The routine may be called by the names f01ekf or nagf_matop_real_gen_matrix_fun_std.
3Description
$f\left(A\right)$, where $f$ is either the exponential, sine, cosine, sinh or cosh, is computed using the Schur–Parlett algorithm described in Higham (2008) and Davies and Higham (2003).
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: $\mathbf{fun}$ – Character(*)Input
On entry: indicates which matrix function will be computed.
${\mathbf{fun}}=\text{'EXP'}$
The matrix exponential, ${e}^{A}$, will be computed.
${\mathbf{fun}}=\text{'SIN'}$
The matrix sine, $\mathrm{sin}\left(A\right)$, will be computed.
${\mathbf{fun}}=\text{'COS'}$
The matrix cosine, $\mathrm{cos}\left(A\right)$, will be computed.
${\mathbf{fun}}=\text{'SINH'}$
The hyperbolic matrix sine, $\mathrm{sinh}\left(A\right)$, will be computed.
${\mathbf{fun}}=\text{'COSH'}$
The hyperbolic matrix cosine, $\mathrm{cosh}\left(A\right)$, will be computed.
Constraint:
${\mathbf{fun}}=\text{'EXP'}$, $\text{'SIN'}$, $\text{'COS'}$, $\text{'SINH'}$ or $\text{'COSH'}$.
2: $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint:
${\mathbf{n}}\ge 0$.
3: $\mathbf{a}({\mathbf{lda}},*)$ – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array a
must be at least
${\mathbf{n}}$.
On entry: the $n\times n$ matrix $A$.
On exit: the $n\times n$ matrix, $f\left(A\right)$.
4: $\mathbf{lda}$ – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f01ekf is called.
Constraint:
${\mathbf{lda}}\ge {\mathbf{n}}$.
5: $\mathbf{imnorm}$ – Real (Kind=nag_wp)Output
On exit: if $A$ has complex eigenvalues, f01ekf will use complex arithmetic to compute the matrix function. 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, f01ekf uses real arithmetic and ${\mathbf{imnorm}}=0$.
6: $\mathbf{ifail}$ – 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).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
A Taylor series failed to converge.
${\mathbf{ifail}}=2$
An unexpected internal error occurred when evaluating the function at a point. Please contact NAG.
${\mathbf{ifail}}=3$
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.
${\mathbf{ifail}}=6$
The linear equations to be solved are nearly singular and the Padé approximant used to compute the exponential may have no correct figures. Note: this failure should not occur and suggests that the routine has been called incorrectly.
${\mathbf{ifail}}=-1$
On entry, ${\mathbf{fun}}=\u27e8\mathit{\text{value}}\u27e9$ was an illegal value.
${\mathbf{ifail}}=-2$
On entry, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=-4$
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 ${A}^{\mathrm{T}}A=A{A}^{\mathrm{T}}$), 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.
For further discussion of the Schur–Parlett algorithm see Section 9.4 of Higham (2008).
8Parallelism and Performance
f01ekf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f01ekf 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
The integer allocatable memory required is $n$. If $A$ has real eigenvalues then up to $9{n}^{2}$ of real allocatable memory may be required. If $A$ has complex eigenvalues then up to $9{n}^{2}$ 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; see Algorithm 9.6 of Higham (2008).
If the matrix exponential is required then it is recommended that f01ecf be used. f01ecf uses an algorithm which is, in general, more accurate than the Schur–Parlett algorithm used by f01ekf.
If estimates of the condition number of the matrix function are required then f01jaf should be used.
f01fkf can be used to find the matrix exponential, sin, cos, sinh or cosh of a complex matrix.
10Example
This example finds the matrix cosine of the matrix