NAG Library Routine Document
F01JHF
1 Purpose
F01JHF computes the Fréchet derivative $L\left(A,E\right)$ of the matrix exponential of a real $n$ by $n$ matrix $A$ applied to the real $n$ by $n$ matrix $E$. The matrix exponential ${e}^{A}$ is also returned.
2 Specification
INTEGER 
N, LDA, LDE, IFAIL 
REAL (KIND=nag_wp) 
A(LDA,*), E(LDE,*) 

3 Description
The Fréchet derivative of the matrix exponential of
$A$ is the unique linear mapping
$E\u27fcL\left(A,E\right)$ such that for any matrix
$E$
The derivative describes the firstorder effect of perturbations in $A$ on the exponential ${e}^{A}$.
F01JHF uses the algorithms of
Al–Mohy and Higham (2009a) and
Al–Mohy and Higham (2009b) to compute
${e}^{A}$ and
$L\left(A,E\right)$. The matrix exponential
${e}^{A}$ is computed using a Padé approximant and the scaling and squaring method. The Padé approximant is then differentiated in order to obtain the Fréchet derivative
$L\left(A,E\right)$.
4 References
Al–Mohy A H and Higham N J (2009a) A new scaling and squaring algorithm for the matrix exponential SIAM J. Matrix Anal. 31(3) 970–989
Al–Mohy A H and Higham N J (2009b) Computing the Fréchet derivative of the matrix exponential, with an application to condition number estimation SIAM J. Matrix Anal. Appl. 30(4) 1639–1657
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, twentyfive years later SIAM Rev. 45 3–49
5 Parameters
 1: $\mathrm{N}$ – INTEGERInput

On entry: $n$, the order of the matrix $A$.
Constraint:
${\mathbf{N}}\ge 0$.
 2: $\mathrm{A}\left({\mathbf{LDA}},*\right)$ – REAL (KIND=nag_wp) arrayInput/Output

Note: the second dimension of the array
A
must be at least
${\mathbf{N}}$.
On entry: the $n$ by $n$ matrix $A$.
On exit: the $n$ by $n$ matrix exponential ${e}^{A}$.
 3: $\mathrm{LDA}$ – INTEGERInput

On entry: the first dimension of the array
A as declared in the (sub)program from which F01JHF is called.
Constraint:
${\mathbf{LDA}}\ge {\mathbf{N}}$.
 4: $\mathrm{E}\left({\mathbf{LDE}},*\right)$ – REAL (KIND=nag_wp) arrayInput/Output

Note: the second dimension of the array
E
must be at least
${\mathbf{N}}$.
On entry: the $n$ by $n$ matrix $E$
On exit: the Fréchet derivative $L\left(A,E\right)$
 5: $\mathrm{LDE}$ – INTEGERInput

On entry: the first dimension of the array
E as declared in the (sub)program from which F01JHF is called.
Constraint:
${\mathbf{LDE}}\ge {\mathbf{N}}$.
 6: $\mathrm{IFAIL}$ – INTEGERInput/Output

On entry:
IFAIL must be set to
$0$,
$1\text{ or}1$. If you are unfamiliar with this parameter you should refer to
Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
$1\text{ or}1$ is recommended. If the output of error messages is undesirable, then the value
$1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
$0$.
When the value $\mathbf{1}\text{ or}\mathbf{1}$ 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).
6 Error Indicators and Warnings
If on entry
${\mathbf{IFAIL}}={\mathbf{0}}$ or
${{\mathbf{1}}}$, 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.
 ${\mathbf{IFAIL}}=2$

${e}^{A}$ has been computed using an IEEE double precision Padé approximant, although the arithmetic precision is higher than IEEE double precision.
 ${\mathbf{IFAIL}}=3$

An unexpected internal error has occurred. Please contact
NAG.
 ${\mathbf{IFAIL}}=1$

On entry, ${\mathbf{N}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{N}}\ge 0$.
 ${\mathbf{IFAIL}}=3$

On entry, ${\mathbf{LDA}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{N}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{LDA}}\ge {\mathbf{N}}$.
 ${\mathbf{IFAIL}}=5$

On entry, ${\mathbf{LDE}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{N}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{LDE}}\ge {\mathbf{N}}$.
 ${\mathbf{IFAIL}}=99$
An unexpected error has been triggered by this routine. Please
contact
NAG.
See
Section 3.8 in the Essential Introduction for further information.
 ${\mathbf{IFAIL}}=399$
Your licence key may have expired or may not have been installed correctly.
See
Section 3.7 in the Essential Introduction for further information.
 ${\mathbf{IFAIL}}=999$
Dynamic memory allocation failed.
See
Section 3.6 in the Essential Introduction for further information.
7 Accuracy
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 nonnormal matrices. See Section 10.3 of
Higham (2008),
Al–Mohy and Higham (2009a) and
Al–Mohy and Higham (2009b) for details and further discussion.
8 Parallelism and Performance
F01JHF is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
F01JHF 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 implementationspecific information.
The cost of the algorithm is
$O\left({n}^{3}\right)$ and the real allocatable memory required is approximately
$9{n}^{2}$; see
Al–Mohy and Higham (2009a) and
Al–Mohy and Higham (2009b).
If the matrix exponential alone is required, without the Fréchet derivative, then
F01ECF should be used.
If the condition number of the matrix exponential is required then
F01JGF should be used.
As well as the excellent book
Higham (2008), the classic reference for the computation of the matrix exponential is
Moler and Van Loan (2003).
10 Example
This example finds the matrix exponential
${e}^{A}$ and the Fréchet derivative
$L\left(A,E\right)$, where
10.1 Program Text
Program Text (f01jhfe.f90)
10.2 Program Data
Program Data (f01jhfe.d)
10.3 Program Results
Program Results (f01jhfe.r)