The routine may be called by the names f01ecf or nagf_matop_real_gen_matrix_exp.
3Description
${e}^{A}$ is computed using a Padé approximant and the scaling and squaring method described in Al–Mohy and Higham (2009).
4References
Al–Mohy A H and Higham N J (2009) A new scaling and squaring algorithm for the matrix exponential SIAM J. Matrix Anal.31(3) 970–989
Higham N J (2005) The scaling and squaring method for the matrix exponential revisited SIAM J. Matrix Anal. Appl.26(4) 1179–1193
Higham N J (2008) Functions of Matrices: Theory and Computation SIAM, Philadelphia, PA, USA
Moler C B and Van Loan C F (2003) Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later SIAM Rev.45 3–49
5Arguments
1: $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint:
${\mathbf{n}}\ge 0$.
2: $\mathbf{a}({\mathbf{lda}},*)$ – 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 exponential ${e}^{A}$.
3: $\mathbf{lda}$ – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f01ecf is called.
Constraint:
${\mathbf{lda}}\ge {\mathbf{n}}$.
4: $\mathbf{ifail}$ – 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:
${\mathbf{ifail}}=1$
The linear equations to be solved for the Padé approximant are singular; it is likely that this routine has been called incorrectly.
${\mathbf{ifail}}=2$
The linear equations to be solved are nearly singular and the Padé approximant probably has no correct figures; it is likely that this routine has been called incorrectly.
${\mathbf{ifail}}=3$
${e}^{A}$ has been computed using an IEEE double precision Padé approximant, although the arithmetic precision is higher than IEEE double precision.
${\mathbf{ifail}}=4$
An unexpected internal error has 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, ${\mathbf{lda}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
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 computed matrix, ${e}^{A}$, is guaranteed to be close to the exact matrix, that is, the method is forward stable. No such guarantee can be given for non-normal matrices. See Al–Mohy and Higham (2009) and Section 10.3 of Higham (2008) for details and further discussion.
If estimates of the condition number of the matrix exponential are required then f01jgf should be used.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f01ecf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f01ecf 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, and the real allocatable memory required is approximately $6\times {{\mathbf{n}}}^{2}$.
The cost of the algorithm is $O\left({n}^{3}\right)$; see Section 5 of of Al–Mohy and Higham (2009). The real allocatable memory required is approximately $6\times {n}^{2}$.
If the Fréchet derivative of the matrix exponential is required then f01jhf should be used.
As well as the excellent book cited above, the classic reference for the computation of the matrix exponential is Moler and Van Loan (2003).
10Example
This example finds the matrix exponential of the matrix