The scalar function $f$, and the derivatives of $f$, are returned by the subroutine f which, given an integer $m$, should evaluate ${f}^{\left(m\right)}\left({z}_{\mathit{i}}\right)$ at a number of points ${z}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{n}_{z}$, on the complex plane. f01fmf is, therefore, appropriate for functions that can be evaluated on the complex plane and whose derivatives, of arbitrary order, can also be evaluated on the complex plane.
4References
Davies P I and Higham N J (2003) A Schur–Parlett algorithm for computing matrix functions SIAM J. Matrix Anal. Appl.25(2) 464–485
Higham N J (2008) Functions of Matrices: Theory and Computation SIAM, Philadelphia, PA, USA
On exit: iflag should either be unchanged from its entry value of zero, or may be set nonzero to indicate that there is a problem in evaluating the function $f\left(z\right)$; for instance $f\left({z}_{i}\right)$ may not be defined for a particular ${z}_{i}$. If iflag is returned as nonzero then f01fmf will terminate the computation, with ${\mathbf{ifail}}={\mathbf{2}}$.
3: $\mathbf{nz}$ – IntegerInput
On entry: ${n}_{z}$, the number of function or derivative values required.
On exit: the ${n}_{z}$ function or derivative values.
${\mathbf{fz}}\left(\mathit{i}\right)$ should return the value ${f}^{\left(m\right)}\left({z}_{\mathit{i}}\right)$, for $\mathit{i}=1,2,\dots ,{n}_{z}$.
7: $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) arrayUser Workspace
f is called with the arguments iuser and ruser as supplied to f01fmf. You should use the arrays iuser and ruser to supply information to f.
f must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which f01fmf is called. Arguments denoted as Input must not be changed by this procedure.
Note:f should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by f01fmf. If your code inadvertently does return any NaNs or infinities, f01fmf is likely to produce unexpected results.
6: $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) arrayUser Workspace
iuser and ruser are not used by f01fmf, but are passed directly to f and may be used to pass information to this routine.
7: $\mathbf{iflag}$ – IntegerOutput
On exit: ${\mathbf{iflag}}=0$, unless iflag has been set nonzero inside f, in which case iflag will be the value set and ifail will be set to ${\mathbf{ifail}}={\mathbf{2}}$.
8: $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $\mathrm{-1}$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $\mathrm{-1}$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $\mathrm{-1}$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).
6Error Indicators and Warnings
If on entry ${\mathbf{ifail}}=0$ or $\mathrm{-1}$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
There was an error whilst reordering the Schur form of $A$. Note: this failure should not occur and suggests that the routine has been called incorrectly.
${\mathbf{ifail}}=4$
The routine was unable to compute the Schur decomposition of $A$. Note: this failure should not occur and suggests that the routine has been called incorrectly.
${\mathbf{ifail}}=5$
An unexpected internal error occurred. Please contact NAG.
${\mathbf{ifail}}=-1$
On entry, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=-3$
On entry, argument lda is invalid.
Constraint: ${\mathbf{lda}}\ge {\mathbf{n}}$.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please
contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
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.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
7Accuracy
For a normal matrix $A$ (for which ${A}^{\mathrm{H}}A=A{A}^{\mathrm{H}}$), the Schur decomposition is diagonal and the algorithm reduces to evaluating $f$ at the eigenvalues of $A$ and then constructing $f\left(A\right)$ using the Schur vectors. This should give a very accurate result. In general, however, no error bounds are available for the algorithm. See Section 9.4 of Higham (2008) for further discussion of the Schur–Parlett algorithm.
8Parallelism and Performance
f01fmf 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. The user workspace arrays iuser and ruser are classified as OpenMP shared memory and use of iuser and ruser has to take account of this in order to preserve thread safety whenever information is written back to either of these arrays. If at all possible, it is recommended that these arrays are only used to supply read-only data to the user functions when a multithreaded implementation is being used.
f01fmf 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.
9Further Comments
Up to $6{n}^{2}$ of complex allocatable memory is required.
The cost of the Schur–Parlett algorithm depends on the spectrum of $A$, but is roughly between $28{n}^{3}$ and ${n}^{4}/3$ floating-point operations. There is an additional cost in evaluating $f$ and its derivatives.
If the derivatives of $f$ are not known analytically, then f01flf can be used to evaluate $f\left(A\right)$ using numerical differentiation.
If $A$ is complex Hermitian then it is recommended that f01fff be used as it is more efficient and, in general, more accurate than f01fmf.
Note that $f$ must be analytic in the region of the complex plane containing the spectrum of $A$.
For further information on matrix functions, see Higham (2008).
If estimates of the condition number of the matrix function are required then f01kcf should be used.
f01emf can be used to find the matrix function $f\left(A\right)$ for a real matrix $A$.