The scalar function $f$, and the derivatives of $f$, are returned by the function f which, given an integer $m$, should evaluate ${f}^{\left(m\right)}\left({z}_{\mathit{i}}\right)$ at a number of (generally complex) points ${z}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{n}_{z}$. For any $z$ on the real line, $f\left(z\right)$ must also be real. f01emc 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
5Arguments
1: $\mathbf{order}$ – Nag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by ${\mathbf{order}}=\mathrm{Nag\_RowMajor}$. See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint:
${\mathbf{order}}=\mathrm{Nag\_RowMajor}$ or $\mathrm{Nag\_ColMajor}$.
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 f01emc will terminate the computation, with ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$NE_INT, NE_INT_2 or NE_USER_STOP.
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.
${\mathbf{fz}}\left[\mathit{i}-1\right]$ should return the value ${f}^{\left(m\right)}\left({z}_{\mathit{i}}\right)$, for $\mathit{i}=1,2,\dots ,{n}_{z}$. If ${z}_{i}$ lies on the real line, then so must ${f}^{\left(m\right)}\left({z}_{i}\right)$.
6: $\mathbf{comm}$ – Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to f.
user – double *
iuser – Integer *
p – Pointer
The type Pointer will be void *. Before calling f01emc you may allocate memory and initialize these pointers with various quantities for use by f when called from f01emc (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
Note:f should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by f01emc. If your code inadvertently does return any NaNs or infinities, f01emc is likely to produce unexpected results.
6: $\mathbf{comm}$ – Nag_Comm *
The NAG communication argument (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
7: $\mathbf{iflag}$ – Integer *Output
On exit: ${\mathbf{iflag}}=0$, unless iflag has been set nonzero inside f, in which case iflag will be the value set and fail will be set to ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$NE_INT, NE_INT_2 or NE_USER_STOP.
8: $\mathbf{imnorm}$ – double *Output
On exit: if $A$ has complex eigenvalues, f01emc will use complex arithmetic to compute $f\left(A\right)$. The imaginary part is discarded at the end of the computation, because it will theoretically vanish. imnorm contains the $1$-norm of the imaginary part, which should be used to check that the function has given a reliable answer.
If $A$ has real eigenvalues, f01emc uses real arithmetic and ${\mathbf{imnorm}}=0$.
9: $\mathbf{fail}$ – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error 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 $\u27e8\mathit{\text{value}}\u27e9$ had an illegal value.
NE_CONVERGENCE
A Taylor series failed to converge.
NE_INT
On entry, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{n}}\ge 0$.
NE_INT_2
On entry, ${\mathbf{pda}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{pda}}\ge {\mathbf{n}}$.
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 unexpected internal error occurred when ordering the eigenvalues of $A$. Please contact NAG.
There was an error whilst reordering the Schur form of $A$. Note: this failure should not occur and suggests that the function has been called incorrectly.
The routine was unable to compute the Schur decomposition of $A$. Note: this failure should not occur and suggests that the function has been called incorrectly.
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.
For a normal matrix $A$ (for which ${A}^{\mathrm{T}}A=A{A}^{\mathrm{T}}$), 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
f01emc 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. You must also ensure that you use the NAG communication argument comm in a thread safe manner, which is best achieved by only using it to supply read-only data to the user functions.
f01emc 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.
9Further Comments
If $A$ has real eigenvalues then up to $6{n}^{2}$ of double allocatable memory may be required. If $A$ has complex eigenvalues then up to $6{n}^{2}$ of Complex allocatable memory may be 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 f01elc can be used to evaluate $f\left(A\right)$ using numerical differentiation.
If $A$ is real symmetric then it is recommended that f01efc be used as it is more efficient and, in general, more accurate than f01emc.
For any $z$ on the real line, $f\left(z\right)$ must be real. $f$ must also be complex analytic on the spectrum of $A$. These conditions ensure that $f\left(A\right)$ is real for real $A$.
For further information on matrix functions, see Higham (2008).
If estimates of the condition number of the matrix function are required then f01jcc should be used.
f01fmc can be used to find the matrix function $f\left(A\right)$ for a complex matrix $A$.