f01jaf computes an estimate of the absolute condition number of a matrix function at a real 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 f01jaf or nagf_matop_real_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 . f01jaf 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: – Real (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 f01jaf 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 evaluating the matrix function . Please contact NAG.
An internal error occurred when estimating the norm of the Fréchet derivative of at . Please contact NAG.
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.
f01jaf uses the norm estimation routine f04ydf to estimate a quantity , where and . For further details on the accuracy of norm estimation, see the documentation for f04ydf.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f01jaf 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.
f01jaf 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.
The matrix function is computed using one of three underlying matrix function routines: