f01kaf 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.
The routine may be called by the names f01kaf or nagf_matop_complex_gen_matrix_cond_std.
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 . f01kaf 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).
Higham N J (2008) Functions of Matrices: Theory and Computation SIAM, Philadelphia, PA, USA
1: – Character(*)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.
, , , , or .
2: – IntegerInput
On entry: , the order of the matrix .
3: – Complex (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array a
must be at least
On entry: the matrix .
On exit: the matrix, .
4: – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f01kaf is called.
5: – Real (Kind=nag_wp)Output
On exit: an estimate of the absolute condition number of at .
6: – Real (Kind=nag_wp)Output
On exit: the -norm of .
7: – Real (Kind=nag_wp)Output
On exit: the -norm of .
8: – IntegerInput/Output
On entry: ifail must be set to , or to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of means that an error message is printed while a value of means that it is not.
If halting is not appropriate, the value or is recommended. If message printing is undesirable, then the value is recommended. Otherwise, the value is recommended. When the value or is used it is essential to test the value of ifail on exit.
On exit: unless the routine detects an error or a warning has been flagged (see Section 6).
6Error Indicators and Warnings
If on entry or , explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
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 f01fcf,f01fjforf01fkf with the matrix .
An unexpected error has been triggered by this routine. Please
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
f01kaf uses the norm estimation routine f04zdf to estimate a quantity , where and . For further details on the accuracy of norm estimation, see the documentation for f04zdf.
8Parallelism and Performance
f01kaf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library. In these implementations, this routine may make calls to the user-supplied functions from within an OpenMP parallel region. Thus OpenMP directives 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.
f01kaf 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 routine. 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 f01fcf,f01fjforf01fkf.
f01kaf returns the matrix function . This is computed using f01fcf if , f01fjf if and f01fkf otherwise. If only is required, without an estimate of the condition number, then it is far more efficient to use f01fcf,f01fjforf01fkf directly.
f01jaf can be used to find the condition number of the exponential, logarithm, sine, cosine, sinh or cosh at a real matrix.
This example estimates the absolute and relative condition numbers of the matrix sinh function for