f01elc computes the matrix function, $f\left(A\right)$, of a real $n\times n$ matrix $A$. Numerical differentiation is used to evaluate the derivatives of $f$ when they are required.
The function may be called by the names: f01elc or nag_matop_real_gen_matrix_fun_num.
3Description
$f\left(A\right)$ is computed using the Schur–Parlett algorithm described in Higham (2008) and Davies and Higham (2003). The coefficients of the Taylor series used in the algorithm are evaluated using the numerical differentiation algorithm of Lyness and Moler (1967).
The scalar function $f$ is supplied via function f which evaluates $f\left({z}_{i}\right)$ at a number of points ${z}_{i}$.
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
Lyness J N and Moler C B (1967) Numerical differentiation of analytic functions SIAM J. Numer. Anal.4(2) 202–210
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 $f\left({z}_{i}\right)$; for instance $f\left({z}_{i}\right)$ may not be defined. If iflag is returned as nonzero then f01elc will terminate the computation, with ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$NE_INT, NE_INT_2 or NE_USER_STOP.
2: $\mathbf{nz}$ – IntegerInput
On entry: ${n}_{z}$, the number of function values required.
On exit: the ${n}_{z}$ function values.
${\mathbf{fz}}\left[\mathit{i}-1\right]$ should return the value $f\left({z}_{\mathit{i}}\right)$, for $\mathit{i}=1,2,\dots ,{n}_{z}$. If ${z}_{i}$ lies on the real line, then so must $f\left({z}_{i}\right)$.
5: $\mathbf{comm}$ – 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 f01elc you may allocate memory and initialize these pointers with various quantities for use by f when called from f01elc (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 f01elc. If your code inadvertently does return any NaNs or infinities, f01elc is likely to produce unexpected results.
5: $\mathbf{comm}$ – Nag_Comm *
The NAG communication argument (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
6: $\mathbf{iflag}$ – 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.
7: $\mathbf{imnorm}$ – double *Output
On exit: if $A$ has complex eigenvalues, f01elc 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 routine has given a reliable answer.
If a has real eigenvalues, f01elc uses real arithmetic and ${\mathbf{imnorm}}=0$.
8: $\mathbf{fail}$ – 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 after $40$ terms. Further Taylor series coefficients can no longer reliably be obtained by numerical differentiation.
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.
The function was unable to compute the Schur decomposition of $A$. Note: this failure should not occur and suggests that the function has been called incorrectly.
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.
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. See Section 9.4 of Higham (2008) for further discussion of the Schur–Parlett algorithm, and Lyness and Moler (1967) for a discussion of numerical differentiation.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f01elc 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.
f01elc 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
The Integer allocatable memory required is $n$. 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 numerically differentiating $f$, in order to obtain the Taylor series coefficients. If the derivatives of $f$ are known analytically, then f01emc can be used to evaluate $f\left(A\right)$ more accurately. If $A$ is real symmetric then it is recommended that f01efc be used as it is more efficient and, in general, more accurate than f01elc.
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 f01jbc should be used.
f01flc can be used to find the matrix function $f\left(A\right)$ for a complex matrix $A$.