What's new in version 3.0?

Version 3.0 of LAPACK introduces new routines, as well as extending the functionality of existing routines. The most significant new routines and functions are:

1.

a faster singular value decomposition (SVD), computed by divide-and-conquer (xGESDD)

2.

faster routines for solving rank-deficient least squares problems:

• using QR with column pivoting (xGELSY, based on xGEQP3)
• using the SVD based on divide-and-conquer (xGELSD)

3.

new routines for the generalized symmetric eigenproblem:

• xHEGVD/xSYGVD, xHPGVD/xSPGVD, xHBGVD/xSBGVD: faster routines based on divide-and-conquer
• xHEGVX/xSYGVX, xHPGVX/xSPGVX, xHBGVX/xSBGVX: routines based on bisection/inverse iteration to more efficiently compute a subset of the eigenvalues and/or eigenvectors

4.

faster routines for the symmetric eigenproblem using the ``relative robust representation'' algorithm (xSYEVR/xHEEVR, xSTEVR, xSTEGR)

5.

new simple and expert drivers for the generalized nonsymmetric eigenproblem (xGGES, xGGEV, xGGESX, xGGEVX), including error bounds

6.

a solver for the generalized Sylvester equation (xTGSYL), used in 5)

7.

computational routines (xTGEXC, xTGSEN, xTGSNA) used in 5)

8.

a blocked version of xTZRQF (xTZRZF), and associated xORMRZ/xUNMRZ

One of the primary design features of the LAPACK library is that all releases are backward compatible. A user's program calling LAPACK will never fail because of a new release of the library. As a result, however, the calling sequences (or amount of workspace required) to existing routines cannot be altered. Therefore, if a performance enhancement requires a modification of this type, a new routine must be created. There are several routines included in LAPACK, version 3.0, that fall into this category. Specifically,

• xGEGS is deprecated and replaced by routine xGGES
• xGEGV is deprecated and replaced by routine xGGEV
• xGELSX is deprecated and replaced by routine xGELSY
• xGEQPF is deprecated and replaced by routine xGEQP3
• xTZRQF is deprecated and replaced by routine xTZRZF
• xLATZM is deprecated and replaced by routines xORMRZ/xUNMRZ

The ``old'' version of the routine is still included in the library but the user is advised to upgrade to the ``new'' faster version. References to the ``old'' versions are removed from this users' guide.

In addition to replacing the above list of routines, there are a number of other significantly faster new driver routines that we recommend in place of their older counterparts listed below. We continue to include the older drivers in this users' guide because the old drivers may use less workspace than the new drivers, and because the old drivers may be faster in certain special cases (we will continue to improve the new drivers in a future release until they completely replace their older counterparts):

• xSYEV/xHEEV and xSYEVD/xHEEVD should be replaced by xSYEVR/xHEEVR
• xSTEV and xSTEVD should be replaced by xSTEVR
• xSPEV/xHPEV should be replaced by xSPEVD/xHPEVD
• xSBEV/xHBEV should be replaced by xSBEVD/xHBEVD
• xGESVD should be replaced by xGESDD
• xSYGV/xHEGV should be replaced by xSYGVD/xHEGVD
• xSPGV/xHPGV should be replaced by xSPGVD/xHPGVD
• xSBGV/xHBGV should be replaced by xSBGVD/xHBGVD

This release of LAPACK introduces routines that exploit IEEE arithmetic. We have a prototype running of a new algorithm (xSTEGR), which may be the ultimate solution for the symmetric eigenproblem on both parallel and serial machines. This algorithm has been incorporated into the drivers xSYEVR, xHEEVR and xSTEVR for the symmetric eigenproblem, and will be propagated into the generalized symmetric definite eigenvalue problems, the SVD, the generalized SVD and the SVD-based least squares solver. Refer to section 2.4.4 for further information. We expect to also propagate this algorithm into ScaLAPACK.

We have also incorporated the LWORK=-1 query capability into this release of LAPACK, whereby a user can request the amount of workspace required for a routine. For complete details, refer to section 5.1.8.