Numerical examples demonstrate that the resulting algorithm for computing the CS decomposition enjoys excellent numerical stability. |
Year | DOI | Venue |
|---|---|---|
2018 | 10.1137/18M1182747 | SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS |
Keywords | Field | DocType |
CS decomposition,polar decomposition,eigendecomposition,Zolotarev,generalized singular value decomposition,simultaneous diagonalization,backward stability | Partial isometry,Matrix (mathematics),Algorithm,Polar decomposition,Orthonormal basis,Sign function,Eigendecomposition of a matrix,Hermitian matrix,Mathematics,Numerical stability | Journal |
Volume | Issue | ISSN |
39 | 3 | 0895-4798 |
Citations | PageRank | References |
0 | 0.34 | 5 |
Authors | ||
3 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Evan S. Gawlik | 1 | 5 | 3.60 |
| Yuji Nakatsukasa | 2 | 97 | 17.74 |
| Brian D. Sutton | 3 | 56 | 9.12 |