NAG Library Function Document
nag_matop_real_gen_matrix_cond_std (f01jac) computes an estimate of the absolute condition number of a matrix function at a real by matrix in the -norm, where is either the exponential, logarithm, sine, cosine, hyperbolic sine (sinh) or hyperbolic cosine (cosh). The evaluation of the matrix function, , is also returned.
||nag_matop_real_gen_matrix_cond_std (Nag_MatFunType fun,
The absolute condition number of
is given by the norm of the Fréchet derivative of
, which is defined by
is the Fréchet derivative in the direction
is linear in
and can therefore be written as
operator stacks the columns of a matrix into one vector, so that
. nag_matop_real_gen_matrix_cond_std (f01jac) computes an estimate
. The relative condition number can then be computed via
The algorithm used to find
is detailed in Section 3.4 of Higham (2008)
Higham N J (2008) Functions of Matrices: Theory and Computation SIAM, Philadelphia, PA, USA
fun – Nag_MatFunTypeInput
: indicates which matrix function will be used.
- The matrix exponential, , will be used.
- The matrix sine, , will be used.
- The matrix cosine, , will be used.
- The hyperbolic matrix sine, , will be used.
- The hyperbolic matrix cosine, , will be used.
- The matrix logarithm, , will be used.
, , , , or .
n – IntegerInput
On entry: , the order of the matrix .
a – doubleInput/Output
the dimension, dim
, of the array a
must be at least
The th element of the matrix is stored in .
On entry: the by matrix .
On exit: the by matrix, .
pda – IntegerInput
: the stride separating matrix row elements in the array a
conda – double *Output
On exit: an estimate of the absolute condition number of at .
norma – double *Output
On exit: the -norm of .
normfa – double *Output
On exit: the -norm of .
fail – NagError *Input/Output
The NAG error argument (see Section 3.6
in the Essential Introduction).
6 Error Indicators and Warnings
Allocation of memory failed.
On entry, argument had an illegal value.
On entry, .
On entry, and .
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG
An internal error occurred when estimating the norm of the Fréchet derivative of
. Please contact NAG
An internal error occurred when evaluating the matrix function
You can investigate further by calling nag_real_gen_matrix_exp (f01ecc)
, nag_matop_real_gen_matrix_log (f01ejc)
or nag_matop_real_gen_matrix_fun_std (f01ekc)
with the matrix
nag_matop_real_gen_matrix_cond_std (f01jac) uses the norm estimation function nag_linsys_real_gen_norm_rcomm (f04ydc)
to estimate a quantity
. For further details on the accuracy of norm estimation, see the documentation for nag_linsys_real_gen_norm_rcomm (f04ydc)
8 Parallelism and Performance
nag_matop_real_gen_matrix_cond_std (f01jac) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_matop_real_gen_matrix_cond_std (f01jac) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
In these implementations, this may make calls to the user supplied functions from within an OpenMP parallel region. Thus OpenMP directives
within the user functions should be avoided, unless you are using the same OpenMP runtime library (which normally means using the same compiler) as that used to build your NAG Library implementation, as listed in the Installers' Note.
Please consult the Users' Note
for your implementation for any additional implementation-specific information.
The matrix function is computed using one of three underlying matrix function routines:
Approximately of real allocatable memory is required by the routine, in addition to the memory used by these underlying matrix function routines.
If only is required, without an estimate of the condition number, then it is far more efficient to use the appropriate matrix function routine listed above.
can be used to find the condition number of the exponential, logarithm, sine, cosine, sinh or cosh matrix functions at a complex matrix.
This example estimates the absolute and relative condition numbers of the matrix sinh function where
10.1 Program Text
Program Text (f01jace.c)
10.2 Program Data
Program Data (f01jace.d)
10.3 Program Results
Program Results (f01jace.r)