In this chapter we explain our overall approach to obtaining error bounds and provide enough information to use the software. The comments at the beginning of the individual routines should be consulted for more details. It is beyond the scope of this chapter to justify all the bounds we present. Instead, we give references to the literature. For example, standard material on error analysis can be found in [71, 114, 84, 38].
To make this chapter easy to read, we have labeled parts not essential for a first reading as Further Details. The sections not labeled as Further Details should provide all the information needed to understand and use the main error bounds computed by ScaLAPACK. The Further Details sections provide mathematical background, references, and tighter but more expensive error bounds, and may be read later.
Since ScaLAPACK uses the same overall algorithmic approach as LAPACK, its error bounds are essentially the same as those for LAPACK. Therefore, this chapter is largely analogous to Chapter 4 of the LAPACK Users' Guide [3]. Significant differences between LAPACK and ScaLAPACK include the following:
,
which is available in IEEE standard floating-point arithmetic.
In contrast to its LAPACK analogue, xSYEVX, PxSYEVX allows
the user to trade off orthogonality of computed eigenvectors and
runtime.
In section 6.1 we discuss the sources of numerical error, in particular roundoff error. We also briefly discuss IEEE arithmetic. Section 6.2 discusses the new sources of numerical error specific to parallel libraries, and the restrictions they impose on the reliable use of ScaLAPACK. Section 6.3 discusses how to measure errors, as well as some standard notation. Section 6.4 discusses further details of how error bounds are derived. Sections 6.5 through 6.9 present error bounds for linear equations, linear least squares problems, the symmetric eigenproblem, the singular value decomposition, and the generalized symmetric definite eigenproblem, respectively.