Footnotes
Keywords: slug

...output
  This example program computes the relative machine precision which causes, on some systems, the IEEE floating-point exception flags to be raised. This may result in the printing of a warning message. This is normal.

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...output
  This example program computes the relative machine precision which causes, on some systems, the IEEE floating-point exception flags to be raised. This may result in the printing of a warning message. This is normal.

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...output),
  (Local input or local output) means that the argument may be either a local input argument or a local output argument, depending on the values of other arguments; for example, in the PxyySVX driver routines, some arguments are used either as local output arguments to return details of a factorization, or as local input arguments to supply details of a previously computed factorization.

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...input),
  (local or global input) is used to describe the length of the workspace arguments, e.g., LWORK, where the value can be local input specifying the size of the local WORK array, or global input LWORK=-1 specifying a global query for the amount of workspace required.

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...input)
  (local or global input) is used to describe the length of the workspace arguments, e.g., LWORK, where the value can be local input specifying the size of the local WORK array, or global input LWORK=-1 specifying a global query for the amount of workspace required.

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...size
  The block size must be large enough that the local matrix multiply is efficient. A block size of 64 suffices for most computers that have only one processor per node. Computers that have multiple shared-memory processors on each node may require a larger block size.

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...(DSSL)
  Dakota Scientific Software, Inc., 501 East Saint Joseph Street, Rapid City, SD 57701-3995 USA, (605) 394-2471

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...(DSSL)
  Dakota Scientific Software, Inc., 501 East Saint Joseph Street, Rapid City, SD 57701-3995 USA, (605) 394-2471

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...PSLAHQR/PDLAHQR.
  Strictly speaking, PSLAHQR/PDLAHQR is an auxiliary routine for computing the eigenvalues and optionally the corresponding eigenvectors of the more general case of nonsymmetric matrices.

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...computer.
  The ScaLAPACK sample timer page (contained within the ScaLAPACK examples directory on netlib and on the CD-ROM) has a BLACS port of the message-passing program and instructions for building the BLAS timing program.

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...PxSYEVX.
  See section 3.1.3 for explanation of the naming convention used for ScaLAPACK routines.

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...'E'))
  ICTXT refers to the BLACS CONTEXT parameter. Refer to section 4.1.2 for further details.

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...digit,
  This is the case on the Cray C90 and its predecessors and emulators.

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...PCs,
  Important machines that do not implement the IEEE standard include the CRAY X-MP, CRAY Y-MP, CRAY 2, CRAY C90, IBM 370, DEC Vax, and their emulators.

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...[#lawn112##1#].
  Running either machine in non-default mode to avoid this problem, either so that the IBM RS/6000 flushes denormalized numbers to zero or so that the DEC Alpha handles denormalized numbers correctly by doing gradual underflow, slows down the machine significantly [42].

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....
  Sometimes our algorithms satisfy only 535#535 where both 532#532 and 536#536 are small. This does not significantly change the following analysis.

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...small.
  More generally, we need only Lipschitz continuity of f and may use the Lipschitz constant in place of f' in deriving error bounds.

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...exist).
  This is a different use of the term ill-posed from that used in other contexts. For example, to be well-posed (not ill-posed) in the sense of Hadamard, it is sufficient for f to be continuous, whereas we require Lipschitz continuity.

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...described
  There are some caveats to this statement. When computing the inverse of a matrix, the backward error E is small, taking the columns of the computed inverse one at a time, with a different E for each column [62]. The same is true when computing the eigenvectors of a nonsymmetric matrix. When computing the eigenvalues and eigenvectors of 550#550, 551#551 or 552#552, with A symmetric and B symmetric and positive definite (using PxSYGVX or PxHEGVX), the method may not be backward normwise stable if B has a large condition number 553#553, although it has useful error bounds in this case too (see section 6.9). Solving the Sylvester equation AX+XB=C for the matrix X may not be backward stable, although there are again useful error bounds for X [83].

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...error.
  For other algorithms, the answers (and computed error bounds) are as accurate as though the algorithms were componentwise relatively backward stable, even though they are not. These algorithms are called componentwise relatively forward stable.

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...bound
  As discussed in section 6.3, this approximate error bound may underestimate the true error by a factor p(n), which is a modestly growing function of the problem dimension n. Often 569#569.

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...vectors
  These bounds are special cases of those in section 6.7 since the singular values and vectors of A are simply related to the eigenvalues and eigenvectors of the Hermitian matrix 700#700 [71, p. 427,].

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...magnitude:
  This bound is guaranteed only if the Level 3 BLAS are implemented in a conventional way, not in a fast way.

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...large.
  Another interpretation of chordal distance is as half the usual Euclidean distance between the projections of 714#714 and 637#637 on the Riemann sphere, i.e., half the length of the chord connecting the projections.

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