NAG CL Interface
f01kac (complex_gen_matrix_cond_std)
1
Purpose
f01kac computes an estimate of the absolute condition number of a matrix function of a complex 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.
2
Specification
The function may be called by the names: f01kac or nag_matop_complex_gen_matrix_cond_std.
3
Description
The absolute condition number of
at
,
is given by the norm of the Fréchet derivative of
,
, which is defined by
where
is the Fréchet derivative in the direction
.
is linear in
and can, therefore, be written as
where the
operator stacks the columns of a matrix into one vector, so that
is
.
f01kac computes an estimate
such that
, where
. The relative condition number can then be computed via
The algorithm used to find
is detailed in Section 3.4 of
Higham (2008).
4
References
Higham N J (2008) Functions of Matrices: Theory and Computation SIAM, Philadelphia, PA, USA
5
Arguments
-
1:
– Nag_MatFunType
Input
-
On entry: 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.
Constraint:
, , , , or .
-
2:
– Integer
Input
-
On entry: , the order of the matrix .
Constraint:
.
-
3:
– Complex
Input/Output
-
Note: the dimension,
dim, of the array
a
must be at least
.
The th element of the matrix is stored in .
On entry: the matrix .
On exit: the matrix, .
-
4:
– Integer
Input
-
On entry: the stride separating matrix row elements in the array
a.
Constraint:
.
-
5:
– double *
Output
-
On exit: an estimate of the absolute condition number of at .
-
6:
– double *
Output
-
On exit: the -norm of .
-
7:
– double *
Output
-
On exit: the -norm of .
-
8:
– NagError *
Input/Output
-
The NAG error argument (see
Section 7 in the Introduction to the NAG Library CL Interface).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_INT
-
On entry, .
Constraint: .
- NE_INT_2
-
On entry, and .
Constraint: .
- NE_INTERNAL_ERROR
-
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 for assistance.
See
Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
An internal error occurred when estimating the norm of the Fréchet derivative of
at
. Please contact
NAG.
An internal error occurred when evaluating the matrix function
. You can investigate further by calling
f01fcc,
f01fjc or
f01fkc with the matrix
.
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
See
Section 8 in the Introduction to the NAG Library CL Interface for further information.
7
Accuracy
f01kac uses the norm estimation function
f04zdc to estimate a quantity
, where
and
. For further details on the accuracy of norm estimation, see the documentation for
f04zdc.
8
Parallelism and Performance
Background information to multithreading can be found in the
Multithreading documentation.
f01kac is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library. In these implementations, this function may make calls to the user-supplied functions from within an OpenMP parallel region. Thus OpenMP pragmas within the user functions can only be used if you are compiling the user-supplied function and linking the executable in accordance with the instructions in the
Users' Note for your implementation.
f01kac 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.
Please consult the
X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the
Users' Note for your implementation for any additional implementation-specific information.
Approximately
of complex allocatable memory is required by the routine, in addition to the memory used by the underlying matrix function routines
f01fcc,
f01fjc or
f01fkc.
f01kac returns the matrix function
. This is computed using
f01fcc if
,
f01fjc if
and
f01fkc otherwise. If only
is required, without an estimate of the condition number, then it is far more efficient to use
f01fcc,
f01fjc or
f01fkc directly.
f01jac can be used to find the condition number of the exponential, logarithm, sine, cosine, sinh or cosh at a real matrix.
10
Example
This example estimates the absolute and relative condition numbers of the matrix sinh function for
10.1
Program Text
10.2
Program Data
10.3
Program Results