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### Solution of Common Numerical Linear Algebra Problems

This section contains performance numbers for selected driver routines. These routines provide complete solutions for common linear algebra problems.

- Solve a general N-by-N system of linear equations with one right-hand side using the routine PSGESV/PDGESV.
- Solve a symmetric positive definite N-by-N system of linear equations with one right-hand side, using PSPOSV/PDPOSV.
- Solve an N-by-N linear least squares problem with one right-hand side using the routine PSGELS/PDGELS.
- Find the eigenvalues and optionally the corresponding eigenvectors of an N-by-N symmetric matrix, using the routine PSSYEVX/PDSYEVX.
- Find the eigenvalues and optionally the corresponding eigenvectors of an N-by-N symmetric matrix, using the routine PSSYEV/PDSYEV.
- Find the singular values and optionally the corresponding right and left singular vectors of an N-by-N matrix, using PSGESVD/PDGESVD.
- Find the eigenvalues and optionally the corresponding right eigenvectors of an N-by-N Hessenberg matrix, using the routine PSLAHQR/PDLAHQR.

Table 5.8 presents ``standard'' floating-point operation costs () for selected ScaLAPACK drivers for matrices of order N. Approximate values of the constants and defined in section 5.2.3 are also provided.

Table 5.8: ``Standard'' floating-point operation () and communication costs (, ) for selected ScaLAPACK drivers

The operation counts given for the eigenvalue and SVD drivers are incomplete. They do not include any of the computation costs (i.e., the entire tridiagonal eigendecomposition is ignored in PxxxEVX). Furthermore, the reductions involved require matrix-vector multiplies, which are less efficient than the matrix-matrix multiplies required by the other drivers listed here. Hence this table greatly underestimates the execution time of the eigenvalue and SVD drivers, especially the expert symmetric eigensolver drivers. For PxLAHQR, when only eigenvalues are computed, and look the same as the full Schur form case, in terms of ``order of magnitude''. There is actually to the number of messages/volume depending on the circumstances.

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